In the year $2012,12 \%$ of the residents of the US moved to a different residence. Suppose that several census workers took random samples of $n=150$ residents from the population and computed the proportion $\hat{p}$ of residents in each sample who had moved in 2012.

What would be the shape of the sampling distribution of $\hat{p}$ ?

Choose 1 answer:
(A) Skewed to the left
B) Skewed to the right
(C) Approximately normal
D) Uniform

Question

In the year $2012,12 \%$ of the residents of the US moved to a different residence. Suppose that several census workers took random samples of $n=150$ residents from the population and computed the proportion $\hat{p}$ of residents in each sample who had moved in 2012.

What would be the shape of the sampling distribution of $\hat{p}$ ?

Choose 1 answer:
(A) Skewed to the left
B) Skewed to the right
(C) Approximately normal
D) Uniform

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ To determine the shape of the sampling distribution, we need to consider the Central Limit Theorem (CLT). ‖ ‖ step 2 ⋮ The CLT states that for a sufficiently large sample size, the sampling distribution of the sample proportion will be approximately normal. ‖ ‖ step 3 ⋮ Given that the number of trials $n = 150$ and the probability of success $p = 0.12$, we can consider the sample size to be large enough for the CLT to apply. ‖ ‖ step 4 ⋮ The mean of the sampling distribution is given by $\mu = np = 150 imes 0.12 = 18$. ‖ ‖ step 5 ⋮ The standard deviation of the sampling distribution is given by $\sigma = \sqrt{np(1-p)} = \sqrt{150 imes 0.12 imes (1 - 0.12)} \approx 3.97995$. ‖ ‖ step 6 ⋮ Since the sample size is large ($n = 150$), the sampling distribution of $\hat{p}$ will be approximately normal according to the CLT. ‖ ***C*** ∻Key Concept∻ ⚹Central Limit Theorem⚹ ∻Explanation∻ ⚹The Central Limit Theorem states that the sampling distribution of the sample mean (or proportion) will be approximately normal if the sample size is large enough, regardless of the population's distribution. In this case, with $n = 150$, the sampling distribution of $\hat{p}$ is approximately normal.⚹◊What conditions need to be met for the sampling distribution of $\hat{p}$ to be approximately normal?⍭ Generate me a similar question◊

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