9 Printed Page 586 SECTION 9.3 Summary Significance tests for the mean of a Normal population are based on the sampling distribution of the sample μ mean X. Due to the central limit theorem, the resulting procedures are approximately correct for other population distributions when the sample is large. If we somehow know we can use a test statistic and the standard Normal distribution to perform z σ, calculations. In practice, we typically do not know Then, we use the one-sample statistic σ. with calculated from the distribution with 1 degrees of freedom. P-values − The is approximately correct when Random The data were produced by random sampling or a randomized experiment. Normal The population distribution is Normal OR the sample size is large (n 30). ≥ Independent Individual observations are independent. When sampling without replacement, check that the population is at least 10 times as large as the sample. Confidence intervals provide additional information that significance do not—namely, a range of plausible values for the parameter A two-sided test of = at significance level gives the same H0: μ. μ μ0 α conclusion as a 100(1 confidence interval for − α)% μ. Analyze by first taking the difference within each pair to produce a single sample. Then use one-sample procedures. Very small differences can be highly significant (small when a test is based on a large sample. A P-value) statistically significant difference need not be practically important. Plot the data to display the difference you are seeking, and use confidence intervals to estimate the actual values of parameters. On the other hand, lack of significance does not imply that is true. Even a large difference can fail to be H0 significant when a test is based on a small sample. Significance tests are not always valid. Faulty data collection, outliers in the data, and other practical problems can invalidate a test. Many tests run at once will probably produce some significant results by chance alone, even if all the null hypotheses are true.
Printed Page 586 SECTION 9.3 Summary Significance tests for the mean μ of a Normal population are based on the sampling distribution of the sample mean X. Due to the central limit theorem, the resulting procedures are approximately correct for other population distributions when the sample is large. If we somehow know σ, we can use a z test statistic and the standard Normal distribution to perform calculations. In practice, we typically do not know σ. Then, we use the one-sample statistic with P-values calculated from the distribution with − 1 degrees of freedom. The is approximately correct when Random The data were produced by random sampling or a randomized experiment. Normal The population distribution is Normal OR the sample size is large (n ≥ 30). Independent Individual observations are independent. When sampling without replacement, check that the population is at least 10 times as large as the sample. Confidence intervals provide additional information that significance do not—namely, a range of plausible values for the parameter μ. A two-sided test of H0: μ = μ0 at significance level α gives the same conclusion as a 100(1 − α)% confidence interval for μ. Analyze by first taking the difference within each pair to produce a single sample. Then use one-sample procedures. Very small differences can be highly significant (small P-value) when a test is based on a large sample. A statistically significant difference need not be practically important. Plot the data to display the difference you are seeking, and use confidence intervals to estimate the actual values of parameters. On the other hand, lack of significance does not imply that H0 is true. Even a large difference can fail to be significant when a test is based on a small sample. Significance tests are not always valid. Faulty data collection, outliers in the data, and other practical problems can invalidate a test. Many tests run at once will probably produce some significant results by chance alone, even if all the null hypotheses are true. 9
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