Question

a random sample of 40 adults with no children under the age of 18 years results in a mean daily leisure time of 5 47 hours, with a standard deviation of 2 33 hours. a random sample of 40 adults with children under the age of 18 results in a mean daily leisure time of 4 06 hours, with a standard deviation of 1 82 hours. construct and interpret a 90% confidence interval for the mean difference in leisure time between adults with no children and adults with children (μ₁ - μ₂)
let μ₁ represent the mean leisure hours of adults with no children under the age of 18 and μ₂ represent the mean leisure hours of adults with children under the age of 18.
the 90% confidence interval for (μ₁ - μ₂) is the range from 0.64 hours to 2.18 hours
(round to two decimal places as needed.)
what is the interpretation of this confidence interval?
○ a. there is 90% confidence that the difference of the means is in the interval. conclude that there is a significant difference in the number of leisure hours.
○ b. there is a 90% probability that the difference of the means is in the interval. conclude that there is a significant difference in the number of leisure hours.
○ c. there is 90% confidence that the difference of the means is in the interval. conclude that there is insufficient evidence of a significant difference in the number of leisure hours.
○ d. there is a 90% probability that the difference of the means is in the interval. conclude that there is insufficient evidence of a significant difference in the number of leisure hours.

Explanation:

A confidence interval gives a range of values within which the true population parameter is likely to lie. A 90% confidence interval means that if we were to take many samples and construct confidence intervals in the same way, about 90% of those intervals would contain the true difference in population means. Since the given confidence interval (0.64 to 2.18) does not contain 0, it indicates a significant difference. The term "confidence" is used (not "probability" for a specific interval as the interval either contains the true value or not - it's a frequentist interpretation).

Answer:

A. There is 90% confidence that the difference of the means is in the interval. Conclude that there is a significant difference in the number of leisure hours.