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