An inspector inspects a shipment of medications to determine the efficacy in terms of the proportion p in the shipment that failed to retain full potency after 60 days of production. Unless there is clear evidence that this proportion is less than 0.05, she will reject the shipment. To reach a decision, she will test the following hypotheses using the large-sample test for a population proportion: H₀: p = 0.05, H_a: p

Question

An inspector inspects a shipment of medications to determine the efficacy in terms of the proportion p in the shipment that failed to retain full potency after 60 days of production. Unless there is clear evidence that this proportion is less than 0.05, she will reject the shipment. To reach a decision, she will test the following hypotheses using the large-sample test for a population proportion:
H₀: p = 0.05, H_a: p < 0.05
To do so, she selects a simple random sample of 200 pills. Suppose that 8 of the pills have failed to retain their full potency. Using appropriate technology, what is a 95% plus four confidence interval for the true proportion of pills in the shipment that have failed to retain full potency based on these data?

Quizplus · Accepted Answer

We use the formula for the confidence interval for a population proportion:

$\hat{p} \pm z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$

where $\hat{p}$ is the sample proportion, $n$ is the sample size, and $z_{\alpha/2}$ is the appropriate z-score from the standard normal distribution.

In this case, $\hat{p} = \frac{8}{200} = 0.04$ and $n = 200$. Since we want a 95% confidence interval, $\alpha = 0.05$ and $\alpha/2 = 0.025$, and we need to find the z-score such that the area to the right is 0.025. Using a standard normal table or calculator, we find $z_{0.025} = 1.96$.

Plugging in the values, we get:

$0.04 \pm 1.96 \sqrt{\frac{0.04(0.96)}{200}}$

$= 0.04 \pm 0.0296$

So the 95% confidence interval is $0.049 \pm 0.0296$, which gives us the range (0.0194, 0.0786). Therefore, we can be 95% confident that the true proportion of pills in the shipment that have failed to retain full potency is between 0.0194 and 0.0786. The best choice is C.

An Inspector Inspects a Shipment of Medications to Determine the Efficacy