Question

From a sample of 64 motorists working in Cape Town, the sample average time that it takes them to drive to work is 36 minutes. It is known that the population standard deviation is 20 minutes. Construct a 99% confidence interval for the actual average time it takes all motorists in Cape Town to drive to work. What is the upper limit of the interval?


43.98


43.56


41.87


42.44

Answer 1

To construct a 99% confidence interval for the average time it takes all motorists in Cape Town to drive to work, we can utilize the formula for the confidence interval for a population mean when the population standard deviation is known:

\[
\text{Confidence Interval} = \bar{x} \pm z \left( \frac{\sigma}{\sqrt{n}} \right)
\]

Where:
- \(\bar{x}\) = sample mean
- \(z\) = z-score corresponding to the desired confidence level
- \(\sigma\) = population standard deviation
- \(n\) = sample size

Given:
- Sample mean (\(\bar{x}\)) = 36 minutes
- Population standard deviation (\(\sigma\)) = 20 minutes
- Sample size (\(n\)) = 64

### Step 1: Find the z-score for 99% confidence level
For a 99% confidence interval, the z-score (critical value) can be found using a z-table. The z-score for a 99% confidence level is approximately 2.576.

### Step 2: Calculate the standard error (SE)
\[
SE = \frac{\sigma}{\sqrt{n}} = \frac{20}{\sqrt{64}} = \frac{20}{8} = 2.5
\]

### Step 3: Calculate the margin of error (ME)
\[
ME = z \cdot SE = 2.576 \cdot 2.5 = 6.44
\]

### Step 4: Construct the confidence interval
The lower and upper limits of the confidence interval can now be calculated.

- Lower limit:
\[
\bar{x} - ME = 36 - 6.44 = 29.56
\]

- Upper limit:
\[
\bar{x} + ME = 36 + 6.44 = 42.44
\]

### Conclusion
The upper limit of the 99% confidence interval for the actual average time it takes all motorists in Cape Town to drive to work is **42.44**.

Therefore, the correct answer is: **42.44**.