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.
tinterval
(85.74,91.76)
\\( \overline { x } = 88.75 \\)
\\( s x = 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.
85.74 min < \\( \mu \\) < 91.76 min
(round to two decimal places as needed.)
b. identify the best point estimate of \\( \mu \\) and the margin of error.
the point estimate of \\( \mu \\) is \\( \square \\) minutes.
(round to two decimal places as needed)

Explanation:

Part a

Step1: Recall confidence interval format

A confidence interval for the population mean \(\mu\) can be written as \(L < \mu < U\), where \(L\) is the lower limit and \(U\) is the upper limit.
Given \(L = 85.74\) and \(U=91.76\) from the TI - Interval \((85.74,91.76)\)

Step2: Write the interval

The confidence interval in the “less than” symbol format is \(85.74\space\text{min}<\mu < 91.76\space\text{min}\)

Part b

Step1: Point estimate of \(\mu\)

The best point estimate of the population mean \(\mu\) is the sample mean \(\bar{x}\).
Given \(\bar{x}=88.75\) from the TI - Interval output.

Step2: Margin of error \(E\)

The margin of error \(E\) for a confidence interval \((L, U)\) is given by \(E=\frac{U - L}{2}\)
Substitute \(L = 85.74\) and \(U = 91.76\)
\(E=\frac{91.76-85.74}{2}=\frac{6.02}{2}=3.01\)

Answer:

a. \(85.74\space\text{min}<\mu < 91.76\space\text{min}\)
b. The point estimate of \(\mu\) is \(88.75\) minutes. The margin of error \(E = 3.01\) minutes.