Question 20 of 31 Step 4 of 4

Noise levels at 4 volcanoes were measured in decibels yielding the following data:
\[
152,157,157,158
\]

Construct the $90 \%$ confidence interval for the mean noise level at such locations. Assume the population is approximately normal.
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Step 4 of 4 : Construct the $90 \%$ confidence interval. Round your answer to one decimal place.
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Lower endpoint:
Upper endpoint:

Step 1 ：First, we calculate the sample mean of the noise levels. The noise levels are \(152, 157, 157, 158\). The sample mean is calculated as the sum of the noise levels divided by the number of noise levels, which gives us a mean of \(156.0\).

Step 2 ：Next, we calculate the standard error of the mean. The standard error of the mean is calculated as the standard deviation divided by the square root of the sample size. The standard error of the mean is \(1.35400640077266\).

Step 3 ：We then use the z-score for a 90% confidence interval. The z-score for a 90% confidence interval is \(1.6448536269514722\) for a one-tailed test.

Step 4 ：We calculate the margin of error by multiplying the z-score by the standard error of the mean. The margin of error is \(2.2271423392264182\).

Step 5 ：Finally, we construct the 90% confidence interval for the mean noise level. The confidence interval is the sample mean plus and minus the margin of error. The 90% confidence interval for the mean noise level at such locations is \(\boxed{[153.8, 158.2]}\).