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Hypothesis Testing Using Z- And T-tests
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Assignment Sem 1-Maths
Weaver (27-May-2011)
z- and t-tests ... 1
In hypothesis testing, one attempts to answer the following question: If the null hypothesis is assumed to be true, what is the probability of obtaining the observed result, or any more extreme result that is favourable to the alternative hypothesis?1 In order to tackle this question, at least in the context of z- and t-tests, one must first understand two important concepts: 1) sampling distributions of statistics, and 2) the central limit theorem. Sampling Distributions Imagine drawing (with replacement) all possible samples of size n from a population, and for each sample, calculating a statistic--e.g., the sample mean. The frequency distribution of those sample means would be the sampling distribution of the mean (for samples of size n drawn from that particular population). Normally, one thinks of sampling from relatively large populations. But the concept of a sampling distribution can be illustrated with a small population. Suppose, for example, that our population consisted of the following 5 scores: 2, 3, 4, 5, and 6. The population mean = 4, and the population standard deviation (dividing by N) = 1.414. If we drew (with replacement) all possible samples of 2 from this population, we would end up with the 25 samples shown in Table 1. Table 1: All possible samples of n=2 from a population of 5 scores.
First Score 2 2 2 2 2 3 3 3 3 3 4 4 4 Second Score 2 3 4 5 6 2 3 4 5 6 2 3 4 Sample Mean 2 2.5 3 3.5 4 2.5 3 3.5 4 4.5 3 3.5 4 First Score 4 4 5 5 5 5 5 6 6 6 6 6 Second Score 5 6 2 3 4 5 6 2 3 4 5 6 Sample Mean 4.5 5 3.5 4 4.5 5 5.5 4 4.5 5 5.5 6
Sample # 1 2 3 4 5 6 7 8 9 10 11 12 13
Sample # 14 15 16 17 18 19 20 21 22 23 24 25
Mean of the sample means = SD of the sample means =
(SD calculated with division by N)
That probability is called a p-value. It is a really a conditional probability--it is conditional on the null hypothesis being true.
B. Weaver (27-May-2011)
The 25 sample means from Table 1 are plotted below in Figure 1 (a histogram). This distribution of sample means is called the sampling distribution of the mean for samples of n=2 from the population of interest (i.e., our population of 5 scores).
Distribution of Sample Means*
1 0 2.00 2.50 3.00 3.50 4.00 4.50 5.00 5.50 6.00
Std. Dev = 1.02 Mean = 4.00 N = 25.00
* or "Sampling Distribution of the Mean"
Figure 1: Sampling distribution of the mean for samples of n=2 from a population of N=5.
I suspect the first thing you noticed about this figure is peaked in the middle, and symmetrical about the mean. This is an important characteristic of sampling distributions, and we will return to it in a moment. You may have also noticed that the standard deviation reported in the figure legend is 1.02, whereas I reported SD = 1.000 in Table 1. Why the discrepancy? Because I used the population SD formula (with division by N) to compute SD = 1.000 in Table 1, but SPSS used the sample SD formula (with division by n-1) when computing the SD it plotted alongside the histogram. The population SD is the correct one to use in this case, because I have the entire population of 25 samples in hand. The Central Limit Theorem (CLT) If I were a mathematical statistician, I would now proceed to work through derivations, proving the following statements:
1. The mean of the sampling distribution of the mean = the population mean 2. The SD of the sampling distribution of the mean = the standard error (SE) of the mean = the population standard deviation divided by the square root of the sample size Putting these statements into symbols:
X = X { mean of the sample means = the population mean } X = X
{ SE of mean = population SD over square root of n }
But alas, I am not a mathematical statistician. Therefore, I will content myself with telling you that these statements are true (those of you who do not trust me, or are simply curious, may consult a mathematical stats textbook), and pointing to the example we started this chapter with. For that population of 5 scores, = 4 and = 1.414 . As shown in Table 1, X = = 4 , and X = 1.000 . According to equation (1.2), if we divide the population SD by the square root of the sample size, we should obtain the standard error of the mean. So let's give it a try:
= .9998 1
When I performed the calculation in Excel and did not round off to 3 decimals, the solution worked out to 1 exactly. In the Excel worksheet that demonstrates this, you may also change the values of the 5 population scores, and should observe that X = n for any set of 5 scores you choose. Of course, these demonstrations do not prove the CLT (see the aforementioned mathstats books if you want proof), but they should reassure you that it does indeed work.
What the CLT tells us about the shape of the sampling distribution The central limit theorem also provides us with some very helpful information about the shape of the sampling distribution of the mean. Specifically, it tells us the conditions under which the sampling distribution of the mean is normally distributed, or at least approximately normal, where approximately means close enough to treat as normal for practical purposes. The shape of the sampling distribution depends on two factors: the shape of the population from which you sampled, and sample size. I find it useful to think about the two extremes: 1. If the population from which you sample is itself normally distributed, then the sampling distribution of the mean will be normal, regardless of sample size. Even for sample size = 1, the sampling distribution of the mean will be normal, because it will be an exact copy of the population distribution.
2. If the population from which you sample is extremely non-normal, the sampling distribution of the mean will still be approximately normal given a large enough sample size (e.g., some authors suggest for sample sizes of 300 or greater). So, the general principle is that the more the population shape departs from normal, the greater the sample size must be to ensure that the sampling distribution of the mean is approximately normal. This tradeoff is illustrated in the following figure, which uses colour to represent the shape of the sampling distribution (purple = non-normal, red = normal, with the other colours representing points in between).
Does n have to be 30? Some textbooks say that one should have a sample size of at least 30 to ensure that the sampling distribution of the mean is approximately normal. The example we started with (i.e., samples of n = 2 from a population of 5 scores) suggests that this is not correct (see Figure 1). Here is another example that makes the same point. The figure on the left, which shows the age distribution for all students admitted to the Northern Ontario School of Medicine in its first 3 years of operation, is treated as the population. The figure on the right shows the distribution of means for 10,000 samples of size 16 drawn from that population. Notice that despite the severe
positive skew in the population, the distribution of sample means is near enough to normal for the normal approximation to be useful.
What is the rule of 30 about then? In the olden days, textbook authors often did make a distinction between small-sample and largesample versions of t-tests. The small- and large-sample versions did not differ at all in terms of how t was calculated. Rather, they differed in how/where one obtained the critical value to which they compared their computed t-value. For the small-sample test, one used the critical value of t, from a table of critical t-values. For the large-sample test, one used the critical value of z, obtained from a table of the standard normal distribution. The dividing line between small and large samples was usually n = 30 (or sometimes 20). Why was this done? Remember that in that era, data analysts did not have access to desktop computers and statistics packages that computed exact p-values. Therefore, they had to compute the test statistic, and compare it to the critical value, which they looked up in a table. Tables of critical values can take up a lot of room. So when possible, compromises were made. In this particular case, most authors and statisticians agreed that for n 30, the critical value of z (from the standard normal distribution) was close enough to the critical value of t that it could be used as an approximation. The following figure illustrates this by plotting critical values of t with
alpha = .05 (2-tailed) as a function of sample size. Notice that when n 30 (or even 20), the critical values of t are very close to 1.96, the critical value of z.
Nowadays, we typically use statistical software to perform t-tests, and so we get a p-value computed using the appropriate t-distribution, regardless of the sample size. Therefore the distinction between small- and large-sample t-tests is no longer relevant, and has disappeared from most modern textbooks. The sampling distribution of the mean and z-scores When you first encountered z-scores, you were undoubtedly using them in the context of a raw score distribution. In that case, you calculated the z-score corresponding to some value of X as follows: z= X = X X
And, if the distribution of X was normal, or at least approximately normal, you could then take that z-score, and refer it to a table of the standard normal distribution to figure out the proportion of scores higher than X, or lower than X, etc. Because of what we learned from the central limit theorem, we are now in a position to compute a z-score as follows:
z=
X n
This is the same formula, but with X in place of X, and X in place of X . And, if the sampling distribution of X is normal, or at least approximately normal, we may then refer this value of z to the standard normal distribution, just as we did when we were using raw scores. (This is where the CLT comes in, because it tells the conditions under which the sampling distribution of X is approximately normal.) An example. Here is a (fictitious) newspaper advertisement for a program designed to increase intelligence of school children2:
As an expert on IQ, you know that in the general population of children, the mean IQ = 100, and the population SD = 15 (for the WISC, at least). You also know that IQ is (approximately) normally distributed in the population. Equipped with this information, you can now address questions such as: If the n=25 children from Dundas are a random sample from the general population of children, a) What is the probability of getting a sample mean of 108 or higher? b) What is the probability of getting a sample mean of 92 or lower? c) How high would the sample mean have to be for you to say that the probability of getting a mean that high (or higher) was 0.05 (or 5%)? d) How low would the sample mean have to be for you to say that the probability of getting a mean that low (or lower) was 0.05 (or 5%)?
I cannot find the original source for this example, but I believe I got it from Dr. Geoff Norman, McMaster University.
The solutions to these questions are quite straightforward, given everything we have learned so far in this chapter. If we have sampled from the general population of children, as we are assuming, then the population from which we have sampled is at least approximately normal. Therefore, the sampling distribution of the mean will be normal, regardless of sample size. Therefore, we can compute a z-score, and refer it to the table of the standard normal distribution. So, for part (a) above: z= X X
X X 108 100 8 = = = 2.667 X 153 n 25
And from a table of the standard normal distribution (or using a computer program, as I did), we can see that the probability of a z-score greater than or equal to 2.667 = 0.0038. Translating that back to the original units, we could say that the probability of getting a sample mean of 108 (or greater) is .0038 (assuming that the 25 children are a random sample from the general population). For part (b), do the same, but replace 108 with 92: z= X X = X X 92 100 8 = == 2.667 X 153 n 25 (1.7)
Because the standard normal distribution is symmetrical about 0, the probability of a z-score equal to or less than -2.667 is the same as the probability of a z-score equal to or greater than 2.667. So, the probability of a sample mean less than or equal to 92 is also equal to 0.0038. Had we asked for the probability of a sample mean that is either 108 or greater, or 92 or less, the answer would be 0.0038 + 0.0038 = 0.0076. Part (c) above amounts to the same thing as asking, "What sample mean corresponds to a z-score of 1.645?", because we know that p ( z 1.645) = 0.05 . We can start out with the usual z-score formula, but need to rearrange the terms a bit, because we know that z = 1.645, and are trying to determine the corresponding value of X . X X
{ cross-multiply to get to next line }
z X = X X { add X to both sides } (1.8) z X + X = X { switch sides }
X = z X + X = 1.645 15 25 + 100 = 104.935
So, had we obtained a sample mean of 105, we could have concluded that the probability of a mean that high or higher was .05 (or 5%). For part (d), because of the symmetry of the standard normal distribution about 0, we would use the same method, but substituting -1.645 for 1.645. This would yield an answer of 100 - 4.935 = 95.065. So the probability of a sample mean less than or equal to 95 is also 5%. The single sample z-test It is now time to translate what we have just been doing into the formal terminology of hypothesis testing. In hypothesis testing, one has two hypotheses: The null hypothesis, and the alternative hypothesis. These two hypotheses are mutually exclusive and exhaustive. In other words, they cannot share any outcomes in common, but together must account for all possible outcomes. In informal terms, the null hypothesis typically states something along the lines of, "there is no treatment effect", or "there is no difference between the groups".3 The alternative hypothesis typically states that there is a treatment effect, or that there is a difference between the groups. Furthermore, an alternative hypothesis may be directional or non-directional. That is, it may or may not specify the direction of the difference between the groups. H 0 and H1 are the symbols used to represent the null and alternative hypotheses respectively. (Some books may use H a for the alternative hypothesis.) Let us consider various pairs of null and alternative hypotheses for the IQ example we have been considering. Version 1: A directional alternative hypothesis H 0 : 100 H1 : > 100 This pair of hypotheses can be summarized as follows. If the alternative hypothesis is true, the sample of 25 children we have drawn is from a population with mean IQ greater than 100. But if the null hypothesis is true, the sample is from a population with mean IQ equal to or less than 100. Thus, we would only be in a position to reject the null hypothesis if the sample mean is greater than 100 by a sufficient amount. If the sample mean is less than 100, no matter by how much, we would not be able to reject H 0 . How much greater than 100 must the sample mean be for us to be comfortable in rejecting the null hypothesis? There is no unequivocal answer to that question. But the answer that most
I said it typically states this, because there may be cases where the null hypothesis specifies a difference between groups of a certain size rather than no difference between groups. In such a case, obtaining a mean difference of zero may actually allow you to reject the null hypothesis.
z- and t-tests ... 10
disciplines use by convention is this: The difference between X and must be large enough that the probability it occurred by chance (given a true null hypothesis) is 5% or less. The observed sample mean for this example was 108. As we saw earlier, this corresponds to a zscore of 2.667, and p( z 2.667) = 0.0038 . Therefore, we could reject H 0 , and we would act as if the sample was drawn from a population in which mean IQ is greater than 100. Version 2: Another directional alternative hypothesis H 0 : 100 H1 : < 100 This pair of hypotheses would be used if we expected the Dr. Duntz's program to lower IQ, and if we were willing to include an increase in IQ (no matter how large) under the null hypothesis. Given a sample mean of 108, we could stop without calculating z, because the difference is in the wrong direction. That is, to have any hope of rejecting H 0 , the observed difference must be in the direction specified by H1 .
Version 3: A non-directional alternative hypothesis H 0 : = 100 H1 : 100 In this case, the null hypothesis states that the 25 children are a random sample from a population with mean IQ = 100, and the alternative hypothesis says they are not--but it does not specify the direction of the difference from 100. In the first directional test, we needed to have X > 100 by a sufficient amount, and in the second directional test, X < 100 by a sufficient amount in order to reject H 0 . But in this case, with a non-directional alternative hypothesis, we may reject H 0 if X < 100 or if X > 100 , provided the difference is large enough. Because differences in either direction can lead to rejection of H 0 , we must consider both tails of the standard normal distribution when calculating the p-value--i.e., the probability of the observed outcome, or a more extreme outcome favourable to H1 . For symmetrical distributions like the standard normal, this boils down to taking the p-value for a directional (or 1-tailed) test, and doubling it. For this example, the sample mean = 108. This represents a difference of +8 from the population mean (under a true null hypothesis). Because we are interested in both tails of the distribution, we must figure out the probability of a difference of +8 or greater, or a change of -8 or greater. In other words, p = p( X 108) + p( X 92) = .0038 + .0038 = .0076 .
Single sample t-test (when is not known) In many real-world cases of hypothesis testing, one does not know the standard deviation of the population. In such cases, it must be estimated using the sample standard deviation. That is, s (calculated with division by n-1) is used to estimate . Other than that, the calculations are as we saw for the z-test for a single sample--but the test statistic is called t, not z.
X X t( df = n 1) = sX
where s X =
s n
, and s =
XX n 1
SS X n 1
In equation (1.9), notice the subscript written by the t. It says "df = n-1". The "df" stands for degrees of freedom. "Degrees of freedom" can be a bit tricky to grasp, but let's see if we can make it clear. Degrees of Freedom Suppose I tell you that I have a sample of n=4 scores, and that the first three scores are 2, 3, and 5. What is the value of the 4th score? You can't tell me, given only that n = 4. It could be anything. In other words, all of the scores, including the last one, are free to vary: df = n for a sample mean. To calculate t, you must first calculate the sample standard deviation. The conceptual formula for the sample standard deviation is:
s=
Suppose that the last score in my sample of 4 scores is a 6. That would make the sample mean equal to (2+3+5+6)/4 = 4. As shown in Table 2, the deviation scores for the first 3 scores are -2, -1, and 1. Table 2: Illustration of degrees of freedom for sample standard deviation Deviation from Mean -2 -1 1 x4
Score 2 3 5 --
Mean 4 4 4 --
Using only the information shown in the final column of Table 2, you can deduce that x4 , the 4th deviation score, is equal to -2. How so? Because by definition, the sum of the deviations about the mean = 0. This is another way of saying that the mean is the exact balancing point of the distribution. In symbols:
So, once you have n-1 of the ( X X
deviation scores, the final deviation score is determined.
That is, the first n-1 deviation scores are free to vary, but the final one is not. There are n-1 degrees of freedom whenever you calculate a sample variance (or standard deviation). The sampling distribution of t To calculate the p-value for a single sample z-test, we used the standard normal distribution. For a single sample t-test, we must use a t-distribution with n-1 degrees of freedom. As this implies, there is a whole family of t-distributions, with degrees of freedom ranging from 1 to infinity ( = the symbol for infinity). All t-distributions are symmetrical about 0, like the standard normal. In fact, the t-distribution with df = is identical to the standard normal distribution. But as shown in Figure 2 below, t-distributions with df < infinity have lower peaks and thicker tails than the standard normal distribution. To use the technical term for this, they are leptokurtic. (The normal distribution is said to be mesokurtic.) As a result, the critical values of t are further from 0 than the corresponding critical values of z. Putting it another way, the absolute value of critical t is greater than the absolute value of critical z for all t-distributions with df < :
For df < ,
tcritical > zcritical
Probability density functions
Standard Normal Distribution t-distribution with df = 10
t-distribution with df = 2
0.0 -6 -4 -2 0 2 4 6
Figure 2: Probability density functions of: the standard normal distribution (the highest peak with the thinnest tails); the t-distribution with df=10 (intermediate peak and tails); and the t-distribution with df=2 (the lowest peak and thickest tails). The dotted lines are at -1.96 and +1.96, the critical values of z for a two-tailed test with alpha = .05. For all t-distributions with df < , the proportion of area beyond -1.96 and +1.96 is greater than .05. The lower the degrees of freedom, the thicker the tails, and the greater the proportion of area beyond -1.96 and +1.96.
Table 3 (see below) shows yet another way to think about the relationship between the standard normal distribution and various t-distributions. It shows the area in the two tails beyond -1.96 and +1.96, the critical values of z with 2-tailed alpha = .05. With df=1, roughly 15% of the area falls in each tail of the t-distribution. As df gets larger, the tail areas get smaller and smaller, until the t-distribution converges on the standard normal when df = infinity.
Table 3: Area beyond critical values of + or -1.96 in various t-distributions. The t-distribution with df = infinity is identical to the standard normal distribution.
Degrees of Freedom 1 2 3 4 5 10 15 20 25 30 Area beyond + or -1.96 0.30034 0.18906 0.14485 0.12155 0.10729 0.07844 0.06884 0.06408 0.06123 0.05934 Degrees of Freedom 40 50 100 200 300 400 500 1,000 5,000 10,000 Infinity Area beyond + or -1.96 0.05699 0.05558 0.05278 0.05138 0.05092 0.05069 0.05055 0.05027 0.05005 0.05002 0.05000
Example of single-sample t-test. This example is taken from Understanding Statistics in the Behavioral Sciences (3rd Ed), by Robert R. Pagano.
A researcher believes that in recent years women have been getting taller. She knows that 10 years ago the average height of young adult women living in her city was 63 inches. The standard deviation is unknown. She randomly samples eight young adult women currently residing in her city and measures their heights. The following data are obtained: [64, 66, 68, 60, 62, 65, 66, 63.]
The null hypothesis is that these 8 women are a random sample from a population in which the mean height is 63 inches. The non-directional alternative states that the women are a random sample from a population in which the mean is not 63 inches.
H 0 : = 63 H1 : 63 The sample mean is 64.25. Because the population standard deviation is not known, we must estimate it using the sample standard deviation.
sample SD = s =
(X X )
(1.13) = (64 64.25) 2 + (66 64.25) 2 + ...(63 64.25) 2 = 2.5495 7
We can now use the sample standard deviation to estimate the standard error of the mean: Estimated SE of mean = s X = And finally: t= X 0 64.25 63 == 1.387 0.901sX (1.15) s n = 2.5495 8 = 0.901 (1.14)
This value of t can be referred to a t-distribution with df = n-1 = 7. Doing so, we find that the conditional probability4 of obtaining a t-statistic with absolute value equal to or greater than 1.387 = 0.208. Therefore, assuming that alpha had been set at the usual .05 level, the researcher cannot reject the null hypothesis. I performed the same test in SPSS (Analyze Compare Means One sample t-test), and obtained the same results, as shown below.
T-TEST /TESTVAL=63/* <-/MISSING=ANALYSIS /VARIABLES=height/* <-/CRITERIA=CIN (.95) .
This is where you specify the value of mu */ This is the variable you measured */
T-Test
One-Sample Statistics Std. Error Mean .901
N HEIGHT 8
Mean 64.25
Std. Deviation 2.550
Remember that a p-value is really a conditional probability. It is conditional on the null hypothesis being true.
One-Sample Test Test Value = 63 95% Confidence Interval of the Difference LowerUpper -.883.38
t 1.387
df 7
Sig. (2-tailed) .208
Mean Difference 1.25
Paired (or related samples) t-test Another common application of the t-test occurs when you have either 2 scores for each person (e.g., before and after), or when you have matched pairs of scores (e.g., husband and wife pairs, or twin pairs). The paired t-test may be used in this case, given that its assumptions are met adequately. (More on the assumptions of the various t-tests later). Quite simply, the paired t-test is just a single-sample t-test performed on the difference scores. That is, for each matched pair, compute a difference score. Whether you subtract 1 from 2 or vice versa does not matter, so long as you do it the same way for each pair. Then perform a single-sample t-test on those differences. The null hypothesis for this test is that the difference scores are a random sample from a population in which the mean difference has some value which you specify. Often, that value is zero--but it need not be. For example, suppose you found some old research which reported that on average, husbands were 5 inches taller than their wives. If you wished to test the null hypothesis that the difference is still 5 inches today (despite the overall increase in height), your null hypothesis would state that your sample of difference scores (from husband/wife pairs) is a random sample from a population in which the mean difference = 5 inches. In the equations for the paired t-test, X is often replaced with D , which stands for the mean difference. D D D D (1.16)t== sDsD n
where D = the (sample) mean of the difference scores
D = the mean difference in the population, given a true H 0 {often D =0, but not always}
s D = the sample SD of the difference scores (with division by n-1) n = the number of matched pairs; the number of individuals = 2n sD = the SE of the mean difference df = n 1
Example of paired t-test This example is from the Study Guide to Paganos book Understanding Statistics in The Behavioral Sciences (3rd Edition).
A political candidate wishes to determine if endorsing increased social spending is likely to affect her standing in the polls. She has access to data on the popularity of several other candidates who have endorsed increases spending. The data was available both before and after the candidates announced their positions on the issue [see Table 4].
Table 4: Data for paired t-test example. Popularity Ratings BeforeAfterDifference 42431 41454 50566 52542 58657 3229-3 39467 42486 4847-1 47536
Candidate 1 2 3 4 5 6 7 8 9 10
I entered these BEFORE and AFTER scores into SPSS, and performed a paired t-test as follows:
T-TEST PAIRS= after WITH before (PAIRED) /CRITERIA=CIN(.95) /MISSING=ANALYSIS.
This yielded the following output.
Paired Samples Statistics Std. Error Mean 3.001 2.345
AFTER BEFORE
Mean 48.60 45.10
N 10 10
Std. Deviation 9.489 7.415
Paired Samples Correlations N Pair 1 AFTER & BEFORE 10 Correlation .940 Sig. .000
Paired Samples Test Paired Differences 95% Confidence Interval of the Difference LowerUpper .956.05
AFTER - BEFORE
Mean 3.50
Std. Error Mean 1.128
The first output table gives descriptive information on the BEORE and AFTER popularity ratings, and shows that the mean is higher after politicians have endorsed increased spending. The second output table gives the Pearson correlation (r) between the BEFORE and AFTER scores. (The correlation coefficient is measure of the direction and strength of the linear relationship between two variables. I will say more about it in a later section called Testing the significance of Pearson r.) The final output table shows descriptive statistics for the AFTER BEFORE difference scores, and the t-value with its degrees of freedom and p-value. The null hypothesis for this test states that the mean difference in the population is zero. In other words, endorsing increased social spending has no effect on popularity ratings in the population from which we have sampled. If that is true, the probability of seeing a difference of 3.5 points or more is 0.013 (the p-value). Therefore, the politician would likely reject the null hypothesis, and would endorse increased social spending.
The same example done using a one-sample ttest Earlier, I said that the paired t-test is really just a single-sample t-test done on difference scores. Lets demonstrate for ourselves that this is really so. Here are the BEFORE, AFTER, and DIFF scores from my SPSS file. (I computed the DIFF scores using compute diff = after before.)
BEFORE AFTER 42 41 50 52 58 32 39 42 48 47 43 45 56 54 65 29 46 48 47 53 DIFF 1 4 6 2 7 -3 7 6 -1 6
Number of cases read:
Number of cases listed:
I then ran a single-sample t-test on the difference scores using the following syntax:
T-TEST /TESTVAL=0 /MISSING=ANALYSIS /VARIABLES=diff /CRITERIA=CIN (.95) .
The output is shown below.
One-Sample Statistics Std. Error Mean 1.128
N DIFF 10
One-Sample Test Test Value = 0 95% Confidence Interval of the Difference LowerUpper .956.05
Notice that the descriptive statistics shown in this output are identical to those shown for the paired differences in the first analysis. The t-value, df, and p-value are identical too, as expected. A paired t-test for which H 0 specifies a non-zero difference The following example is from an SPSS syntax file I wrote to demonstrate a variety of t-tests.
* * * * * * * A researcher knows that in 1960, Canadian husbands were 5" taller than their wives on average. Overall mean height has increased since then, but it is not known if the husband/wife difference has changed, or is still 5". The SD for the 1960 data is not known. The researcher randomly selects 25 couples and records the heights of the husbands and wives. Test the null hypothesis that the mean difference is still 5 inches. Mean difference in height (in population) = 5 inches. Mean difference in height does not equal 5 inches.
* H0: * H1:
list all. COUPLE 1.00 2.00 3.00 4.00 5.00 6.00 7.00 HUSBAND 69.22 69.60 65.38 65.27 69.39 69.29 69.87 WIFE 56.73 77.12 54.22 53.01 65.73 70.58 70.80 DIFF 12.49 -7.52 11.16 12.26 3.66 -1.29 -.93
8.00 9.00 10.00 11.00 12.00 13.00 14.00 15.00 16.00 17.00 18.00 19.00 20.00 21.00 22.00 23.00 24.00 25.00
78.79 73.19 67.24 70.98 69.86 69.07 68.81 76.47 71.28 79.77 66.87 71.10 74.68 75.56 67.04 60.80 71.91 79.17
15.29 5.07 -.37 10.38 6.32 3.87 6.88 4.77 3.56 17.74 8.28 2.13 8.78 16.64 2.07 -7.13 8.23 6.09
Number of cases read: * * * *
We cannot use the "paired t-test procedure, because it does not allow for non-zero mean difference under the null hypothesis. Instead, we need to use the single-sample t-test on the difference scores, and specify a mean difference of 5 inches under the null.
T-TEST /TESTVAL=5/* H0: Mean difference = 5 inches (as in past) */ /MISSING=ANALYSIS /VARIABLES=diff/* perform analysis on the difference scores */ /CRITERIA=CIN (.95) .
One-Sample Statistics Std. Error Mean 1.30948
Mean 5.9372
Std. Deviation 6.54740
One-Sample Test Test Value = 5 95% Confidence Interval of the Difference LowerUpper -1.76543.6398
t DIFF .716
df 24
Mean Difference .9372
* The observed mean difference = 5.9 inches. * The null hypothesis cannot be rejected (p = 0.481). * This example shows that the null hypothesis does not always have to * specify a mean difference of 0. We obtained a mean difference of * 5.9 inches, but were unable to reject H0, because it stated that
* the mean difference = 5 inches. * * * * * Suppose we had found that the difference in height between husbands and wives really had decreased dramatically. In that case, we might have found a mean difference close to 0, which might have allowed us to reject H0. An example of this scenario follows below. COUPLE 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00 11.00 12.00 13.00 14.00 15.00 16.00 17.00 18.00 19.00 20.00 21.00 22.00 23.00 24.00 25.00 HUSBAND 68.78 66.09 71.99 74.51 67.31 64.05 66.77 75.33 74.11 75.71 69.01 67.86 66.61 68.64 78.74 71.66 73.43 70.39 70.15 71.53 57.49 68.95 77.60 72.36 72.70 WIFE 75.34 67.57 69.16 69.17 68.11 68.62 70.31 72.92 73.10 62.66 76.83 63.23 72.01 76.10 68.53 62.65 70.46 79.99 64.27 69.07 81.21 69.92 70.70 67.79 67.50 DIFF -6.56 -1.48 2.83 5.34 -.80 -4.57 -3.54 2.41 1.01 13.05 -7.82 4.63 -5.40 -7.46 10.21 9.01 2.97 -9.60 5.88 2.46 -23.72 -.97 6.90 4.57 5.20
Mean .1820
One-Sample Test Test Value = 5 95% Confidence Interval of the Difference LowerUpper -8.0216-1.6144
t -3.104
Mean Difference -4.8180
* The mean difference is about 0.2 inches (very close to 0). * Yet, because H0 stated that the mean difference = 5, we * are able to reject H0 (p = 0.005).
Unpaired (or independent samples) t-test Another common form of the t-test may be used if you have 2 independent samples (or groups). The formula for this version of the test is given in equation (1.17).
t=
X1 X
The left side of the numerator, ( X 1 X 2 ) , is the difference between the means of two (independent) samples, or the difference between group means. The right side of the numerator, ( 1 2 ) , is the difference between the corresponding population means, assuming that H 0 is true. The denominator is the standard error of the difference between two independent means. It is calculated as follows: s2s2pooledpooled= + nn2 1 SS1 + SS 2 SS Within Groups = n1 + n2 2 df Within Groups
s2pooled = pooled variance estimate =
SS1 = SS
( X X ) for Group 1 = ( X X ) for Group 2
n1 = sample size for Group 1 n2 = sample size for Group 2 df = n1 + n2 2
As indicated above, the null hypothesis for this test specifies a value for ( 1 2 ) , the difference between the population means. More often than not, H 0 specifies that ( 1 2 ) = 0. For that reason, most textbooks omit ( 1 2 ) from the numerator of the formula, and show it like this:
tunpaired =
SS1 + SS 2 1 1 + n1 + n2 2 n1 n2
I prefer to include ( 1 2 ) for two reasons. First, it reminds me that the null hypothesis can specify a non-zero difference between the population means. Second, it reminds me that all ttests have a common format, which I will describe in a section to follow. The unpaired (or independent samples) t-test has df = n1 + n2 2 . As discussed under the Single-sample t-test, one degree of freedom is lost whenever you calculate a sum of squares (SS). To perform an unpaired t-test, we must first calculate both SS1 and SS 2 , so two degrees of freedom are lost. Example of unpaired t-test The following example is from Understanding Statistics in the Behavioral Sciences (3rd Ed), by Robert R. Pagano.
A nurse was hired by a governmental ecology agency to investigate the impact of a lead smelter on the level of lead in the blood of children living near the smelter. Ten children were chosen at random from those living near the smelter. A comparison group of 7 children was randomly selected from those living in an area relatively free from possible lead pollution. Blood samples were taken from the children, and lead levels determined. The following are the results (scores are in micrograms of lead per 100 milliliters of blood):
Lead Levels Children LivingChildren Living in Near SmelterUnpolluted Area
18 16 21 14 17 19 22 24 15 18 9 13 8 15 17 12 11
= 0.012tailed , what do you conclude?
I entered these scores in SPSS as follows:
GRP 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 LEAD 18 16 21 14 17 19 22 24 15 18 9 13 8 15 17 12 11
To run an unpaired t-test: Analyze Compare Means Independent Samples t-test. The syntax and output follow.
T-TEST GROUPS=grp(1 2) /MISSING=ANALYSIS /VARIABLES=lead /CRITERIA=CIN(.95) .
Group Statistics Std. Error Mean 1.002 1.204
GRP 1 2
N 10 7
Independent Samples Test Levene's Test for Equality of Variances
t-test for Equality of Means 95% Confidence Interval of the Difference LowerUpper 2.922 2.874 9.593 9.640
F LEAD Equal variances assumed Equal variances not assumed .001
Sig. .972
t 3.998 3.995
df 15 13.028
Sig. (2-tailed) .001 .002
Mean Difference 6.26 6.26
Std. Error Difference 1.565 1.566
The null hypothesis for this example is that the 2 groups of children are 2 random samples from populations with the same mean levels of lead concentration in the blood. The p-value is 0.001, which is less than the 0.01 specified in the question. So we would reject the null hypothesis. Putting it another way, we would act as if living close to the smelter is causing an increase in lead concentration in the blood. Testing the significance of Pearson r Pearson r refers to the Pearson Product-Moment Correlation Coefficient. If you have not learned about it yet, it will suffice to say for now that r is a measure of the strength and direction of the linear (i.e., straight line) relationship between two variables X and Y. It can range in value from -1 to +1. Negative values indicate that there is a negative (or inverse) relationship between X and Y: i.e., as X increases, Y decreases. Positive values indicate that there is a positive relationship: as X increases, Y increases. If r=0, there is no linear relationship between X and Y. If r = -1 or +1, there is a perfect linear relationship between X and Y: That is, all points fall exactly on a straight line. The further r is from 0, the better the linear relationship. Pearson r is calculated using matched pairs of scores. For example, you might calculate the correlation between students' scores in two different classes. Often, these pairs of scores are a sample from some population of interest. In that case, you may wish to make an inference about the correlation in the population from which you have sampled. The Greek letter rho is used to represent the correlation in the population. It looks like this: . As you may have already guessed, the null hypothesis specifies a value for . Often that value is 0. In other words, the null hypothesis often states that there is no linear relationship between X and Y in the population from which we have sampled. In symbols: H0 : = 0 H1 : 0 { there is no linear relationship between X and Y in the population } { there is a linear relationship between X and Y in the population }
A null hypothesis which states that = 0 can be tested with a t-test, as follows5: r t= sr Example 1 r2 where sr = n2
and df = n 2
If the null hypothesis specifies a non-zero value of rho, Fisher's r-to-z transformation may need to be applied (depending on how far from zero the specified value is). See Howell (1997, pp. 261-263) for more information.
You may recall the following output from the first example of a paired t-test (with BEFORE and AFTER scores:
The number in the Correlation column is a Pearson r. It indicates that there is a very strong and positive linear relationship between BEFORE and AFTER scores for the 10 politicians. The p-value (Sig.) is for a t-test of the null hypothesis that there is no linear relationship between BEFORE and AFTER scores in the population of politicians from which the sample was drawn. The p-value indicates the probability of observing a correlation of 0.94 or greater (or 0.94 or less, because its two-tailed) if the null hypothesis is true. General format for all z- and t-tests You may have noticed that all of the z- and t-tests we have looked at have a common format. The formula always has 3 components, as shown below: z or t = statistic - parameter|H SEstatistic
The numerator always has some statistic (e.g., a sample mean, or the difference between two independent sample means) minus the value of the corresponding parameter, given that H 0 is true. The denominator is the standard error of the statistic in the numerator. If the population standard deviation is known (and used to calculate the standard error), the test statistic is z. if the population standard deviation is not known, it must be estimated with the sample standard deviation, and the test statistic is t with some number of degrees of freedom. Table 5 lists the 3 components of the formula for the t-tests we have considered in this chapter. For all of these tests but the first, the null hypothesis most often specifies that the value of the parameter is equal to zero. But there may be exceptions.
Similarity of standard errors for single-sample and unpaired t-tests People often fail to see any connection between the formula for the standard error of the mean, and the standard error of the difference between two independent means. Nevertheless, the two formulae are very similar, as shown below. s s2 =sX = = n n s2 n is a fairly
Given how s X is expressed in the preceding Equation (1.22), it is clear that s straightforward extension of s X (see Table 5).
Table 5: The 3 components of the t-formula for t-tests described in this chapter.
Name of Test
Parameter|H0
SE of the Statistic sX = s n sD n
Single-sample t-test
Paired t-test
sD =
Independent samples t-test
( X1 X 2 )
s2s2pooledpooled= + nn2 1 1 r2 sr = n2
Test of significance of Pearson r
Assumptions of t-tests All t-ratios are of the form t = (statistic parameter under a true H 0 ) / SE of the statistic. The key requirement (or assumption) for any t-test is that the statistic in the numerator must have a sampling distribution that is normal. This will be the case if the populations from which you have sampled are normal. If the populations are not normal, the sampling distribution may still be approximately normal, provided the sample sizes are large enough. (See the discussion of the Central Limit Theorem earlier in this chapter.) The assumptions for the individual t-tests we have considered are given in Table 6. A little more about assumptions for t-tests Many introductory statistics textbooks list the following key assumptions for t-tests:
1. 2. 3. The data must be sampled from a normally distributed population (or populations in case of a two-sample test). For two-sample tests, the two populations must have equal variances. Each score (or difference score for the paired t-test) must be independent of all other scores.
The third of these is by far the most important assumption. The first two are much less important than many people realize.
Table 6: Assumptions of various t-tests.
Type of t-test
Single-sample Independent samples ttest
You have a single sample of scores All scores are independent of each other The sampling distribution of X is normal (the Central Limit Theorem tells you when this will be the case) You have matched pairs of scores (e.g., two measures per person, or matched pairs of individuals, such as husband and wife) Each pair of scores is independent of every other pair The sampling distribution of D is normal (see Central Limit Theorem) You have two independent samples of scoresi.e., there is no basis for pairing of scores in sample 1 with those in sample 2 All scores within a sample are independent of all other scores within that sample The sampling distribution of X 1 X 2 is normal The populations from which you sampled have equal variances The sampling distribution of r is normal H 0 states that the correlation in the population = 0
Test for significance of Pearson r
Lets stop and think about this for a moment in the context of an unpaired t-test. A normal distribution ranges from minus infinity to positive infinity. So in truth, none of us who are dealing with real data ever sample from normally distributed populations. Likewise, it is a virtual impossibility for two populations (at least of the sort that would interest us as researchers) to have exactly equal variances. The upshot is that we never really meet the assumptions of normality and homogeneity of variance. Therefore, what the textbooks ought to say is that if one was able to sample from two normally distributed populations with exactly equal variances (and with each score being independent of all others), then the unpaired t-test would be an exact test. That is, the sampling distribution of t under a true null hypothesis would be given exactly by the t-distribution with df = n1 + n2 2. Because we can never truly meet the assumptions of normality and homogeneity of variance, ttests on real data are approximate tests. In other words, the sampling distribution of t under a true null hypothesis is approximated by the t-distribution with df = n1 + n2 2. So, rather than getting ourselves worked into a lather over normality and homogeneity of variance (which we know are not true), we ought to instead concern ourselves with the conditions under which the approximation is good enough to use (much like we do when using other approximate tests, such as Pearsons chi-square). Two guidelines
The t-test approximation will be poor if the populations from which you have sampled are too far from normal. But how far is too far? Some folks use statistical tests of normality to address this issue (e.g., Kolmogorov-Smirnov test; Shapiro-Wilks test). However, statistical tests of normality are ill advised. Why? Because the seriousness of departure from normality is inversely related to sample size. That is, departure from normality is most grievous when sample sizes are small, and becomes less serious as sample sizes increase.6 Tests of normality have very little power to detect departure from normality when sample sizes are small, and have too much power when sample sizes are large. So they are really quite useless. A far better way to test the shape of the distribution is to ask yourself the following simple question: Is it fair and honest to describe the two distributions using means and SDs? If the answer is YES, then it is probably fine to proceed with your t-test. If the answer is NO (e.g., due to severe skewness, or due to the scale being too far from interval), then you should consider using another test, or perhaps transforming the data. Note that the answer may be YES even if the distributions are somewhat skewed, provided they are both skewed in the same direction (and to the same degree). The second guideline, concerning homogeneity of variance, is that if the larger of the two variances is no more than 4 times the smaller,7 the t-test approximation is probably good enoughespecially if the sample sizes are equal. But as Howell (1997, p. 321) points out,
It is important to note, however, that heterogeneity of variance and unequal sample sizes do not mix. If you have reason to anticipate unequal variances, make every effort to keep your sample sizes as equal as possible.
Note that if the heterogeneity of variance is too severe, there are versions of the independent groups t-test that allow for unequal variances. The method used in SPSS, for example, is called the Welch-Satterthwaite method. See the appendix for details. All models are wrong Finally, let me remind you of a statement that is attributed to George Box, author of a wellknown book on design and analysis of experiments:
All models are wrong. Some are useful.
This is a very important pointand as one of my colleagues remarked, a very liberating statement. When we perform statistical analyses, we create models for the data. By their nature, models are simplifications we use to aid our understanding of complex phenomena that cannot be grasped directly. And we know that models are simplifications (e.g., I dont think any of the early atomic physicists believed that the atom was actually a plum pudding.) So we should not find it terribly distressing that many statistical models (e.g., t-tests, ANOVA, and all of the other
As sample sizes increase, the sampling distribution of the mean approaches a normal distribution regardless of the shape of the original population. (Read about the central limit theorem for details.) 7This translates to the larger SD being no more than twice as large as the smaller SD.
parametric tests) are approximations. Despite being wrong in the strictest sense of the word, they can still be very useful.
References Howell, D.C. (1997). Statistics for psychology (4th Ed). Belmont, CA: Duxbury. Pagano, R.R. (1990). Understanding statistics in the behavioral sciences (3rd Ed). St. Paul, MN: West Publishing Company.
Appendix: The Unequal Variances t-test in SPSS
Denominator for the independent groups t-test, equal variances assumed
2(n1 1) s12 + (n2 1) s2 1 1 SE = + n1 + n2 2 n1 n2
11 + n1 + n2 2 n1 n2
within _ groups
1 1 = s2 + pooled n1 n2
(A-1)
s2 s2 = pooled + nn2 1
pooled
Denominator for independent t-test, unequal variances assumed
s12 s2 SE = + n1 n2
Suppose the two sample sizes are quite different If you use the equal variance version of the test, the larger sample contributes more to the pooled variance estimate (see first line of equation A-1) But if you use the unequal variance version, both samples contribute equally to the pooled variance estimate
Sampling distribution of t For the equal variance version of the test, the sampling distribution of the t-value you calculate (under a true null hypothesis) is the t-distribution with df = n1 + n2 2 For the unequal variance version, there is some dispute among statisticians about what is the appropriate sampling distribution of t There is some general agreement that it is distributed as t with df < n1 + n2 2
There have been a few attempts to define exactly how much less. Some books (e.g., Biostatistics: The Bare Essentials describe a method that involves use of the harmonic mean. SPSS uses the Welch-Satterthwaite solution. It calcuates the adjusted df as follows:
df =
ss 2 1 + n1 n2
s12 s2 n1 + n2 n1 1 n2 1
For this course, you dont need to know the details of how the df are computed. It is sufficient to know that when the variances are too heterogeneous, you should use the unequal variances version of the t-test.
Student's T Test
Statistical Hypothesis Testing
Null Hypothesis
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Kind Of Blue is ranked as the best album by Miles Davis.
Members who like this album also like: The Black Saint And The Sinner Lady by Charles Mingus, Mingus Ah Um by Charles Mingus and The Freewheelin' Bob Dylan by Bob Dylan.
Related links: top tracks by Miles Davistop tracks of the 1950s, top tracks of 1959.
Showing latest 5 ratings for this album. | Show all 1,792 ratings for this album.
I have to admit my ignorance. I don't get the hype about "Kind of Blue"
The most influential album of all time. A timeless masterpiece.
Hugely influential, the tone is quite understated but lots of emotion on display.
Jazz is surely the most important musical genre of the 20th century. And the album "Kind Of Blue" is by far the best jazz album of all time. Even if you do not like jazz, you have to love this incomparable masterpiece. The album is probably one of the coolest album of all time (still today, while this album at over 50 years old). Miles Davis conveys emotions through this record such as melancholy, romance, joy ... This album is also revolutionary for jazz, it is the inventor of modal jazz and the New Age movement. Miles Davis is actually an artist who touches me. This is my favorite musician: these melodies are so beautiful. He is a really important artist. I think this album is the greatest musical emotion of all my life. I even learned all the pieces of the record at the piano so much I'm a fan. I recommend this classic to 10000%.
I'm just waiting for Jessica Rabbit to walk in with her red dress trailing behind her!
All joking aside, this is a superb album with a real sleazy lounge bar feel about it in parts. That isn't a criticism by the way!
On the right occasion, I love it and the rest of the time I engulf it!
Kind of Blue is jazz for everyone. Not in a demeaning way to say that they're sellouts, but in that, it is very accessible and is highly effective at creating dreamy, jazzy moods. I can't say it moves me emotionally, but it is nonetheless highly enjoyable. Favourites: So What, Freddie Freeloader. There's no denying the incredible talent of every group member, and I think everyone should listen to this simply due to the massive significance this has if nothing else.
Genial! Jazz absoluto, técnico e sentimental.
While this album is something awesome, great tunes, great instrumentalists with a lot of groove, I dont consider it his best work. In a Silent Way remains Miles Davis best work for me... just cant get enough of that vibe that comes from it.
Top Tracks: "So What", "Freddie Freeloader", "Blue in Green"
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Love your card design and color choices. It is a very cheerful card.
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text:
James Edward "Tug" Price
July 11, 1949 - October 24, 2021
James Edward "Tug" Price passed away peacefully Sunday, October 24, 2021 at the WVU Medicine Hospital in Wheeling, WV at the age of 72. He was born to Sarah Price and the late Luther Price on July 11, 1949 in Madison, WV. Tug was an accomplished athlete excelling in football, basketball, and baseball at Scott High School until graduating in 1968. He then took his athletic talents to Institute, WV to play football for West Virginia State College. Tug was a part of the 1968 Championship team. On October 6, 2018, he along with three former Scott Skyhawk teammates were honored by WVSU into the Football Hall of Fame. He was a retired union coal miner and mine foreman, having begun his career at Madison Mine in Ashford, WV, then moved on to Island Creek Kelly No. 5 in Blair, WV. Bethlehem Steel and Peabody Coal were other places he was employed. During retirement, Tug enjoyed his paper route for the Charleston Newspapers for many years. Some of his favorite pastimes were catching up with friends, playing softball, and hanging with his dog Cheyanne. Tug is survived by his wife Connie Price and her daughter Robin Dingess both of Madison, WV; daughter, Erica (Patrick) Dues of Jacksonville, NC; daughter, Stephanie Price of Madison, WV; mother, Sarah Price of Madison, WV; sisters, Bonnie (Arthur) Jackson of Dearborn, MI, Linda (Cornelius) Rodgers of Detroit, MI, and Brenda (Gary) DeGrange of Babson Park, FL; brothers, Wesley (June) Price of Madison, WV, Danny (Shelia) Price of Morgantown, WV, Rex (Shelly) Price of Winter Park, FL, and William (Samantha) Price of Hurricane, WV; uncles, Albert Price of Madison, WV, Joseph "JC" Ford of Detroit, MI; aunt, Virginia Curry of Newport News, VA; granddaughters, Trista (Zach) Price of Madison, WV, Jordan (Shane) Huhn of Jacksonville, NC, Bailey Price of Madison, WV; great grandchildren, Tristin, Elizabeth, Scarlett, Adelyn; and a host of nieces, nephews, cousins, and an abundance of friends. Visitation will begin at 6 p.m. on Friday, November 5, 2021 at Handley Funeral Home in Danville, WV with Reflections from family and friends beginning at 7 p.m. until 8 p.m. Funeral service will be 2 p.m. Saturday at the funeral home with burial following in Memory Gardens on Low Gap Road, Madison, WV.
James Edward "Tug" Price passed away peacefully Sunday, October 24, 2021 at the WVU Medicine Hospital in Wheeling, WV at the age of 72. He was born to Sarah Price and the late Luther Price on July 11, 1949 in Madison, WV. Tug was... View Obituary & Service Information
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text:
Sumup Pos Stand Card Reader Ipad 2023
SumUp POS is one of the quickest and most convenient ways . Sumup Pos Stand Card Reader Ipad. enhancing your organization, with the added bonus offer of little in the method of hoops to leap through. The UK-based mobile point of sale (mPOS) issue is already popular with many companies thanks to its other services and products, which include the Air Card Reader and the 3G Card Reader.
The former lets you use this natty bit of tech by mating it with your smartphone in order to accept mPOS payments, which comes in handy during the coronavirus crisis. The latter permits you to process payments on the go using mobile network connectivity. Another choice nevertheless, is this, the point of sale hardware, which can be teamed up with the app to intensify the selling power of your organization.
International payments fintech is responding to the cost-of-living crisis by releasing a brand-new light-weight suite of point-of-sale (POS) services, which it says will produce "an entry-level POS product" for the tiniest independent companies.
The brand-new service, called POS Lite, consists of a 13in splash-proof touchscreen and stand, card reader, and subscription to SumUp's software. Created for little and nano-sized companies, it will allow merchants to sign up card and money payments, arrange their item catalogue, track earnings and more. It is based on the fintech's existing POS Pro system but created to be more cost efficient; in recent months, small businesses have actually been struck the hardest by increasing fuel costs, spiralling inflation and higher business rates.
has quite a lot of appeal as it does not carry a month-to-month cost. In the UK, where is headquartered, you get the simple alternative of one charge per deal and with any type of card. For that reason, you get no regular monthly expenses( opens in new tab) and simply a 1.69% deal cost.
Including more weight to the potential of SumApp is the app and POS hardware( opens in new tab) mix that offers you a physical gadget for processing payments, which presently costs �,� 139 and includes a Stand, Air Card reader and charging station. Sold on its own, the SumUp Air Card( opens in brand-new tab) reader, which permits you to take payments in tandem with your mobile phone expenses �,� 59 and if you want that with the accompanying cradle then it's �,� 69.
The 3G card reader( opens in new tab) is currently �,� 99 while the docking station, which lets you print receipts and keep your card reader charged is �,� 169 (though currently �,� 129 for a limited time deal). All prices exclude barrel.
In the US, there's a counterpart, the SumUp Point of Sale, which says the business is powered by the app. Again, there are no regular monthly expenses and a 2.65% charge per transaction.
has actually constructed the hardware for the POS Lite system entirely in house, drawing on feedback from small merchants to make the suite as easy-to-use as possible. The company states it's designed for services that require to take cashless payments however do not need a fully fledged POS option right now, and it will cost merchants a one-time cost of �,� 199.
co-founder Marc-Alexander Christ remarks: "SumUp has actually consistently provided innovative innovation, which supports and serves the worldwide little and nano organization neighborhood. By broadening our point-of-sale services, continues to offer merchants with the tools they require to make a success of their company. Development is at our core, and is really proud to support over 4mn merchants around the globe."
In a declaration, SumUp continues: "Unlike what the market offers today, it's ready to use immediately, features totally free pre-installed software, and [has] no subscription charges."
has likewise unveiled brand-new functions to POS Pro– its advanced POS item that lets merchants, especially bars and restaurants, send out several orders to the kitchen area at once. The new features will be readily available in the UK and Ireland, France, Spain and Germany. The London-based business supports more than 3.5 mn merchants in over 30 markets around the world and is doing its bit to utilize its success for good: has promised to donate 1% of future net earnings to ecological causes in the fight against environment change.
he SumUp POS terminal idea, and undoubtedly the other items in the variety, certainly makes a modification from much of the other humdrum portions of hardware out there. There's a stand, which in effect is a sustainably sourced and neatly styled bit of kit to hold a tablet.
There's likewise the Air Card Reader, which will let you accept major debit and credit cards, consisting of contactless Chip & PIN, Apple Pay and Google Pay. You can likewise utilize the charging mentioning, which ensures your terminal is ready to address any offered time. In terms of functionality then the SumUp POS revolves around the supporting app, which if you trigger the Product Brochure allows you to use it as a point of sale service.
A POS system (point of sale system) is the hardware used to process orders and payments, along with the software application operating on those systems. 'Point of sale' describes the time and place a consumer communicates with a merchant to buy products or services.
Processing payments, POS systems– such as Point of Sale Lite– typically enable the merchant to take care of administrative jobs, like evaluating deal reports and handling their products and prices from one single platform.
" As soon as I switched it on, everything just worked! We've all found it extremely intuitive to use. Thank you for making transactions more secure and easier." Sumup Pos Stand Card Reader Ipad
The Product Catalogue lies at the heart of the system as it lets you develop, handle and personalize all of the products and classifications in your business. The functionality for that reason includes everything required to itemize your stock, such as rates, descriptions and photos.
Establishing Point of Sale Lite couldn't be easier. Simply follow these basic actions:
Does Sum Up have a POS? Sumup Pos Stand Card Reader Ipad
Turn it on using the power button below the logo
Now you have access to all the functions– get started by developing an item catalogue with all your items or access your existing item brochure saved in your profile
To take card payments, merely add an item to your shopping cart and pick the payment approach "Card Reader", then follow the guidelines
Presently, Point of Sale Lite works with the SumUp Air card reader. You can get it here if you don't have an Air card reader yet. If you already have an Air card reader, you can purchase the standalone Point of Sale Lite tablet here.
Once visited, you have immediate access to your profile, your individual item brochure, your sales history and business Account.
Point of Sale Lite is compatible with invoice printers from Star Micronics and Epson that have the ability to communicate over USB, Bluetooth or Wi-Fi. Please keep in mind that the Solo Printer can currently not be utilized with Point of Sale Lite.
Before you subscribe, you can sign up for a 30-day free trial on the Goodtill site. You'll require to do this online and not through the app. The sign-up is quite quick and basic– it's primarily about selecting a username and password and providing basic contact information.
Your account is developed immediately, after which asks for more comprehensive business details and leads you to the Backoffice account (browser-based).
You won't have the ability to test the POS app till you have actually included items, settings and main details to your account. This might take a little while, given that it's unclear in the control panel where to begin. The most user-friendly POS apps on the market tend to have a detailed guide to help you begin quickly, whereas SumUp Point of Sale just points you to an introduction video.
You are likewise encouraged to contact among their onboarding personnel to help with setup and learn more about the software application. If you 'd rather get on with it by yourself, there's an assistance chat in Backoffice that can point you to the ideal guides in their (incomplete) POS resource area.
We searched for answers to different queries without luck and got in touch with support through the messaging chat. In all cases, we were described an e-mail address considering that the support personnel didn't know the response. This could be an issue when you just wish to get going quickly, especially as there are few explainers in the Backoffice interface.
Rest assured: the Core POS module is a complete POS system on its own. If you work in hospitality or retail, nevertheless, you will probably need the corresponding add-on modules too. Let's look at what you're getting.
Checkout options: This is part of the Core POS module. The checkout screen in the app shows the item layout, categories (in different tabs) and a left-hand view of the present, itemised bill. You'll require to modify the item screen in the back workplace– this can not be performed in the app.
It provides you an option to accept payment over the phone, however you'll need to manage this through your picked payment processors due to the fact that the till app only gives you the choice to register it as 'Phone'. The same applies to tapping 'Other', 'Gift Card/ Voucher', 'Account' on the payment screen– there is no screen for going into a reference number, so you'll require to have a system for tracking these yourself.
With an integrated card machine, the POS app will operate in conjunction with it for a smooth checkout. If incorporated with payments, the POS app connects with the Air card reader and allows you to send out payment links or generate a QR code for the transaction directly on the screen.
Receipts: The till app lets you send out an e-mail invoice after each sale or print a sales receipt through a linked printer. We've seen reports from users that the app automatically prints a receipt each time, even when you tap that it's not required, so you can't in fact minimize invoice paper up until this bug is repaired.
Stock library: With the Core POS module, you have basic product management capabilities. This includes stock levels, low stock alerts, supplier lists, bulk item import and an in-depth items library.
Each item can be connected to a classification and have variants, characteristics (added layer of variants), modifiers, tags, SKUs, barrel rates, an image and more. After items are contributed to your account, you can add them to the till user interface through the a little clumsy 'Offering Design' in the back workplace.
Integrated and automated order
Customers and loyalty: The Core POS module enables you to add customers with contact details to a list, divisible into client groups. They can be published to the system to save time if you currently have a spreadsheet of individuals.
Sumup Pos For Food Trucks 2023
Sumup Pos Faq 2023


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The Super Bowl Champs are Going to Disneyland!
15 Feb The Super Bowl Champs are Going to Disneyland!
Posted at 08:40h in Disney Updates by Katie Fissore 0 Comments
February 13, 2022, The Los Angeles Rams became Super Bowl Champions against a close game to the Cincinnati Bengals. Super Bowl LVI was hosted in the SoFi Stadium in California, and ESPN was broadcasting from Disneyland Resorts. After the successful Super Bowl victory, Quarterback Matthew Stafford, MVP Cooper Kupp, and Aaron Donald all answered the famous question, "What are you going to do next?" with the best answer possible- "I'm going to Disneyland!"
The Walt Disney World Company celebrates the Super Bowl champs with a parade down Main Street with our favorite Disney characters and the game's MVP. This is a tradition that these players hold in their hearts. The team's champions take their families to the most magical place on earth, after what we can only imagine is the most magical game of their life. Playing professional football places a ton of sacrifices on these families, and to celebrate the hard work played on the field, what better way is there than to put on some Mickey Ears and laugh with Goofy and Donald Duck.
The "I'm going to Disney World" tradition, like every Disney story has a once upon a time…
Michael Eisner (Disney CEO 1984-2005) and his wife, Jeana Eisner recall the birth of the famous slogan in a Sports Illustrated interview-
"In January 1987, we were launching Disneyland's Star Tours, an attraction based on Star Wars. After the ribbon-cutting ceremony, my wife, Jane, and I had dinner with George Lucas, as well as Dick Rutan and Jeana Yeager, who had just become the first people to fly around the world without stopping. It was late and the conversation hit a lull as we waited for our food. So I asked Dick and Jeana, 'Well, now that you've accomplished the pinnacle of your aspirations, what could you possibly do next?' Rutan responded, without hesitation, 'I'm going to Disneyland.' And of course I go, 'Wow, that's cool! You made the right choice.' But my wife interjects: 'You know, that's a good slogan.'"
At the time, Super Bowl XXI was around the corner, where the New York Giants and Denver Broncos were set to play each other. Could there be a better time to air this commercial than after the Super Bowl? Disney realized quickly the pixie dust that was on this idea and reached out to both quarterbacks, Phil Simms and John Elway. Disney presented both quarterbacks with the same offer: $75k after the Super Bowl when asked "What you are going to do?" reply into a Disney camera with "I'm going to Disney World" and "I'm going to DisneyLand." Both players would be asked to say it twice for airings on both sides of the coasts.
John Elway signed Disney's contract right away, while Phil Simms was on the fence about it… until he caught wind that Elway eagerly signed the offer. It was a good thing that Phil Simms did, because he took the Giants to a Super Bowl victory defeating the Broncos 39-20. He was the first athlete in history to say, "I'm going to Disney!" This slogan is now used in response to huge victories from NBA MVP's to American Idol winners. The famous commercial usually has the "When You Wish Upon a Star" song in the background, and an outline of Cinderella or Sleeping Beauty's Castle, and can bring a tear to anyone's eye.
As DVC members, we hold this belief; when in doubt, going to Disney. This is a tradition that has been going strong for 35 years now, and we don't see it stopping anytime soon.


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As restrictions lift and vaccination diminishes the threat of COVID-19, life returns to Toronto, the city that always sleeps
It was never destined to end how it started: with a bang and a burst and a sudden change to every aspect of our lives. Instead, it is now what it was always going to be, a slow unsteady drip, a growing pool, a gradual return to what we used to think of as the every day.
On Friday, Ontario's new chief medical officer of health, Dr. Kieran Moore, announced that the province, almost 16 months into the pandemic shutdown, would be ready to move into a Stage 3 reopening next Friday, days ahead of schedule.
The Star's Robert Benzie described Moore as being "visibly relieved" at the announcement. "Key indicators are trending in the right direction," he said.
It isn't fully over, in other words.
It would be hard to define what "over" even means now.
But it is ending in a way, here, after so many stumbles and false starts.
"It's a pretty special moment," said Dr. Isaac Bogoch, an infectious disease epidemiologist at the University of Toronto.
"You can watch the life return to the city in real time."
After lockdowns and stay-at-home orders, after lost businesses and missed friends, lonely deaths and so much fear, the province, thanks to an unprecedented vaccination campaign, will be ready next week to reopen museums and gyms, indoor restaurants, movie theatres, spectator sports, concerts and more.
If the same trends — these are falling cases and rising vaccines rates, primarily — continue, Ontario could move to a form of post-pandemic normal sometime in August. No more capacity limits. No more social circles or broken bubbles. No more wondering who to leave off the funeral list to keep the numbers down.
"It's noticeable," Bogoch said. "There's just life returning to the streets. There's people flooding patios, and soon there will be indoor dining. And I don't know, it's just wonderful to see. It really is wonderful to see the city and the province revitalize."
As much as anyone not in government, Bogoch, a telegenic and willing interview, has been a public face of COVID-19 in Ontario. He's been ever-present on TV and in the newspapers, serving as a kind of unofficial tonic to the province's official top doctor through much of the pandemic, the muddled and often harried Dr. David Williams.
COVID-19 has been his life, in other words, in the hospital and at home, for 16 months. It's been his everything and he's ready to move on.
"I plan on eating in every single restaurant in the Toronto area and I'm not joking," he said.
"I want to make a shirt that says 'Eat My Way Through The GTA.'
"And I really, truly want to go to every Ma-and-Pa shop that's open and try every type of food …. Mississauga, Etobicoke, Scarborough: I want to try everything, go everywhere, eat everything.
"I can't wait to do that."
(For the record, he plans on starting in Kensington Market with about "18 types of tacos.")
But at the same time, what Bogoch is feeling now isn't simple; it isn't all joy and release. "As good as it feels, it's not lost on me that 9,000 people in Ontario have died," he said. "It's not lost on me that 25,000 Canadians died. It's not lost on me that kids missed significant portions of two school years. It's not lost on me that business owners, both small and large, got pummeled and people are out of work and have either felt this thing financially, or from a health standpoint. It's not lost on me that many people have lost someone very close to them.
"But it's also not lost on me that we're really close to the finish line.
"The worst is behind us. And it's OK to move forward and it's exciting to move forward. But we still can't forget what happened over the last 16 months."
Like so much else about the last year and a half, what Bogoch is feeling, what the whole city is feeling right now, is unprecedented. We have no model for coming out of a pandemic. There is no guide on how to deal with trauma so collective and uneven, while also celebrating the renewed joy of a movie at the TIFF Lightbox, or a favourite painting at the AGO, or just a quiet drink with a magazine at a restaurant you're thrilled is still there.
An Tran was supposed to open his first restaurant in Toronto in March 2020. He saved for years, borrowed money, and poured everything he had into the project, a restaurant, bakery and coffee shop called Ba Noi, or "grandmother," in Vietnamese.
He wasn't open yet when COVID-19 hit. Technically, he had no income and no revenue to lose. So, for months, he couldn't get bridge loans, wage subsidies of even Canada Emergency Response Benefit (CERB). He had a new baby. He was bleeding cash. He didn't know if, financially, he could survive.
Eventually, Tran got his ovens working and he started selling bread, butter tarts and other baked goods out of his front window. He's been doing well since, relatively speaking. But he isn't ready to move on. Not even close.
For Tran, the pandemic doesn't feel over, not yet, no matter what the government says.
"Regardless of everyone being vaxed and things opening up, I don't really know," he said. "I don't want to be stuck in a situation where I invest in a lot of money in something and then I have to shut down again."
The pandemic changed Tran. It changed his plans and his dreams and his outlook on the future. "I'm taking baby steps now," he said. "I'm very conservative about my movements in the business …. Because I don't know what's going to happen.
"I'd rather just wait and see."
Kim Tait, a mother of five and the top rock expert at the Royal Ontario Museum (she is the curator of mineralogy), feels much the same way. She knows the rules have changed. She knows things are looking good. But practically speaking, her life feels pretty much the same.
Her children aren't vaccinated.
She's worried about new variants.
"I don't know if I'm feeling any different. I'm not going to lie," she said. "Canada's Wonderland is open, but I don't know if we're there yet."
In an odd way, what's happening now is the inverse of last March. Back then, doctors and public health experts were begging the city to take COVID-19 seriously. Now many doctors, such as Bogoch, sound more relaxed than cautious citizens wary about another false dawn.
In the earliest days of the pandemic, Dr. David Carr, an emergency room physician in Toronto, was one of the first in the city to openly, and loudly, share his fears about what was coming. It was a new role for Carr. Doctors aren't used to telling the world that they're afraid. But he grew into it.
He felt like he was telling Toronto what it needed to hear.
Sixteen months on, Carr feels very differently.
In a word, he feels safe.
"The biggest thing, obviously, is, I would argue, probably the greatest medical advance in the last 100 years, since insulin: the development of an unbelievable set of vaccines."
It's not that Carr thinks COVID-19 is over; he just doesn't think it's the danger it once was. "There are a lot of variants. There are going to be new variants, and we're all going to learn the Greek letters of the alphabet," he said.
But for the vaccinated, he believes, the risks are low.
There are even signs that life as it was before is starting to come back. In the ER recently, one of Carr's colleagues treated someone who had been hit in the head with a beer bottle, something none of them had seen in more than a year.
"People are vaccinated," Carr said. "People are going to have the fun they used to have. And then that's when we'll start seeing those problems again."
Richard Warnica is a Toronto-based business feature writer for the Star. Reach him via email: [email protected]
Majority of Canadians think COVID-19 U.S. border restrictions should lift this year: Nanos
New wildfire ignites close to Vernon, B.C., as firefighters contain nearby blaze in Coldstream


Primarily aims to develop the student's academic language skills in pursuing college studies. Provides rich and interesting materials for language learning strategies to improve listening and reading comprehension, as well as the ability to compose effective paragraphs.
Published in June 1997. Revised Edition September 1997. Seventh printing 2006. Winner, 1998 Gintong Aklat Award, for Textbook.


He seems take pride to make such fatuous remarks.
Television commercials seem more smug and fatuous than ever.
They're all fatuous, self-obsessed, undirected, confused, emotional retards.
faucet	fatty	fatuous king	fatuous jokes	fatuous fires	fatuous comments	fatuous hopes.


Well four inches of snow of course!! Blame it on the hills, the political leaders or the scared snow averse city dwellers, but four inches of snow is enough to keep most people at home glued to the television. I think the news anchors play to the fears of the public and have enough influence to talk even the most devoted employees into skipping work. I however am not married to one of those types, so while he worked I walked around the city checking out the many street closed signs, empty stalls at Pike Place Market and slushy streets. While strolling around Pike Place Market I even was approached by a man and asked if I wanted to go sledding. Random, but entertaining nonetheless.
Visibility was poor as the snow swirled around all day long.
It was beginning to look a lot like Christmas. If only it was December 23rd.
Usually a busy area, buses had to share the roads only with snow plows.
With steep hills all over the city, our best response to the winter weather is to close the streets down.
Beecher's tomato soup= the perfect snow day meal.
The snow kept away almost all the vendors at Pike Place Market.
Signs like this are entertaining, especially because they lend themselves to showing a society dependent on spell check.
Rachel the pig even was snowed on a tad.
Colors pop on a white background and everything is prettier with snow.
A rare sight of an empty Pike Place market in the middle of the day.
A good day for hot chocolate and warm pastries.
Ship yard cranes in the background.
Roads were slushy and dirty from sand, which made me glad our snow typically does not stay around too very long.
When in doubt, make a snowman and head off sledding.


text:
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redpajama_set_name: RedPajamaGithub


How's your 2018 going so far? If you've let some of your New Year's intentions slide, that's totally normal—it turns out 80 percent of New Year's resolutions fail by February. It could be that the goal isn't specific enough, is overly ambitious, or simply doesn't fit in with your lifestyle. Whatever the reason, even the highest achievers among us can start to fizzle out right about now. The good news is, it's not too late to turn your progress around!
In the leadup to National Green Juice Day on January 26, mbg and Evolution Fresh will be offering small, actionable challenges to help you keep your clean eating intentions in check using—you guessed it—greens!
On the first day of the #GotMyGreens challenge, we shared how to bring more plants, specifically greens, into your life, and now we're talking about something you should take out of your diet to help them work their magic. Yep, we're reducing refined sugar. But don't panic, we've got your back.
Day 2 Challenge: Cut back on refined sugar.
Why cut back on the sweet stuff? Our taste buds quickly learn to crave sugar, which is what makes this challenge so important and so, well, challenging.
But let's talk benefits. Reducing your sugar intake puts less stress on your body, clearing the way for improved health. The best part? You'll have way more room for good, whole foods that nourish you from the inside out.
You might think that cutting back just means avoiding cookies and cake for the day. But unfortunately sugar is lurking is many less-than-obvious packaged foods that you wouldn't necessarily be able to pick out of a sugar-laden lineup. The easiest way to avoid added sugars is to read the labels, prepare your own meals, make your own sauces and salad dressings, and shop the perimeter of the supermarket since refined sugars are mostly hiding in the packaged stuff.
To avoid cravings, make sure you're filling the sugar gaps with lots of fresh, whole foods and fluids like water, herbal tea, and veggie juices. Evolution Fresh green juices have no added sweeteners and are filled with high-quality produce, which makes them the perfect way to get the nutrition of greens in without the hidden harms of sugar. We like to think of them as a shortcut to our clean-eating goals since drinking them boosts all the other positive changes you're making this week.
By reducing sugar and bringing in more nutrient-dense foods and juices in its place, you'll reset your system and get on the right track for all your healthy New Year's intentions.
We're so excited to see what kind of sugar-free meals you guys come up with for this challenge. Don't forget to share them using the #GotMyGreens hashtag and check out our sweepstakes for a chance to win some awesome prizes, just for eating healthy (see the rules here, while you're at it). Tune in tomorrow for your next challenge.


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Q: how do I structure a jquery .after statement with a table and multiple classes? I am trying to create a table in jQuery using .after. Here is my code
$( nearestRow ).after( "

" );

The problem that I am having is that the dom is not generating the table correctly.
The dom is generating it without spaces. How can I fix this?

A: Not sure what spaces you're expecting, but I'm guessing you're trying to set table, responsive and bordered classes on the table. However, since they're not quoted in your string, which is actually:

You just have class=table, and responsive and bordered attributes.  You want to say:
$( nearestRow ).after( "

" );

To set those three classes.

redpajama_set_name: RedPajamaStackExchange


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Paris, France | AFP | Paris Saint-Germain brushed aside bottom club Metz 5-0 in Ligue 1 on Saturday to ease some of the pain of their Champions League exit at the hands of Real Madrid in midweek.
Twenty-year-old Christopher Nkunku scored twice for PSG at the Parc des Princes, with Thomas Meunier, Kylian Mbappe and Thiago Silva also on target.
The result restores PSG's 14-point lead over reigning champions Monaco at the top of the table with just nine games left, keeping them on course to reclaim the title.
A clean sweep of the domestic trophies for the third time in four seasons is on the cards, although that will not completely make up for the disappointment of another underwhelming campaign in the Champions League.
Tuesday's 2-1 home defeat against Madrid condemned them to a second consecutive exit from that competition in the last 16 and effectively ended any prospect of Spanish coach Unai Emery still being in charge next season.
Emery, who was jeered by some sections of the home support, had to somehow lift his side for this game, with PSG missing suspended leading scorer Edinson Cavani as well as the injured Neymar.
"Everyone knows that we are disappointed, that we disappointed the fans, but we are trying to move forward again," said Mbappe.
Belgian right-back Meunier, who came in for Dani Alves and opened the scoring inside five minutes, added: "It was an excellent reaction after the elimination.
"It is very positive. Mentally it does us good, and it allows us to put Tuesday almost definitively behind us.
"There is no point living with regrets. We have three trophies to go and win still and that is the most important thing," added Meunier, whose side are through to the French Cup semi-finals and play Monaco in the League Cup final at the end of this month.
After Meunier's opener, Nkunku put the game beyond Metz before the half-hour mark with two low finishes.
He was unlucky not to get a hat-trick, a header from Angel di Maria's cross on the stroke of half-time disallowed for a marginal offside call.
However, Mbappe did make it 4-0 before the interval by scoring after being set up by Marco Verratti.
Captain Silva headed in late on to complete the scoring, as Paris maintained their perfect record at home in the league this season.
Second-placed Monaco won 3-1 at Strasbourg on Friday with Stevan Jovetic, Rony Lopes and Fabinho getting their goals.
Third-placed Marseille go to Toulouse and fourth-placed Lyon host Caen on Sunday.
Elsewhere, hundreds of angry Lille supporters invaded the pitch at their Stade Pierre-Mauroy after their struggling side drew 1-1 at home to Montpellier.
Some of the pitch invaders tried to attack the young forward Nicolas Pepe, who had scored their goal, before making their way towards the players tunnel, below where club directors were watching on.
Stewards eventually managed to keep back the supporters, who are unhappy at the way their team's season is going — Lille are in the relegation zone, with Christophe Galtier struggling to turn around the fortunes of the side after Marcelo Bielsa was sacked late last year.
Nantes moved back up to fifth, which is likely to be enough for a European place, after beating Troyes 1-0 with a late Emiliano Sala goal, his 12th of the season.
They replace local rivals Rennes, who came from behind to draw 1-1 with Saint-Etienne.


text:
Q: kmeans clustering python There are coordinates that I want to cluster. The result of clustering using the kmeans
[[0, 107], [0, 108], [0, 109], [0, 115], [0, 116],
[0, 117], [0, 118], [0, 125], [0, 126], [0, 127],
[0, 128], [0, 135], [0, 136], [0, 194], [0, 195],
[1, 107], [1, 108], [1, 109], [1, 110], [1, 114],
[1, 115], [1, 116], [1, 117], [1, 118], [1, 119]...]

The result of clustering using the kmeans
from sklearn.cluster import KMeans
num_clusters = 9
km = KMeans(n_clusters=num_clusters)
km_fit = km.fit(nonzero_pred_sub)

>>>array([7, 7, 7, 1, 1, 1, 1, 5, 5, 5, 5, 3, 3, 0, 0, 7, 7, 7, 7, 1, 1, 1,
   1, 1, 1, 5, 5, 5...]

I want to know the coordinates of i-th cluster
for example, i need elements of 1st cluster and i can assume [0, 107], [0, 108], [0, 109] was clustered into the 7-th cluster.
How can i get coordinates from cluster?

A: I assume you want the coordinates affected to the 7th cluster. You can do so by storing you result in a dictionary :
from sklearn.cluster import KMeans
km = KMeans(n_clusters=9)
km_fit = km.fit(nonzero_pred_sub)

d = dict() # dictionary linking cluster id to coordinates
for i in range(len(km_fit)):
  cluster_id = km_fit[i]

  if cluster_id not in d:
    d[cluster_id] = []

  d[cluster_id].append(nonzero_pred_sub[i])

# that way you can access the 7th cluster coordinates like this
d[7]

>>> [[0, 107], [0, 108], [0, 109], [1, 107], [1, 108], [1, 109], [1, 110], ...]

To remove the "if" section in the loop, you can try looking into defaultdict objects.
You can also surely manage that with pandas dataframes to make manipulating more complex results easier.
If I misunderstood you question, and what you want is the coordinates of the center of the i-th cluster, you can get this by calling km_fit.cluster_centers_[i] (cf. doc).


text:
Richard Cockburn Maclaurin ( ; June 5, 1870 – January 15, 1920) was a Scottish-born U.S. educator and mathematical physicist. He was made president of MIT in 1909, and held the position until his death in 1920.

During his tenure as president of MIT, the Institute moved across the Charles River from Boston to its present campus in Cambridge. In Maclaurin's honor, the buildings that surround Killian Court on the oldest part of the campus are sometimes called the Maclaurin Buildings.

Earlier, he was a foundation professor of the then Victoria College of the University of New Zealand from 1899 to 1907. A collection of lecture theatres at the Kelburn campus of that university were named after him. He was also a professor at Columbia University from 1907 to 1908.

Personal
Maclaurin was born in Scotland, and was related to the noted Scottish mathematician Colin Maclaurin. He emigrated to New Zealand with his family at the age of four. In 1904 he married Alice Young of Auckland, and they had two sons. His brother James Scott Maclaurin (1864–1939) was a noted chemist, who invented a process for extracting gold with cyanide.

Education
 University Entrance Scholar, 1887, Auckland Grammar School
 B.Sc. (Hons), Mathematics, 1890, Auckland University College.
 BA, 1895 (12th wrangler); LL.D., 1904, St John's College, University of Cambridge.

Publications
 On the Nature and Evidence of Title to Realty, 1901
 Treatise on the Theory of Light, 1908

Honors
 Smith's Prize in Mathematics, 1896
 Yorke Prize in Law, University of Cambridge, 1898

References

External links

 'MACLAURIN, Richard Cockburn', from An Encyclopaedia of New Zealand, edited by A. H. McLintock, originally published in 1966.
 'Richard Cockburn Maclaurin, 1870–1920', from History of the Office of the MIT President, Institute Archives, MIT Libraries, October 2004.
Maclaurin in Mathematics at Victoria University College

1870 births
1920 deaths
Presidents of the Massachusetts Institute of Technology
University of Auckland alumni
People educated at Auckland Grammar School
Academic staff of the Victoria University of Wellington
Alumni of St John's College, Cambridge
Richard
American mathematicians

redpajama_set_name: RedPajamaWikipedia


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