Question

Math

Posted 6 months ago

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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
```

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Answer from Sia

Posted 6 months ago

Solution by Steps

step 2

The sample size given is $n=150$. The population proportion of US residents who moved is $p=0.12$

step 3

According to the Central Limit Theorem, the sampling distribution of the sample proportion $\hat{p}$ will be approximately normal if the sample size is large enough

step 4

The rule of thumb for the sample size being large enough is that both $np$ and $n(1-p)$ should be greater than 10

step 5

Calculating these values: $np = 150 \times 0.12 = 18$ and $n(1-p) = 150 \times (1-0.12) = 150 \times 0.88 = 132$

step 6

Since both $np$ and $n(1-p)$ are greater than 10, the sampling distribution of $\hat{p}$ will be approximately normal

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, typically if both $np$ and $n(1-p)$ are greater than 10.

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