Question

at a gym, the time people spend on the treadmill is normally distributed with a mean of 45 minutes and a standard deviation of 6 minutes. which of the following best describes a treadmill time of 54 minutes? a the time is 1.5 standard deviations above the mean. this means that this time is longer than 93.32% of the times. b the time is 1.5 standard deviations above the mean. this means that this time is longer than 6.68% of the times. c the time is 1.5 standard deviations below the mean. this means that this time is shorter than 6.68% of the times. d the time is 1.5 standard deviations below the mean. this means that this time is shorter than 93.32% of the times.

Explanation:

Step1: Recall normal - distribution properties

In a normal distribution, about 68% of the data lies within 1 standard deviation of the mean, about 95% lies within 2 standard deviations, and about 99.7% lies within 3 standard deviations. If a value is \(z = 1.5\) standard deviations above the mean, we use the properties of the standard normal distribution (\(Z - \)distribution). The cumulative - distribution function of the standard normal distribution \(\varPhi(z)\) gives the proportion of data less than \(z\). Looking up \(z = 1.5\) in the standard normal table, \(\varPhi(1.5)=0.9332\), which means that the proportion of data less than a value that is 1.5 standard deviations above the mean is 0.9332 or 93.32%. So a value that is 1.5 standard deviations above the mean is longer than 93.32% of the times.

Step2: Analyze each option

• Option A: A time that is 1.5 standard deviations above the mean is longer than 93.32% of the times, so this option is correct.
• Option B: A time 1.5 standard deviations above the mean is not longer than 6.68% of the times. In fact, it is longer than 93.32% of the times.
• Option C: A time 1.5 standard deviations below the mean has a cumulative - distribution value of \(\varPhi(- 1.5)=1 - \varPhi(1.5)=0.0668\), which means it is shorter than 93.32% of the times, not 6.68% as stated in the option.
• Option D: A time 1.5 standard deviations below the mean is shorter than 93.32% of the times, not longer.

Answer:

A. The time is 1.5 standard deviations above the mean. This means that this time is longer than 93.32% of the times.