From a sample of 100 Bloemfontein car commuters, the sample mean time to commute to work daily was found to be 35.8 minutes. It is known that the population standard deviation is 20 minutes.

Find the 90% confidence interval for the actual mean time taken by all car commuters in Bloemfontein to travel to work daily.

1 answer

To find the 90% confidence interval for the actual mean time taken by all car commuters in Bloemfontein to travel to work daily, we can use the formula for the confidence interval for a population mean when the population standard deviation is known:

\[ \text{CI} = \bar{x} \pm Z \left(\frac{\sigma}{\sqrt{n}}\right) \]

where:

  • \(\bar{x}\) is the sample mean (35.8 minutes),
  • \(Z\) is the Z-score corresponding to the desired confidence level (for 90%, \(Z \approx 1.645\)),
  • \(\sigma\) is the population standard deviation (20 minutes),
  • \(n\) is the sample size (100).

Step 1: Calculate the standard error (SE)

\[ SE = \frac{\sigma}{\sqrt{n}} = \frac{20}{\sqrt{100}} = \frac{20}{10} = 2 \]

Step 2: Calculate the margin of error (ME)

\[ ME = Z \cdot SE = 1.645 \cdot 2 = 3.29 \]

Step 3: Calculate the confidence interval

\[ \text{Lower limit} = \bar{x} - ME = 35.8 - 3.29 = 32.51 \] \[ \text{Upper limit} = \bar{x} + ME = 35.8 + 3.29 = 39.09 \]

Conclusion

The 90% confidence interval for the actual mean time taken by all car commuters in Bloemfontein to travel to work daily is:

\[ (32.51 , \text{minutes}, 39.09 , \text{minutes}) \]