Question 7.2: Car Commuter Time Study From a random sample of 100 Cape Tow......

(a) Given \bar{x}=35.8 minutes, \sigma =11 minutes and n = 100 commuters.

• Find the standard error of the sample mean.

\sigma_{\bar{x} }=\frac{\sigma }{\sqrt{n} }=\frac{11}{\sqrt{100} }=1.1 min

• From the z-table, the 95% confidence level equates to z-limits of ±1.96.
• Compute the margin of error = 1.96 × 1.1 = 2.156
• Thus the lower limit is: 35.8 – 1.96(1.1) = 35.8 – 2.156
  = 33.64 minutes

and the upper limit is: 35.8 + 1.96(1.1) = 35.8 + 2.156 = 37.96 minutes.

Thus the 95% confidence interval is defined as 33.64 ≤ μ ≤ 37.96 minutes.

Management Interpretation

There is a 95% chance that a Cape Town car commuter takes, on average, between 33.64 and 37.96 minutes to travel to work daily.

(b) The measures of \bar{x} = 35.8 minutes, σ = 11 minutes, n = 100 commuters and the standard error = 1.1 minutes are all unchanged. Only the z-limits will change.

• From the z-table, the 90% confidence level equates to z-limits of ±1.645.
• Compute the margin of error = 1.645 × 1.1 = 1.81
• Thus the lower limit is: 35.8 – 1.645(1.1) = 35.8 – 1.81 = 33.99 minutes

and the upper limit is: 35.8 + 1.645(1.1) = 35.8 + 1.81 = 37.61 minutes.

Thus the 90% confidence interval is defined as 33.99 ≤ μ ≤ 37.61 minutes.

Management Interpretation

There is a 90% chance that a Cape Town car commuter takes, on average, between 33.99 and 37.61 minutes to travel to work daily.