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: Identify population proportion

The population proportion $p$ of on-time shipments is given as 90%, so $p = 0.90$.

Step2: Calculate mean of $\hat{p}$

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

Step3: Calculate standard deviation of $\hat{p}$

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

Step4: Interpret the standard deviation

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

Answer:

1. Mean of the sampling distribution of $\hat{p}$: $0.90$
2. Standard deviation of the sampling distribution of $\hat{p}$: $0.03$
3. Interpretation: For simple random samples of 100 orders, the sample proportion of on-time shipments typically varies by about 0.03 (or 3 percentage points) from the true population proportion of 0.90.