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.How large a sample of U.S. adults is needed in order to estimate μ with a 95% confidence interval of length 2.4 miles? 111 49 196 784
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.How large a sample of U.S. adults is needed in order to estimate μ with a 95% confidence interval of length 2.4 miles? 111 49 196 784
Solution
To answer this question, we need to use the formula for the sample size in a confidence interval estimation, which is:
n = (Z*σ/E)^2
where:
- n is the sample size
- Z is the Z-score, which corresponds to the desired confidence level (for a 95% confidence level, Z = 1.96)
- σ is the population standard deviation
- E is the desired margin of error (half the length of the confidence interval)
Given in the problem, we have:
- σ = 8.4 miles
- E = 2.4 miles / 2 = 1.2 miles
Substituting these values into the formula, we get:
n = (1.96*8.4/1.2)^2
Calculating this gives us a sample size of approximately 196.
So, the answer is 196.
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