Suppose that a customer service center claims that the telephone calls is receives last 112 seconds on average, with a standard deviation of 9 seconds. If you took a sample of 324 telephone calls, which of the following mean times would be within the 95% confidence interval?
A. 115.5 seconds
B. 112.5 seconds
C. 114.5 seconds
D. 113.5 seconds

Question

Suppose that a customer service center claims that the telephone calls is receives last 112 seconds on average, with a standard deviation of 9 seconds. If you took a sample of 324 telephone calls, which of the following mean times would be within the 95% confidence interval?
A. 115.5 seconds
B. 112.5 seconds
C. 114.5 seconds
D. 113.5 seconds

holsteremission · Answer

Assuming the duration of calls is normally distributed, the confidence interval takes the form $\bar x\pm Z_{\alpha/2}\dfrac{\sigma}{\sqrt n}$, where $\bar x$ is the mean, $\sigma$ is the standard deviation, $n$ is the sample size, and $Z_{\alpha/2}$ is the critical value at $(1-\alpha)\times100\%$ confidence. In other words, $\mathbb P(|Z|<Z_{\alpha/2})=95\%$, which mean $Z_{\alpha/2}\approx1.96$.

So the confidence interval is $(111.02,112.98)$.