From a sample of 100 Bloemfontein car commuters, the sample mean time to commute to work daily was found to be 35.8 minutes. It is known that the population standard deviation is 20 minutes.

Find the 99% confidence interval for the actual mean time taken by all car commuters in Bloemfontein to travel to work daily.

1 answer

To find the 99% confidence interval for the mean commute time, we can use the formula for the confidence interval given by the following equation:

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

where:

  • \(\bar{x}\) is the sample mean,
  • \(z\) is the z-value for the desired confidence level,
  • \(\sigma\) is the population standard deviation, and
  • \(n\) is the sample size.

Given:

  • Sample mean (\(\bar{x}\)) = 35.8 minutes
  • Population standard deviation (\(\sigma\)) = 20 minutes
  • Sample size (\(n\)) = 100
  • Confidence level = 99%
  1. Find the z-value for 99% confidence level: The z-value that corresponds to a 99% confidence level can be found in z-tables or using statistical software. For a 99% confidence level, the critical z-value is approximately 2.576.

  2. Compute the standard error (SE): \[ SE = \frac{\sigma}{\sqrt{n}} = \frac{20}{\sqrt{100}} = \frac{20}{10} = 2 \]

  3. Calculate the margin of error (ME): \[ ME = z \cdot SE = 2.576 \cdot 2 = 5.152 \]

  4. Construct the confidence interval: \[ \text{Confidence Interval} = \bar{x} \pm ME = 35.8 \pm 5.152 \]

Calculating the lower and upper bounds:

  • Lower bound = \(35.8 - 5.152 = 30.648\)
  • Upper bound = \(35.8 + 5.152 = 40.952\)

Thus, the 99% confidence interval for the actual mean time taken by all car commuters in Bloemfontein to travel to work daily is:

\[ (30.648, 40.952) \]

In conclusion, we are 99% confident that the true mean commute time for all car commuters in Bloemfontein lies within the interval \(30.648\) to \(40.952\) minutes.