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?

1 answer

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:

\[ \text{Upper Limit} = 42.44 \text{ minutes} \]