### Problem StatementA variable of two populations has a mean of 55 and a standard deviation of 24 for one of the populations and a mean of 55 and a standard deviation of 40 for the other population. Complete parts (a) through (c).**a.** For independent samples of size 16 and 25, respectively, find the mean and standard deviation of \( \overline{x}_1 - \overline{x}_2 \). (Assume that the sampling is done with replacement or that the population is large enough.)The mean of \( \overline{x}_1 - \overline{x}_2 \) is [ ].(Type an integer or a decimal. Do not round.)**b.** Must the variable under consideration be normally distributed on each of the two populations for you to answer part (a)? Explain your answer.**c.** Can you conclude that the variable \( \overline{x}_1 - \overline{x}_2 \) is normally distributed? Explain your answer.### Explanation#### Part (a): Finding the Mean and Standard DeviationTo find the mean and standard deviation of \( \overline{x}_1 - \overline{x}_2 \), we're given:- Mean and standard deviation for Population 1: μ1 = 55, σ1 = 24- Mean and standard deviation for Population 2: μ2 = 55, σ2 = 40- Sample sizes: n1 = 16, n2 = 25To find the mean \( \mu \) of the difference in sample means \( \overline{x}_1 - \overline{x}_2 \):\[ \mu_{\overline{x}_1 - \overline{x}_2} = \mu_1 - \mu_2 \]To find the standard deviation \( \sigma \) of the difference in sample means \( \overline{x}_1 - \overline{x}_2 \):\[ \sigma_{\overline{x}_1 - \overline{x}_2} = \sqrt{\left(\frac{\sigma_1^2}{n1}\right) + \left(\frac{\sigma_2^2}{n2}\right)} \]#### Part (b): Normal Distribution in Each PopulationDiscuss whether the variable needs to be normally distributed in each of the two populations to determine whether the sample means \( \overline{x}_1 \

Name: ### Problem StatementA variable of two populations has a mean of 55 a
Uploaded: 2024-03-20T07:06:16.000Z
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Description: ### Problem StatementA variable of two populations has a mean of 55 and a standard deviation of 24 for one of the populations and a mean of 55 and a standard deviation of 40 for the other population. Complete parts (a) through (c).**a.** For indepen