A study was conducted to estimate μ, the mean commute distance that all employed U.S. adults travel to work. Suppose a random sample of 49 employed U.S. adults gives a mean commute distance of 22 miles and that from prior studies, the population standard deviation is assumed to be σ = 8.4 miles.We are 95% confident that the mean commute distance to work of all employed U.S. adults falls between which of the following intervals? 5.2 to 38.8 19.6 to 24.4 20.8 to 23.2 18.4 to 25.6
Question
A study was conducted to estimate μ, the mean commute distance that all employed U.S. adults travel to work. Suppose a random sample of 49 employed U.S. adults gives a mean commute distance of 22 miles and that from prior studies, the population standard deviation is assumed to be σ = 8.4 miles.We are 95% confident that the mean commute distance to work of all employed U.S. adults falls between which of the following intervals? 5.2 to 38.8 19.6 to 24.4 20.8 to 23.2 18.4 to 25.6
Solution
To answer this question, we need to calculate the 95% confidence interval for the mean commute distance. The formula for a confidence interval is:
X̄ ± Z * (σ/√n)
where:
- X̄ is the sample mean
- Z is the Z-score (for a 95% confidence interval, the Z-score is approximately 1.96)
- σ is the population standard deviation
- n is the sample size
Substituting the given values into the formula, we get:
22 ± 1.96 * (8.4/√49)
This simplifies to:
22 ± 1.96 * (8.4/7)
Which further simplifies to:
22 ± 1.96 * 1.2
Finally, we get:
22 ± 2.352
So, the 95% confidence interval is from 19.648 to 24.352.
Therefore, the correct answer is 19.6 to 24.4.
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