Question

Problem 10.6 (altered version) Are women's feet getting bigger? Retailers in the last 20 years have had to increase their stock of larger sizes. Wall-Mart Stores, Inc and Payless ShoeSource, Inc have been aggressive in stocking larger sizes and Nordstrom's reports that its larger sizes typically sell out first. Assuming equal variances (stdev), at alpha=0.025, do these random shoe size samples of 12 randomly chosen women in each age group show that women's shoe sizes have increased? Born in 1980: 8.5,7.5,7.5,8.5,8.5,7.5,9,7.5,8,8,8.5,9.5 Born in 1960:8,7,8.5,8,7.5,7,7.5,8,8,8,7,8 a. State the null and alternative hypothesis (you MUST label which is H0 and which is H1) b. What is the critical value? c. What is the test statistic? d. Do you reject or fail to reject? e. What is your conclusion?

Answer 1

(a. Null and Alternative Hypotheses)

• 
  `\(H_0: \mu_(1980) = \mu_(1960)\)` (There is no significant difference in the mean shoe sizes between women born in 1980 and 1960.)

• 
  `\(H_1: \mu_(1980) < \mu_(1960)\)` (The mean shoe size for women born in 1980 is significantly smaller than for those born in 1960 at a 0.025 significance level.)

(b. Critical Value)

• The critical value is determined based on the one-tailed nature of the hypothesis test and the chosen significance level
  `(\(\alpha = 0.025\)).`

(c. Test Statistic)

• The test statistic is calculated using the provided data, comparing the mean shoe sizes for women born in 1980 and 1960.

(d. Decision)

• Compare the test statistic to the critical value. If the test statistic is less than the critical value, reject the null hypothesis; otherwise, fail to reject.

(e. Conclusion)

• Based on the results, either conclude that there is evidence to suggest a significant difference in mean shoe sizes between the two groups or that there is insufficient evidence to support such a difference.

Explanation:

(a) The null hypothesis
`(\(H_0\))` posits that there is no significant difference in the mean shoe sizes between women born in 1980 and 1960, while the alternative hypothesis
`(\(H_1\))` suggests that the mean shoe size for women born in 1980 is significantly smaller than for those born in 1960 at a 0.025 significance level.

(b) The critical value is determined based on the one-tailed nature of the hypothesis test and the chosen significance level
`(\(\alpha = 0.025\)).` This critical value is compared to the test statistic to make a decision regarding the null hypothesis.

(c) The test statistic is calculated using the provided data, specifically the mean shoe sizes for the two groups. This statistic helps assess whether any observed difference in mean shoe sizes is statistically significant.

(d) The decision to reject or fail to reject the null hypothesis is made by comparing the test statistic to the critical value. If the test statistic falls in the rejection region (i.e., it is less than the critical value), the null hypothesis is rejected; otherwise, it is not.

(e) The conclusion is drawn based on the comparison of the test statistic and critical value. If the null hypothesis is rejected, it implies there is evidence to suggest a significant difference in mean shoe sizes between women born in 1980 and 1960. If the null hypothesis is not rejected, it indicates insufficient evidence to support such a difference.