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. 32 46 34 46 24 28 28 40 18 42 16 38 28 37 34 14 71 16 5 57 25 53 32 71 41 32 18 44 35 26 60 85 from past studies, the researchers assume that σ is 18.0 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 (31.5, 42.0). (round to one decimal place as needed.) the 99% confidence interval is ( ) (round to one decimal place as needed.)

Explanation:

Step1: Calculate sample mean

First, find the sum of all data values. Then divide by the sample size \(n = 32\).
Let \(x_1,x_2,\cdots,x_{32}\) be the data values. \(\bar{x}=\frac{\sum_{i = 1}^{32}x_i}{32}\). After summing the data ( \(32+46 + 34+46+24+28+28+40+18+42+16+38+28+37+34+14+71+16+5+57+25+53+32+71+41+32+18+44+35+26+60+85\)) \(\sum_{i=1}^{32}x_i = 1176\), so \(\bar{x}=\frac{1176}{32}=36.75\)

Step2: Find critical values

For a \(90\%\) confidence interval, \(\alpha=1 - 0.90=0.10\), and \(\alpha/2=0.05\). The critical value \(z_{\alpha/2}=z_{0.05}\). From the standard normal table, \(z_{0.05} = 1.645\)
For a \(99\%\) confidence interval, \(\alpha=1 - 0.99=0.01\), and \(\alpha/2=0.005\). The critical value \(z_{\alpha/2}=z_{0.005}\). From the standard normal table, \(z_{0.005}=2.576\)

Step3: Calculate margin of error

The formula for the margin of error \(E = z_{\alpha/2}\frac{\sigma}{\sqrt{n}}\), where \(\sigma = 18\) and \(n = 32\)
For \(90\%\) confidence: \(E_{90}=1.645\times\frac{18}{\sqrt{32}}\approx1.645\times3.182\approx5.24\)
For \(99\%\) confidence: \(E_{99}=2.576\times\frac{18}{\sqrt{32}}\approx2.576\times3.182\approx8.19\)

Step4: Construct confidence intervals

The confidence interval formula is \(\bar{x}-E<\mu<\bar{x} + E\)
For \(90\%\) confidence: \(36.75-5.24<\mu<36.75 + 5.24\), \(31.5<\mu<42.0\)
For \(99\%\) confidence: \(36.75-8.19<\mu<36.75+8.19\), \(28.6<\mu<44.9\)

Answer:

The \(99\%\) confidence interval \((28.6,44.9)\) is wider.