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)
x = 88.75
sx = 8.897431141
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 values are all rounded to one decimal place.
85.74 min < μ < 91.76 min
(round to two decimal places as needed.)
b. identify the best point estimate of μ and the margin of error.
the point estimate of μ is 88.75 minutes.
(round to two decimal places as needed.)
the margin of error is e = □ minutes.
(round to two decimal places as needed.)

Explanation:

Part (b) - Margin of Error Calculation

Step 1: Recall the formula for margin of error (E) in a confidence interval

The confidence interval is given by \(\bar{x} - E < \mu < \bar{x} + E\), where \(\bar{x}\) is the sample mean (point estimate) and \(E\) is the margin of error. We know the confidence interval is \((85.74, 91.76)\) and \(\bar{x} = 88.75\). We can use the upper limit of the confidence interval to find \(E\): \( \bar{x} + E=\text{Upper Limit} \).

Step 2: Solve for E

Substitute the known values: \( 88.75 + E = 91.76 \).
Subtract \( 88.75 \) from both sides: \( E = 91.76 - 88.75 \).
Calculate the difference: \( E = 3.01 \).

Answer:

The margin of error is \( E = \boldsymbol{3.01} \) minutes.