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Chapter 9: Problem 37

Five hundred randomly selected working adults living in Calgary, Canada were asked how long, in minutes, their typical daily commute was (Calgary Herald Traffic Study, Ipsos, September 17,2005 ). The resulting sample mean and standard deviation of commute time were \(28.5\) minutes and \(24.2\) minutes, respectively. Construct and interpret a \(90 \%\) confidence interval for the mean commute time of working adult Calgary residents.

Short Answer

Expert verified
The \(90\% \) confidence interval for the mean commute time is approximately \([26.72, 30.28] \) minutes. This means we are 90% confident that the mean commute time of all working adults in Calgary falls within this interval.

Step by step solution

01

Identifying Given Values

The problem provides us with a sample size \( n=500 \), a sample mean commute time \( \bar{X}=28.5 \) minutes, a standard deviation \( s=24.2 \) minutes, and a confidence level of \(90\% \). We are tasked with finding the confidence interval for the mean commute time.
02

Find the Critical Value

Given a \(90\% \) confidence level, this implies that the level of significance \( \alpha=0.1 \). Since this is a two tailed test, we divide \( \alpha \) by 2 which yields \( \alpha/2 = 0.05 \). Looking this value up in the t-table with degrees of freedom \( df=n-1=499 \), we can find the t-value for \(90\% \) confidence level, \( t_{0.05,499} \). However, due to large sample size (>30), we can use Z score instead for simplicity. From standard Z table, we have Z score for \(90\% \) confidence interval = 1.645.
03

Compute the Standard Error

The standard error (SE) of the sample mean is the standard deviation divided by square root of the sample size, expressed as \( SE = \frac{s}{\sqrt{n}} \). This is approximately \( \frac{24.2}{\sqrt{500}} = 1.082 \) minutes.
04

Construct the Confidence Interval

The confidence interval can be constructed using the formula: \( \bar{X} 卤 (Z\space score 脳 SE) \), where \( \bar{X} \) is the sample mean, Z score is the critical value, and SE is the standard error. Thus, they can compute the lower limit and upper limit for the interval. Lower limit will be \( 28.5 - (1.645 脳 1.082) \) and upper limit will be \( 28.5 + (1.645 脳 1.082) \).
05

Interpret the Confidence Interval

Once the confidence interval is computed, we need to interpret this interval in the context of the problem. This involves explaining what the interval means in terms of the commute times of adult workers in Calgary.

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Most popular questions from this chapter

An article in the Chicago Tribune (August 29,1999 ) reported that in a poll of residents of the Chicago suburbs, \(43 \%\) felt that their financial situation had improved during the past year. The following statement is from the article: "The findings of this Tribune poll are based on interviews with 930 randomly selected suburban residents. The sample included suburban Cook County plus DuPage, Kane, Lake, McHenry, and Will Counties. In a sample of this size, one can say with \(95 \%\) certainty that results will differ by no more than 3 percent from results obtained if all residents had been included in the poll." Comment on this statement. Give a statistical argument to justify the claim that the estimate of \(43 \%\) is within \(3 \%\) of the true proportion of residents who feel that their financial situation has improved.

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Data consistent with summary quantities in the article referenced in Exercise \(9.3\) on total calorie consumption on a particular day are given for a sample of children who did not eat fast food on that day and for a sample of children who did eat fast food on that day. Assume that it is reasonable to regard these samples as representative of the population of children in the United States. No Fast Food \(\begin{array}{llllllll}2331 & 1918 & 1009 & 1730 & 1469 & 2053 & 2143 & 1981 \\ 1852 & 1777 & 1765 & 1827 & 1648 & 1506 & 2669 & \\ \text { Fast Food } & & & & & & \\ 2523 & 1758 & 934 & 2328 & 2434 & 2267 & 2526 & 1195 \\ 890 & 1511 & 875 & 2207 & 1811 & 1250 & 2117 & \end{array}\) a. Use the given information to estimate the mean calorie intake for children in the United States on a day when no fast food is consumed. b. Use the given information to estimate the mean calorie intake for children in the United States on a day when fast food is consumed. c. Use the given information to estimate the produce estimates of the standard deviations of calorie intake for days when no fast food is consumed and for days when fast food is consumed.

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