To construct a 99% confidence interval for the actual average time it takes all motorists in Cape Town to drive to work, we can use the formula for a confidence interval for the mean when the population standard deviation is known. The formula is given by:
\[ \text{CI} = \bar{x} \pm z \left( \frac{\sigma}{\sqrt{n}} \right) \]
where:
- \(\bar{x}\) = sample mean
- \(z\) = z-score corresponding to the confidence level
- \(\sigma\) = population standard deviation
- \(n\) = sample size
Given:
- \(\bar{x} = 36\) minutes (sample mean)
- \(\sigma = 20\) minutes (population standard deviation)
- \(n = 64\) (sample size)
Step 1: Find the z-score for a 99% confidence level
For a 99% confidence level, the z-score (two-tailed) can be found from the standard normal distribution table. The corresponding z-score for a 99% confidence level is approximately:
\[ z = 2.576 \]
Step 2: Calculate the standard error (SE)
The standard error (SE) is calculated as:
\[ SE = \frac{\sigma}{\sqrt{n}} = \frac{20}{\sqrt{64}} = \frac{20}{8} = 2.5 \]
Step 3: Construct the confidence interval
Now, substituting the values into the confidence interval formula:
\[ \text{CI} = 36 \pm 2.576 \times 2.5 \]
Calculating the margin of error:
\[ \text{Margin of Error} = 2.576 \times 2.5 = 6.44 \]
Now, we can find the confidence interval:
\[ \text{Lower Limit} = 36 - 6.44 = 29.56 \] \[ \text{Upper Limit} = 36 + 6.44 = 42.44 \]
Conclusion
The 99% confidence interval for the average time it takes all motorists in Cape Town to drive to work is:
\[ (29.56, 42.44) \]
Thus, the upper limit of the interval is:
\[ \text{Upper Limit} = 42.44 \text{ minutes} \]
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