Question

the monthly average high temperatures for new york city have the following five number summary: the minimum is 40, the median is 63, the third quartile is 77, the interquartile range is 32, and the maximum is 84. which of these shows the correct box-and-whisker plot for the data? select the correct answer below: a b c d

Explanation:

Step1: Find the first quartile (Q1)

The interquartile range (IQR) is \( Q3 - Q1 \). We know \( Q3 = 77 \) and \( IQR = 32 \), so \( Q1 = Q3 - IQR = 77 - 32 = 45 \)? Wait, no, wait. Wait, the minimum is 40, median is 63, Q3 is 77, IQR is 32, maximum is 84. Wait, first, let's recall the five - number summary for a box - and - whisker plot: minimum, \( Q1 \), median (\( Q2 \)), \( Q3 \), maximum.

We know that \( IQR=Q3 - Q1 \), so \( Q1 = Q3 - IQR \). Given \( Q3 = 77 \) and \( IQR = 32 \), then \( Q1=77 - 32 = 45 \)? Wait, but the minimum is 40. Wait, maybe I made a mistake. Wait, no, the five - number summary is min, \( Q1 \), median, \( Q3 \), max. Let's check the positions:

The box in a box - and - whisker plot spans from \( Q1 \) to \( Q3 \), with a line at the median (\( Q2 \)). The whiskers extend from the minimum to \( Q1 \) and from \( Q3 \) to the maximum.

First, let's find \( Q1 \): \( IQR = Q3 - Q1\Rightarrow Q1=Q3 - IQR \). Given \( Q3 = 77 \), \( IQR = 32 \), so \( Q1 = 77-32 = 45 \). Wait, but the minimum is 40. Then the left whisker is from 40 (min) to \( Q1 = 45 \), the box is from \( Q1 = 45 \) to \( Q3 = 77 \), with the median at 63 inside the box, and the right whisker from \( Q3 = 77 \) to max = 84.

Now let's analyze the options:

• Option A: Let's check the min, \( Q1 \), median, \( Q3 \), max. The left whisker should start at 40, \( Q1 \) should be 45, median at 63, \( Q3 \) at 77, max at 84.
• Option B: The left whisker is too long (starts far left, but our min is 40, \( Q1 \) is 45, so the left whisker should be from 40 to 45, not a long whisker as in B).
• Option C: The box is too far to the right, and the left whisker is from 40 to a high \( Q1 \), which is not correct.
• Option D: The left whisker is at 40, but the box starts at 40, which would mean \( Q1 = 40 \), but we calculated \( Q1 = 45 \), so D is wrong.

Wait, maybe I miscalculated \( Q1 \). Wait, the five - number summary: min = 40, median = 63, \( Q3 = 77 \), IQR = 32, max = 84. Let's re - calculate \( Q1 \): \( IQR=Q3 - Q1\Rightarrow Q1 = Q3 - IQR=77 - 32 = 45 \). So the left whisker is from 40 (min) to \( Q1 = 45 \), the box is from 45 (\( Q1 \)) to 77 (\( Q3 \)), median at 63 inside the box, and the right whisker from 77 (\( Q3 \)) to 84 (max).

Now let's look at the options:

• Option A: The left whisker is from 40 to \( Q1 \) (around 45), the box from \( Q1 \) to \( Q3 \) (45 to 77), median at 63, right whisker to 84. This matches.
• Option B: The left whisker is too long (extends far left, but our min is 40 and \( Q1 \) is 45, so the left whisker should be short from 40 to 45).
• Option C: The box is shifted too far right, and the left whisker is too long.
• Option D: The box starts at 40, which would imply \( Q1 = 40 \), but \( Q1 = 45 \), so D is wrong.

Answer:

A