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 36 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 111.
the 90% confidence interval is ( )
(round to two decimal places as needed.)
the 95% confidence interval is ( )
(round to two decimal places as needed.)
which interval is wider?
the 90% confidence interval
the 95% confidence interval
interpret the results. choose the correct answer below.
a. you can be 90% confident that the population mean number of tornadoes is between the bounds of the 90% confidence interval, and 95% confident for the 95% interval.

Explanation:

Step1: Find \( z - \) values

For \( 90\%\) confidence level, \( z_{\alpha/2}=1.645\) (from standard normal distribution table). For \( 95\%\) confidence level, \( z_{\alpha/2}=1.96\).

Step2: Calculate margin of error for \( 90\%\) confidence interval

The formula for margin of error \( E = z_{\alpha/2}\frac{\sigma}{\sqrt{n}}\). Given \( \sigma = 111\), \( n = 36\).
\( E_{90}=1.645\times\frac{111}{\sqrt{36}}=1.645\times\frac{111}{6}=1.645\times18.5 = 30.4325\)
The confidence interval is \( \bar{x}-E<\mu<\bar{x} + E\). With \( \bar{x}=93\), \( 93 - 30.4325<\mu<93+30.4325\), \( 62.57<\mu<123.43\)

Step3: Calculate margin of error for \( 95\%\) confidence interval

\( E_{95}=1.96\times\frac{111}{\sqrt{36}}=1.96\times\frac{111}{6}=1.96\times18.5 = 36.26\)
The confidence interval is \( 93-36.26<\mu<93 + 36.26\), \( 56.74<\mu<129.26\)

Answer:

The \( 90\%\) confidence interval is \( (62.57,123.43)\)
The \( 95\%\) confidence interval is \( (56.74,129.26)\)
Which interval is wider? The \( 95\%\) confidence interval
Interpret the results: A. You can be \( 90\%\) confident that the population mean number of tornadoes is between the bounds of the \( 90\%\) confidence interval, and \( 95\%\) confident for the \( 95\%\) interval.