Answer to Question #107173 in Statistics and Probability for jasmine

Question

A pharmacist has recently started a new private pharmacy and wanted to estimate the average waiting time to get prescribed medication in his pharmacy. He randomly selected 40 patients for observation and recorded how many minutes each waits. The mean waiting time was 28 minutes. Assuming that the waiting time follow a normal distribution with the standard deviation of 4 minutes, estimate the mean waiting time among all patients using a 95% confidence interval. Interpret this interval.

Answer 1

We need to construct the 95% confidence interval for the population mean $\mu$

The following information is provided: $\bar{X}=28, \sigma=4, n=40.$

The critical value for $\alpha=0.05$ is $z_c=z_{1-\alpha/2}=1.96.$

The corresponding confidence interval is computed as shown below:

$$CI=(\bar{X}-z_c\times{\sigma \over \sqrt{n}},\bar{X}+z_c\times{\sigma \over \sqrt{n}})=$$

$$=(28-1.96\times{4 \over \sqrt{40}},28-1.96\times{4 \over \sqrt{40}})=$$

$$=(26.76, 29.24)$$

Therefore, based on the data provided, the $95\%$  confidence interval for the population mean is $26.76<\mu<29.24,$ which indicates that we are 95% confident that the true population mean $\mu$ is contained by the interval $(26.76, 29.24).$