Question

on-time shipping a large mail-order company advertises that it ships 90% of its orders within 3 working days. you select an srs of 100 of the orders received in the past week for an audit. let \\(\hat{p}\\) = the proportion of orders in the sample that were shipped within 3 working days. calculate the mean and the standard deviation of the sampling distribution of \\(\hat{p}\\). interpret the standard deviation from part (a).

Explanation:

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

The mean of the sampling distribution of the sample proportion $\hat{p}$ equals the population proportion $p$.
Given $p = 0.90$, so $\mu_{\hat{p}} = p = 0.90$

Step2: Check conditions for standard deviation

First, verify $np \geq 10$ and $n(1-p) \geq 10$:
$np = 100 \times 0.90 = 90 \geq 10$
$n(1-p) = 100 \times 0.10 = 10 \geq 10$
Also, assume the population is at least 10 times the sample size (10×100=1000, reasonable for a large company).

Step3: Calculate standard deviation

Use the formula for standard deviation of $\hat{p}$:
$$\sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}$$
Substitute $p=0.90$, $n=100$:
$$\sigma_{\hat{p}} = \sqrt{\frac{0.90 \times (1-0.90)}{100}} = \sqrt{\frac{0.90 \times 0.10}{100}} = \sqrt{\frac{0.09}{100}} = \sqrt{0.0009} = 0.03$$

Step4: Interpret the standard deviation

The standard deviation measures the typical variability of the sample proportion from the true population proportion.

Answer:

1. Mean of the sampling distribution of $\hat{p}$: $\mu_{\hat{p}} = 0.90$
2. Standard deviation of the sampling distribution of $\hat{p}$: $\sigma_{\hat{p}} = 0.03$
3. Interpretation: For simple random samples of 100 orders, the sample proportion of orders shipped within 3 working days typically varies by about 0.03 (3 percentage points) from the true population proportion of 0.90.