Question

Math

Posted 3 months ago

```
A quality control manager takes an SRS of 500 phone batteries from factory A, where $5 \%$ of the batteries have a defect. The manager also takes an independent SRS of 250 batteries from factory $B$, where $9 \%$ of the batteries have a defect. The manager will then look at the difference $(B-A)$ between the proportions of defective batteries in each sample.
What are the mean and standard deviation of the sampling distribution of the difference in sample proportions?
Choose 1 answer:
(A) $\mu_{\hat{p}_{\mathrm{B}}-\hat{p}_{\mathrm{A}}}=0.04$
$\sigma_{\hat{p}_{\mathrm{B}}-\hat{p}_{\mathrm{A}}}=\sqrt{\frac{0.05(0.95)}{500}+\frac{0.09(0.91)}{250}}$
(B) $\mu_{\hat{p}_{\mathrm{B}}-\hat{p}_{\mathrm{A}}}=0.04$
$\sigma_{\hat{p}_{\mathrm{B}}-\hat{p}_{\mathrm{A}}}=\sqrt{\frac{0.05(0.95)}{500}-\frac{0.09(0.91)}{250}}$
(C) $\mu_{\hat{p}_{\mathrm{B}}-\hat{p}_{\mathrm{A}}}=0.07$
$\sigma_{\hat{p}_{\mathrm{B}}-\hat{p}_{\mathrm{A}}}=\sqrt{\frac{0.05(0.95)}{500}+\frac{0.09(0.91)}{250}}$
(D) $\mu_{\hat{p}_{\mathrm{B}}-\hat{p}_{\mathrm{A}}}=0.07$
$\sigma_{\hat{p}_{\mathrm{B}}-\hat{p}_{\mathrm{A}}}=\sqrt{\frac{0.05(0.95)}{500}-\frac{0.09(0.91)}{250}}$
```

Sign in to unlock the answer

Answer from Sia

Posted 3 months ago

Solution by Steps

step 1

Calculate the mean difference in sample proportions: $\mu_{\hat{p}_{B}-\hat{p}_{A}} = p_B - p_A$

step 2

Substitute the given proportions: $\mu_{\hat{p}_{B}-\hat{p}_{A}} = 0.09 - 0.05$

step 3

Compute the mean difference: $\mu_{\hat{p}_{B}-\hat{p}_{A}} = 0.04$

step 4

Calculate the standard deviation of the difference in sample proportions: $\sigma_{\hat{p}_{B}-\hat{p}_{A}} = \sqrt{\frac{p_A(1-p_A)}{n_A} + \frac{p_B(1-p_B)}{n_B}}$

step 5

Substitute the given values into the standard deviation formula: $\sigma_{\hat{p}_{B}-\hat{p}_{A}} = \sqrt{\frac{0.05 \times 0.95}{500} + \frac{0.09 \times 0.91}{250}}$

step 6

Compute the standard deviation using the Asksia-LL calculator result: $\sigma_{\hat{p}_{B}-\hat{p}_{A}} \approx 0.0205572$

[1] Answer

The mean and standard deviation of the sampling distribution of the difference in sample proportions are $\mu_{\hat{p}_{B}-\hat{p}_{A}} = 0.04$ and $\sigma_{\hat{p}_{B}-\hat{p}_{A}} \approx 0.0205572$. The correct answer is (A).

Key Concept

Sampling Distribution of the Difference in Sample Proportions

Explanation

The mean of the sampling distribution of the difference in sample proportions is the difference between the two population proportions. The standard deviation is calculated using the formula that includes the proportions and sample sizes of both populations.

Not the question you are looking for? Ask here!

Enter question by text

Enter question by image

Upgrade to Asksia Pro

Join a AskSia's Pro Plan, and get 24/7 AI tutoring for your reviews, assignments, quizzes and exam preps.

Unlimited chat query usages

Strong algorithms that better know you

Early access to new release features

Study Other Question