Hypothesis tests

Appendices

Margin of Error

In a
confidence interval, the range of values above and below the
sample statistic is called the margin of error.

For example, suppose we wanted to know the percentage of adults that
exercise daily. We could devise a
sample design to ensure that our sample estimate will not
differ from the true population value by more than,
say, 5 percent (the margin of error) 90 percent of the time
(the
confidence level).

How to Compute the Margin of Error

The margin of error can be defined by either of the following
equations.

How to Find the Critical Value

The critical value is a factor used to compute
the margin of error. This section describes how to find the
critical value, when the
sampling distribution
of the statistic is
normal
or nearly normal.

When the sampling distribution is nearly normal, the critical value can be expressed as a
t score or as a
z-score.
To find the critical value, follow these steps.

Compute alpha (α): α = 1 - (confidence level / 100)

Find the critical probability (p*): p* = 1 - α/2

To express the critical value as a z-score, find the z-score
having a
cumulative probability
equal to the critical probability (p*).

To express the critical value as a t statistic, follow these steps.

Find the
degrees of freedom (DF). When estimating a mean score or
a proportion from a single sample, DF is equal to the sample
size minus one. For other applications, the degrees of
freedom may be calculated differently. We will describe
those computations as they come up.

The critical t statistic (t*) is
the t statistic having degrees of freedom equal to DF and a
cumulative probability
equal to the critical probability (p*).

T-Score vs. Z-Score

Should you express the critical value as a t statistic or as a z-score? One way to answer
this question focuses on the population standard deviation.

If the population standard deviation is known, use the z-score.

If the population standard deviation is unknown, use the t statistic.

Another approach focuses on sample size.

If the sample size is large, use the z-score. (The
central limit theorem
provides a useful basis for determining whether a sample is "large".)

If the sample size is small, use the t statistic.

In practice, researchers employ a mix of the above guidelines. On this site, we use z-scores
when the population standard deviation is known and the sample size is large.
Otherwise, we use the t statistics, unless the sample size is small and the underlying
distribution is not normal.

Warning: If the sample size is small and the population distribution is not normal,
we cannot be confident that the sampling distribution of the statistic will be normal. In this
situation, neither the t statistic nor the z-score should be used to compute critical values.

Test Your Understanding

Problem 1

Nine hundred (900) high school freshmen were randomly selected for
a national survey. Among survey participants, the mean grade-point
average (GPA) was 2.7, and the standard deviation was 0.4. What
is the margin of error, assuming a 95% confidence level?

(A) 0.013
(B) 0.025
(C) 0.500
(D) 1.960
(E) None of the above.

Solution

The correct answer is (B). To compute the margin of error, we
need to find the
critical value and the standard error of the mean.
To find the critical value, we take the following steps.

Compute alpha (α):

α = 1 - (confidence level / 100)
= 1 - 0.95 = 0.05

Find the critical probability (p*):

p* = 1 - α/2
= 1 - 0.05/2 = 0.975

Find the degrees of freedom (df):

df = n - 1 = 900 -1 = 899

Find the critical value.
Since we don't know the population standard deviation, we'll express the
critical value as a t statistic. For this problem, it will
be the t statistic having 899 degrees of freedom and
a cumulative probability equal to 0.975. Using the
t Distribution Calculator,
we find that the critical value is 1.96.

Next, we find the standard error of the mean, using the following
equation:

SEx = s / sqrt( n )
= 0.4 / sqrt( 900 ) = 0.4 / 30 = 0.013

And finally, we compute the margin of error (ME).

ME = Critical value x Standard error
= 1.96 * 0.013 = 0.025

This means we can be 95% confident that the mean grade point average
in the population is 2.7 plus or minus 0.025, since the margin of error
is 0.025.

Note: The larger the sample size, the more closely the t distribution
looks like the normal distribution. For this problem, since the sample size is very large, we would have
found the same result with a z-score as we found with a t statistic. That is,
the critical value would still have been 1.96. The choice of t statistic versus z-score does not
make much practical difference when the sample size is very large.