To construct a 99% confidence interval for the average time it takes all motorists in Cape Town to drive to work, we'll use the formula for the confidence interval for the population mean when the population standard deviation is known. The formula is:
\[ \bar{x} \pm z^* \left( \frac{\sigma}{\sqrt{n}} \right) \]
Where:
- \(\bar{x}\) is the sample mean.
- \(z^*\) is the critical value for the desired confidence level.
- \(\sigma\) is the population standard deviation.
- \(n\) is the sample size.
Given Data:
- Sample mean (\(\bar{x}\)) = 36 minutes
- Population standard deviation (\(\sigma\)) = 20 minutes
- Sample size (\(n\)) = 64
- Confidence level = 99%
Step 1: Calculate the standard error (SE)
\[ SE = \frac{\sigma}{\sqrt{n}} = \frac{20}{\sqrt{64}} = \frac{20}{8} = 2.5 \]
Step 2: Find the critical value (\(z^*\))
For a 99% confidence level, we need to find the critical value from the standard normal distribution. This corresponds to the 0.5% tails in each tail of the distribution (1% total in both tails).
Using the Z-table, the critical value for 99% confidence (\(z^*\)) is approximately 2.576.
Step 3: Calculate the confidence interval
Using the formula:
\[ \text{Confidence Interval} = \bar{x} \pm z^* \cdot SE \]
\[ = 36 \pm 2.576 \cdot 2.5 \]
Calculating \(2.576 \cdot 2.5\):
\[ 2.576 \times 2.5 = 6.44 \]
Thus, the confidence interval is:
\[ 36 \pm 6.44 \]
This gives us the interval:
\[ (36 - 6.44, 36 + 6.44) = (29.56, 42.44) \]
Conclusion
The 99% confidence interval for the actual average time it takes all motorists in Cape Town to drive to work is (29.56 minutes, 42.44 minutes).
The critical value of the interval, \(z^*\), is approximately 2.576.