Question

question 18 of 25
these box plots show the basketball scores for two teams.
wolverines
35 55 80 85 96
panthers
33 50 70 90 107
30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110
basketball scores
compare the shapes of the box plots.
a. the wolverines distribution is negatively skewed, but
the panthers distribution is symmetric.
b. both distributions are symmetric.
c. both distributions are negatively skewed.
d. the wolverines distribution is positively skewed, but
the panthers distribution is symmetric.

Explanation:

Step1: Recall Box Plot Skewness

For a box plot, skewness is determined by the position of the median (line inside the box) relative to the quartiles. If the median is closer to the lower quartile, the distribution is positively skewed (tail on the right). If closer to the upper quartile, negatively skewed (tail on the left). A symmetric distribution has the median in the middle of the box, and whiskers of similar length.

Step2: Analyze Wolverines' Box Plot

Wolverines' box: median (line) is closer to the upper quartile (85) than the lower quartile (55). The left whisker (from 35 to 55) is shorter, right whisker (from 85 to 96) is longer? Wait, no—wait, the box is from 55 (Q1) to 85 (Q3), median at 80? Wait, no, the plot: Wolverines have 35 (min), 55 (Q1), 80 (median?), 85 (Q3), 96 (max). Wait, the box is between 55 and 85, median line at 80? Wait, no, maybe I misread. Wait, the Wolverines' box: Q1=55, median=80, Q3=85? Then the left whisker (35 to 55) is length 20, right whisker (85 to 96) is length 11. Wait, no, median is closer to Q3 (85 - 80 = 5, 80 - 55 = 25). So median is closer to Q3, meaning left whisker (from min 35 to Q1 55) is longer (20) than right whisker (Q3 85 to max 96, length 11)? Wait, no, 55 - 35 = 20, 96 - 85 = 11. So left whisker is longer, right shorter? Wait, no, skewness: if median is closer to Q3, the distribution is negatively skewed? Wait, no—wait, skewness: positive skew is when mean (and median) is pulled to the right, so tail on right. So if the median is closer to Q1, then the right whisker is longer (tail on right), positive skew. If median closer to Q3, left whisker longer (tail on left), negative skew. Wait, Wolverines: Q1=55, median=80, Q3=85. So median - Q1 = 25, Q3 - median = 5. So median is closer to Q3, so left whisker (min to Q1: 35 to 55, length 20) is longer than right whisker (Q3 to max: 85 to 96, length 11). So tail on the left? No, wait, min is 35, which is a long left whisker? Wait, no, maybe I messed up. Wait, let's re-express:

Wolverines: min=35, Q1=55, median=80, Q3=85, max=96.

Q1 to median: 80 - 55 = 25

Median to Q3: 85 - 80 = 5

So median is closer to Q3 (since 5 < 25), so the distribution is negatively skewed? Wait, no—wait, negative skew: tail on the left (low values), so the left whisker is longer. But here, min is 35, which is a low value, left whisker from 35 to 55 (length 20), right whisker from 85 to 96 (length 11). So left whisker is longer, median closer to Q3: that's negative skew? Wait, no, maybe I got it reversed. Let's use the rule: in a box plot, if the left whisker is longer than the right, and the median is closer to the upper quartile, the distribution is negatively skewed (tail on the left). If right whisker longer and median closer to lower quartile, positively skewed (tail on the right).

Now Panthers: min=33, Q1=50, median=70, Q3=90, max=107. Box from 50 to 90, median at 70. Q1 to median: 20, median to Q3: 20. So median is in the middle of the box, and whiskers: left whisker (33 to 50, length 17), right whisker (90 to 107, length 17). So whiskers are similar length, median in the middle: symmetric.

Wait, Wolverines: Q1=55, median=80, Q3=85. Q1 to median: 25, median to Q3: 5. So median is closer to Q3 (upper quartile), left whisker (35 to 55, length 20) longer than right whisker (85 to 96, length 11). So that's negative skew? Wait, no, wait the answer options: option A says Wolverines negative, Panthers symmetric. Wait, but let's recheck.

Wait, maybe I misread the Wolverines' median. Let's look at the plot again: Wolverines have min=35, Q1=55, median (line) at 80, Q3=85, max=96. So the box is 55 (Q1) to 85 (Q3), median at 80. So the distance from Q1 to median is 80 - 55 = 25, Q3 to median is 85 - 80 = 5. So median is closer to Q3, so the left whisker (35 to 55) is longer (length 20), right whisker (85 to 96) is shorter (length 11). So the tail is on the left (since left whisker is longer, low values extend more), so that's negative skew. Panthers: Q1=50, median=70, Q3=90, min=33, max=107. Q1 to median: 20, median to Q3: 20. Whiskers: left (33 to 50, length 17), right (90 to 107, length 17). So symmetric. So option A: Wolverines negative skewed, Panthers symmetric. Wait, but let's check the options again.

Wait, option A: "The Wolverines’ distribution is negatively skewed, but the Panthers’ distribution is symmetric." Option D: "The Wolverines’ distribution is positively skewed, but the Panthers’ distribution is symmetric." Wait, maybe I messed up skewness direction. Let's use the correct rule: In a box plot, positive skewness (right skew) occurs when the median is closer to the lower quartile (Q1), so the upper whisker is longer. Negative skewness (left skew) occurs when median is closer to upper quartile (Q3), so lower whisker is longer.

Wolverines: median (80) is closer to Q3 (85) than Q1 (55) (distance 5 vs 25). So median closer to Q3: negative skew (left tail, since lower whisker is longer: from 35 to 55 is 20, upper whisker from 85 to 96 is 11). Panthers: median (70) is exactly in the middle of Q1 (50) and Q3 (90) (distance 20 each), and whiskers are similar length (left: 50 - 33 = 17, right: 107 - 90 = 17). So Panthers are symmetric. So option A is correct? Wait, but let's check the answer options again. Wait, the options:

A. Wolverines negative, Panthers symmetric.

B. Both symmetric.

C. Both negative.

D. Wolverines positive, Panthers symmetric.

Wait, maybe I made a mistake in Wolverines' median. Let's look at the box plot again: the Wolverines' box has the median line at 80? Wait, the plot shows Wolverines: 35 (min), 55 (Q1), 80 (median), 85 (Q3), 96 (max). So Q1=55, median=80, Q3=85. So Q1 to median: 25, Q3 to median: 5. So median is closer to Q3 (upper quartile), so the distribution is negatively skewed (because the left tail is longer: from 35 to 55 is a longer whisker than right from 85 to 96). Panthers: Q1=50, median=70, Q3=90, min=33, max=107. Q1 to median: 20, Q3 to median: 20. Whiskers: left 33-50 (17), right 90-107 (17). So symmetric. So option A is correct. Wait, but let's confirm with the answer options. So the correct answer is A? Wait, no—wait, maybe I mixed up positive and negative. Wait, positive skew: tail on the right (median closer to Q1, right whisker longer). Negative skew: tail on the left (median closer to Q3, left whisker longer). So Wolverines: median closer to Q3 (85), left whisker longer (35-55), so negative skew. Panthers: symmetric. So option A.

Answer:

A. The Wolverines’ distribution is negatively skewed, but the Panthers’ distribution is symmetric.