Section 7.1 Summary

• Significance tests and confidence intervals for the mean of a Normal population are based on the sample mean of an SRS. Because of the central limit theorem, the resulting procedures are approximately correct for other population distributions when the sample is large.

• The standard error of the sample mean is

• The standardized sample mean, or one-sample statistic,

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  has the distribution. If the standard deviation of is replaced by the standard error , the one-sample statistic

  has the distribution with degrees of freedom.

• There is a distribution for every positive degrees of freedom . All are symmetric distributions similar in shape to Normal distributions. The distribution approaches the distribution as increases.

• The margin of error for level confidence is

  where is the value for the density curve with area between and .

• A level confidence interval for the mean of a Normal population is

• Significance tests for are based on the one-sample statistic. -values or fixed significance levels are computed from the distribution.
• A matched pairs analysis is needed when subjects or experimental units are matched in pairs or when there are two measurements on each individual or experimental unit and the question of interest concerns the difference between the two measurements.
• These one-sample procedures are used to analyze matched pairs data by first taking the differences within the matched pairs to produce a single sample.
• One-sample equivalence testing assesses whether a population mean is practically different from a hypothesized mean . This test requires a threshold , which represents the largest difference between and such that the means are considered equivalent.
• The procedures are relatively robust against lack of Normality, especially for larger sample sizes. The procedures are useful for non-Normal data when unless the data show outliers or strong skewness.