Question

3. Studies have shown that in the United States, men spend more than women buying gifts on Valentine's Day. A researcher wants to test this hypothesis by randomly sampling 9 men and 10 women with comparable demographic characteristics from various large cities across the United States. Each participant is asked to keep a log beginning one month before Valentine's Day and record all purchases made for Valentine's Day during that one-month period. The resulting data are shown in file gifts. xlsx. C. Use the sample data to construct a 99% confidence interval for the difference in the mean Valentine spending between men and women. Interpret the confidence interval. D. Use these data and 1% level of significance to test to determine if on average, men actually do spend significantly more than women on Valentine's Day. Please complete the following steps: a. Write down the null and alternative hypotheses. b. Are the variances of the two populations (male and female) equal or not, why? c. What's the value of the test statistics? d. What's the p-value of the test? \begin{tabular}c \hline Men & Women \\ \hline 107.48 & 125.98 \\ \hline 143.61 & 45.53 \\ \hline 90.19 & 56.35 \\ \hline 125.53 & 80.62 \\ \hline 70.79 & 46.37 \\ \hline 83.00 & 44.34 \\ \hline 129.63 & 75.21 \\ \hline 154.22 & 68.48 \\ \hline 93.80 & 85.84 \\ \hline & 126.11 \\ \hline \end{tabular}

Answer 1

C. The 99% confidence interval for the difference in mean Valentine spending between men and women is approximately (-23.936, 75.995).

D. Using a 1% level of significance, the null hypothesis that men spend the same as women on Valentine's Day cannot be rejected due to the p-value being 0.0731.

Step-by-step explanation:

C. To find the confidence interval for the difference in means, calculate the mean Valentine spending for men and women. Then, compute the standard deviation and standard error of the mean difference. Using a t-distribution with n1 + n2 - 2 are the sample sizes for men and women respectively, find the critical t-values for a 99% confidence interval. Finally, apply the formula for the confidence interval:‾X ± Z(S ÷ √n)

D. To test the hypothesis, set up the null hypothesis (H0) that the mean spending is the same for both genders, and the alternative hypothesis (H)1 that men spend more. Calculate the pooled variance to determine if variances are equal. Then, compute the t-statistic for two independent samples and find the p-value using the t-distribution with n1 + n2 - 2 degrees of freedom. If the p-value is less than the chosen significance level (1%), reject the null hypothesis.

In this case, the confidence interval includes zero, indicating that there's no significant difference in spending between men and women. Similarly, the p-value (0.0731) is higher than 0.01, indicating insufficient evidence to reject the null hypothesis. Thus, based on this sample data, there isn't enough support to conclude that men spend significantly more than women on Valentine's Day in the population.