Question

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

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

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.

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.

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.

Construct the Confidence Interval

The confidence interval can be constructed using the formula: \( \bar{X} ± (Z\space score × SE) \), 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 \( 28.5 - (1.645 × 1.082) \) and upper limit will be \( 28.5 + (1.645 × 1.082) \).

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.