Question

10 multiple choice 1 point
suppose we wish to estimate the percentage of students who smoke cigarettes at each of several colleges and universities. two of the colleges are wabash college (enrollment 900) and purdue university (enrollment 36,000). what should the relative size of our samples be from each school if we want the two sampling distributions to have approximately the same standard deviation?
we should select samples that are exactly 10% of each schools enrollment.
we should select a larger number of wabash students because there are fewer of them.
we should select a larger number of purdue students because there are more of them.
we should select the same size sample from each school.

11 multiple choice 1 point
a manufacturer of potato chips claims that only 5% of their bags are under the advertised weight. a store receives a delivery of 500 bags of chips. let $hat{p}$ = the proportion of bags in a sample of 500 bags that are under the advertised weight. assuming the manufacturer is correct that only 5% of their products are defective, what are the mean and standard deviation of the sampling distribution of $hat{p}$?
the mean is 0.05 and the standard deviation is 0.00009.
the mean is 25 and the standard deviation is 4.87.
the mean is 25 and the standard deviation is 1.25.
the mean is 0.05 and the standard deviation is approximately 0.01.

Explanation:

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Question 10

Step1: Recall SE formula for proportions

The standard error (deviation) of a sample proportion is $\sqrt{\frac{p(1-p)}{n} \times \frac{N-n}{N-1}}$, where $N$ is population size, $n$ is sample size, $p$ is population proportion.

Step2: Assume large population approximation

For large $N$, $\frac{N-n}{N-1} \approx 1$, so SE $\approx \sqrt{\frac{p(1-p)}{n}}$. This depends only on $n$ and $p$, not $N$.

Step3: Match SE for both schools

Since $p$ is the same (student smoking rate), equal $n$ (sample size) gives equal SE.

Step1: Calculate mean of $\hat{p}$

The mean of the sampling distribution of $\hat{p}$ equals the population proportion $p$.
<Expression>$\mu_{\hat{p}} = p = 0.05$</Expression>

Step2: Calculate standard deviation of $\hat{p}$

Use the formula $\sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}$, where $n=500$, $p=0.05$.
<Expression>$\sigma_{\hat{p}} = \sqrt{\frac{0.05(1-0.05)}{500}} = \sqrt{\frac{0.0475}{500}} \approx \sqrt{0.000095} \approx 0.01$</Expression>

Answer:

We should select the same size sample from each school.

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Question 11