Question

Your neighborhood gym tracked how many days each of its members worked out over the past week. Let G= the number of days per week a member worked out.

Number of Days 0 1 2 3 4 5 6 7
Probability 0.49 0.12 0.13 0.15 0.06 0.02 0.02 0.01

Calculate the median of G. Use the histogram to verify your choice of the shape of the distribution. Since the shape of the distribution for the number of days is ___, we expect the median to be ___ the mean number of days. This is confirmed when we calculate the median to be compared to the mean of 1.36 days. The shape of the histogram supports the relationship of the mean and the median because the mean is ___

Answer 1

The median of G, the number of days a gym member works out per week, is 2 days, which is less than the mean of 1.36 days. This suggests the distribution is skewed to the right.

Step-by-step explanation:

To find the median of G, we must first understand that the median is the middle value when the data is ordered numerically. Since we have probabilities for each number of days, we can calculate the median by finding the value such that the sum of the probabilities below it is closest to 0.5. Looking at the probabilities given: 0.49 (for 0 days), 0.12 (for 1 day), 0.13 (for 2 days), 0.15 (for 3 days), 0.06 (for 4 days), 0.02 (for 5 days), 0.02 (for 6 days), and 0.01 (for 7 days), we can see that adding the probabilities from 0 to 2 days gives us 0.49 + 0.12 + 0.13 = 0.74, which is already above 0.5. Hence, the median is actually at 2 days, because 0 and 1 days together give us only 0.49 + 0.12 = 0.61, which is closer to 0.5 than 0.74.

Seeing that most members work out fewer than the mean number of days (1.36), this indicates the shape of the distribution is skewed to the right. We expect the median to be less than the mean in a right-skewed distribution. Thus, the histogram would show a longer tail going towards the higher number of work out days. The mean is higher, indicating that a few members working out many days are pulling the mean upward compared to the median.