Question

Suppose Laura, a facilities manager at a health and wellness company, wants to estimate the difference in the average amount of time that men and women spend at the company's fitness centers each week.

Laura randomly selects 14 adult male fitness center members from the membership database and then selects 14 adult female members from the database. Laura gathers data from the past month containing logged time at the fitness center for these members. She plans to use the data to estimate the difference in the time men and women spend per week at the fitness center. The sample statistics are summarized in the table.

Population description male female
Population mean (unknown)
Sample size n = 14 n = 14
Sample mean (min) X1 = 120.1 X2 = 104.5
Sample standard deviation (min) 40.3 25.9
Population M2 df = 22.174

The population standard deviations are unknown and unlikely to be equal, based on the sample data. Laura plans to use the two-sample t-procedures to estimate the difference of the two population means, μ1 - μ2.

First, compute the estimated standard error, SE, of the difference in the sample means that Laura uses to construct the confidence interval. Provide your answer precise to at least three decimal places.
SE = 12.803 min

Next, compute the margin of error, m, for the 95% confidence interval for the difference of the population means using software or a table of t-distributions. If you are using software, you may find some software manuals helpful. Provide your answer precise to at least one decimal place.

Answer 1

The estimated standard error (SE) of the difference in the sample means is SE = 12.803 min. The margin of error (m) for the 95% confidence interval for the difference of the population means is approximately m = 21.6 min.

Step-by-step explanation:

To calculate the estimated standard error (SE), Laura can use the formula:

SE = sqrt((s₁²/n₁) + (s₂²/n₂))

where s₁ and s₂ are the sample standard deviations, and n₁ and n₂ are the sample sizes. Plugging in the given values:

SE = sqrt((40.3²/14) + (25.9²/14))

SE ≈ 12.803 min (rounded to three decimal places).

To calculate the margin of error (m) for the 95% confidence interval, Laura can use the t-distribution with degrees of freedom given by the formula:

m = t * SE

where t is the critical value for a 95% confidence interval with the appropriate degrees of freedom. The value of t can be obtained from a t-distribution table or statistical software. Laura finds that t ≈ 2.145 for df = 22.174, giving:

m ≈ 2.145 * 12.803

m ≈ 21.6 min (rounded to one decimal place).