Question

motorcycle noise a recent study of 45 motorcycles found that the average noise pollution in decibels is 90. if the population standard deviation is 4.6 decibels, find the 97% confidence interval for the mean. round answers to the nearest whole decibel. round intermediate answers to two decimal places. we are 97% confident that the mean noise pollution level is between 89 and 92 decibels, based on a sample of 45 motorcycles.

Explanation:

Step1: Find the z - value

For a 97% confidence interval, the significance level \(\alpha=1 - 0.97=0.03\). Then \(\alpha/2 = 0.015\). Looking up in the standard normal table, \(z_{\alpha/2}=z_{0.015}\approx 2.17\).

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

The formula for the margin of error when the population standard deviation \(\sigma\) is known is \(E = z_{\alpha/2}\frac{\sigma}{\sqrt{n}}\). Given \(\sigma = 4.6\), \(n = 45\), and \(z_{\alpha/2}=2.17\).

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Step3: Calculate the confidence interval

The confidence interval for the population mean \(\mu\) is \(\bar{x}-E<\mu <\bar{x}+E\). Given \(\bar{x}=90\).

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Answer:

We are 97% confident that the mean noise pollution level is between \(89\) and \(91\) decibels, based on a sample of \(45\) motorcycles.