Question

use the information to construct 90% and 99% confidence intervals for the population mean. interpret the results and compare the widths of the confidence intervals. researchers surveyed a random sample of 32 u.s. workers ages 16 years and over who did not work from home and asked how long (in minutes) it takes them to get from home to work. the responses are listed. 31 45 33 43 27 25 25 36 16 45 19 36 28 35 30 13 72 19 7 56 24 51 32 71 42 29 16 40 30 27 59 83 from past studies, the researchers assume that $sigma$ is 17.7 minutes. construct the 90% and 99% confidence intervals for the population mean. which interval is wider? if convenient, use technology to construct the confidence intervals. the 90% confidence interval is (30.6, 40.9). (round to one decimal place as needed.) the 99% confidence interval is ( ). (round to one decimal place as needed.)

Explanation:

Step1: Identify the formula

For a confidence - interval of 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 - score, $\sigma$ is the population standard deviation, and $n$ is the sample size. First, calculate the sample mean $\bar{x}$.
$\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}$, where $n = 32$ and $x_{i}$ are the individual data points.
$\sum_{i=1}^{32}x_{i}=31 + 45+33 + 43+27+25+25+36+16+45+19+36+28+35+30+13+72+19+7+56+24+51+32+71+42+29+16+40+30+27+59+83=1039$
$\bar{x}=\frac{1039}{32}\approx32.5$

Step2: Find $z_{\alpha/2}$ for 99% confidence interval

For a 99% confidence interval, $\alpha=1 - 0.99 = 0.01$, so $\alpha/2=0.005$. The $z$ - score $z_{\alpha/2}=z_{0.005}$. Looking up in the standard normal distribution table, $z_{0.005} = 2.576$.

Step3: Calculate the margin of error $E$

$E = z_{\alpha/2}\frac{\sigma}{\sqrt{n}}$, with $\sigma = 17.7$, $n = 32$, and $z_{\alpha/2}=2.576$.
$E=2.576\times\frac{17.7}{\sqrt{32}}\approx2.576\times\frac{17.7}{5.657}\approx2.576\times3.13=8.06$

Step4: Calculate the confidence - interval

The confidence interval is $\bar{x}\pm E$.
Lower limit: $32.5-8.06 = 24.4$
Upper limit: $32.5 + 8.06=40.6$

Answer:

$(24.4,40.6)$