Question

you are given the sample mean and the population standard deviation. use this information to construct the 90% and 95% confidence intervals for the population mean. interpret the results and compare the widths of the confidence intervals.
from a random sample of 35 months from january 2006 through december 2020, the mean number of tornadoes per month in the united states was about 93. assume the population standard deviation is 100.
the 90% confidence interval is ( , ).
(round to two decimal places as needed )

Explanation:

Step1: Identify the formula for confidence interval

For a confidence interval for the population mean when the population standard deviation \(\sigma\) is known, the formula is \(\bar{x} \pm z_{\alpha/2} \frac{\sigma}{\sqrt{n}}\), where \(\bar{x}\) is the sample mean, \(z_{\alpha/2}\) is the z - critical value, \(\sigma\) is the population standard deviation, and \(n\) is the sample size.

We are given that \(\bar{x}=93\), \(\sigma = 100\), \(n = 35\), and we want a 90% confidence interval. For a 90% confidence interval, \(\alpha=1 - 0.90=0.10\), so \(\alpha/2=0.05\). The \(z\) - critical value \(z_{0.05}\) (from the standard normal distribution table) is approximately 1.645.

Step2: Calculate the margin of error

The margin of error \(E=z_{\alpha/2}\frac{\sigma}{\sqrt{n}}\). Substitute the values: \(z_{\alpha/2} = 1.645\), \(\sigma=100\), \(n = 35\). First, calculate \(\sqrt{n}=\sqrt{35}\approx5.916\). Then \(\frac{\sigma}{\sqrt{n}}=\frac{100}{5.916}\approx16.90\). Then \(E = 1.645\times16.90\approx27.80\)

Step3: Calculate the confidence interval

The lower limit is \(\bar{x}-E=93 - 27.80 = 65.20\) and the upper limit is \(\bar{x}+E=93 + 27.80=120.80\) (We can also calculate it more precisely:

First, \(\frac{\sigma}{\sqrt{n}}=\frac{100}{\sqrt{35}}\approx\frac{100}{5.91608}\approx16.903\)

Then \(z_{\alpha/2}\frac{\sigma}{\sqrt{n}}=1.645\times16.903\approx1.645\times16.903\approx27.80\)

So the confidence interval is \(93\pm27.80\), which is \((65.20,120.80)\) (rounded to two decimal places)

Answer:

\((65.20, 120.80)\)