Inference for a Single Population Mean for AP Statistics

Practice problems for these concepts can be found at:

• Inference for Means and Proportions Multiple Choice Practice Problems for AP Statistics
• Inference for Means and Proportions Free Response Practice Problems for AP Statistics
• Inference for Means and Proportions Review Problems for AP Statistics
• Inference for Means and Proportions Rapid Review for AP Statistics

In step II of the hypothesis-testing procedure, we need to identify the test to be used and justify the conditions needed. This involves calculating a test statistic. All test statistics have the following form:

When doing inference for a single mean, the estimator is , the hypothesized value is μ₀ in the null hypothesis H₀: μ = μ₀, and the standard error is the estimate of the standard deviation of , which is

This can be summarized in the following table.

example: A study was done to determine if 12- to 15-year-old girls who want to be engineers differ in IQ from the average of all girls. The mean IQ of all girls in this age range is known to be about 100 with a standard deviation of 15. A random sample of 49 girls is selected, who state that they want to be engineers and their IQ is measured. The mean IQ of the girls in the sample is 104.5. Does this finding provide evidence, at the 0.05 level of significance, that the mean IQ of 12- to 15-year-old girls who want to be engineers differs from the average? Assume that the population standard deviation is 15 (σ = 15).

solution 1 (test statistic approach): The solution to this problem will be put into a form that emphasizes the format required when writing out solutions on the AP exam.

Notes on the above solution:

• Had the alternative hypothesis been one-sided, the P-value would have been 1 – 0.9821 = 0.0179. We multiplied by 2 in step III because we needed to consider the area in both tails.
• The problem told us that the significance level was 0.05. Had it not mentioned a significance level, we could have arbitrarily chosen one, or we could have argued a conclusion based only on the derived P-value without a significance level.
• The linkage between the P-value and the significance level must be made explicit in part IV. Some sort of statement, such as "Since P < α…" or, if no significance level is stated, "Since the P-value is low…" will indicate that your conclusion is based on the P-value determined in step III.

solution 2 (confidence interval approach—ok since H_A is two-sided):

example: A company president believes that there are more absences on Monday than on other days of the week. The company has 45 workers. The following table gives the number of worker absences on Mondays and Wednesdays for an 8-week period. Do the data provide evidence that there are more absences on Mondays?

solution: Because the data are paired on a weekly basis, the data we use for this problem are the difference between the days of the week for each of the 8 weeks. Adding a row to the table that gives the differences (absences on Monday minus absences on Wednesday), we have:

Practice problems for these concepts can be found at:

• Inference for Means and Proportions Multiple Choice Practice Problems for AP Statistics
• Inference for Means and Proportions Free Response Practice Problems for AP Statistics
• Inference for Means and Proportions Review Problems for AP Statistics
• Inference for Means and Proportions Rapid Review for AP Statistics