Question

1. a sample of 33 broadway shows was examined. each show was categorized as being either a musical or a play, and the run time (or total length of the show, in minutes) was recorded. this data is summarized below with both a graphical display and a table of summary statistics. which one of the following is the most accurate conclusion to draw based on this information? a. at least one musical must have had a run time equal to 142.5 minutes. b. the run times of plays are skewed to the right. c. less than half of the musicals in the sample have run times above 150 minutes. d. the majority of shows in this data set are plays. e. the run times of plays are less variable than the run times of musicals.

Explanation:

Step1: Recall quartile and median concepts

The median divides the data - set into two halves. For a set of data, 50% of the values are below the median and 50% are above it.

Step2: Analyze option C

For musicals, the median run - time is 150 minutes. By the definition of the median, less than half of the musicals in the sample have run - times above 150 minutes.

Step3: Analyze option A

The first quartile ($Q_1$) for musicals is 142.5 minutes. It does not mean that at least one musical has a run - time equal to 142.5 minutes. $Q_1$ is the value such that 25% of the data lies below it.

Step4: Analyze option B

For plays, $Q_3 - Q_2=152.5 - 115 = 37.5$ and $Q_2 - Q_1=115 - 96.5 = 18.5$. Since $Q_3 - Q_2>Q_2 - Q_1$, the data for plays is skewed to the right. But we need to find the most accurate conclusion.

Step5: Analyze option D

There is no information given about the number of musicals and plays in the sample, so we cannot conclude that the majority of shows are plays.

Step6: Analyze option E

The inter - quartile range (IQR) for musicals is $IQR_{musical}=Q_3 - Q_1=157.5 - 142.5 = 15$, and for plays is $IQR_{play}=Q_3 - Q_1=152.5 - 96.5 = 56$. The run - times of plays are more variable than those of musicals.

Answer:

C. Less than half of the musicals in the sample have run times above 150 minutes.