A company has offices in two different countries. Suppose that the average age (in years) of the employees at location $A$ is 40.1 with a standard deviation of 5.5 , and the average at location $B$ is 36.7 with a standard deviation of 6.3 .

Every month, the company takes separate random samples of 50 employees from each location for a survey. Each time, they look at the difference in the mean age sampled from each location $\left(\bar{x}_{\mathrm{A}}-\bar{x}_{\mathrm{B}}\right)$.

What do we know about the shape of the sampling distribution of $\bar{x}_{\mathrm{A}}-\bar{x}_{\mathrm{B}}$, and why?

Choose 1 answer:
(A) It's exactly normal, because both populations are normally distributed.
(B) It's approximately normal, because both sample sizes are at least 30.
CC) It's not normal, because the populations aren't normal, and both sample sizes are less than 30.
(D) The shape cannot be determined since we don't know the shape of either population distribution.

Question

A company has offices in two different countries. Suppose that the average age (in years) of the employees at location $A$ is 40.1 with a standard deviation of 5.5 , and the average at location $B$ is 36.7 with a standard deviation of 6.3 .

Every month, the company takes separate random samples of 50 employees from each location for a survey. Each time, they look at the difference in the mean age sampled from each location $\left(\bar{x}_{\mathrm{A}}-\bar{x}_{\mathrm{B}}\right)$.

What do we know about the shape of the sampling distribution of $\bar{x}_{\mathrm{A}}-\bar{x}_{\mathrm{B}}$, and why?

Choose 1 answer:
(A) It's exactly normal, because both populations are normally distributed.
(B) It's approximately normal, because both sample sizes are at least 30.
CC) It's not normal, because the populations aren't normal, and both sample sizes are less than 30.
(D) The shape cannot be determined since we don't know the shape of either population distribution.

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ To determine the shape of the sampling distribution of $\bar{x}_{\mathrm{A}}-\bar{x}_{\mathrm{B}}$, we consider the Central Limit Theorem (CLT). ‖ ‖ step 2 ⋮ The CLT states that the sampling distribution of the sample mean will be approximately normal if the sample size is large enough, typically n ≥ 30. ‖ ‖ step 3 ⋮ Given that both samples from locations A and B are of size 50, which is greater than 30, the CLT applies to both $\bar{x}_{\mathrm{A}}$ and $\bar{x}_{\mathrm{B}}$ individually. ‖ ‖ step 4 ⋮ Since the sampling distributions of both $\bar{x}_{\mathrm{A}}$ and $\bar{x}_{\mathrm{B}}$ are approximately normal, the distribution of their difference $\bar{x}_{\mathrm{A}}-\bar{x}_{\mathrm{B}}$ is also approximately normal. ‖ ***B*** ∻Key Concept∻ ⚹Central Limit Theorem⚹ ∻Explanation∻ ⚹The Central Limit Theorem states that the sampling distribution of the sample mean will be approximately normal if the sample size is large enough (n ≥ 30), regardless of the population distribution's shape.⚹◊What is the Central Limit Theorem and how does it apply to the sampling distribution of the difference in mean ages between employees at two office locations?⍭ Generate me a similar question◊

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