Question

refer to the accompanying data display that results from a simple random sample of times (minutes) between eruptions of the old faithful geyser. the confidence level of 95% was used. complete parts (a) and (b) below. interval (85.74,91.76) x = 88.75 sx = 8.897431411 n = 36 a. express the confidence interval in the format that uses the \less than\ symbol. round the confidence interval limits given that the original times are all rounded to one decimal place. □ min < μ < □ min (round to two decimal places as needed.)

Explanation:

Step1: Determine the critical value

For a 95% confidence level with \(n = 36\) (degrees of freedom \(df=n - 1=35\)), using a t - distribution table or calculator, the critical value \(t_{\alpha/2}\) is approximately \(t_{0.025,35}=2.030\)

Step2: Calculate the margin of error \(E\)

The formula for the margin of error for a confidence interval for the population mean \(\mu\) (when the population standard deviation \(\sigma\) is unknown) is \(E = t_{\alpha/2}\frac{s}{\sqrt{n}}\)
We are given \(s = 8.897431411\), \(n = 36\), and \(t_{\alpha/2}=2.030\)
\(E=2.030\times\frac{8.897431411}{\sqrt{36}}=2.030\times\frac{8.897431411}{6}\)
\(E = 2.030\times1.482905235\approx3.01\)

Step3: Calculate the confidence interval

The confidence interval for the population mean \(\mu\) is \(\bar{x}-E<\mu<\bar{x} + E\)
We are given \(\bar{x}=88.75\)
\(\bar{x}-E=88.75 - 3.01=85.74\)
\(\bar{x}+E=88.75+3.01 = 91.76\)

Answer:

\(85.74\) min \(<\mu<91.76\) min