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%
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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.
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Compute the standard error (SE): \[ SE = \frac{\sigma}{\sqrt{n}} = \frac{20}{\sqrt{100}} = \frac{20}{10} = 2 \]
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Calculate the margin of error (ME): \[ ME = z \cdot SE = 2.576 \cdot 2 = 5.152 \]
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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.
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