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The following are the ratings of men by women in an experiment involving speed dating. Use the given data to construct a boxplot and identify the​ 5-number summary.

2.0
3.0
3.5
4.5
5.5
5.5
6.5
6.5
6.5
6.5
6.5
7.5
8.5
9.5
9.5
9.5
10.0
10.0
10.0
10.0

Question content area bottom
Part 1
The​ 5-number summary is
  
enter your response here​,
  
enter your response here​,
  
enter your response here​,
  
enter your response here​, and
  
enter your response here.
​(Use ascending order. Type integers or decimals. Do not​ round.)
Part 2
Which of the following boxplots best represents the​ data?
A.
0
2
4
6
8
10
Ratings

A boxplot titled Ratings is plotted above a horizontal scale from 0 to 10 in increments of 1. The boxplot consists of a box that extends from 4.5 to 9.75, a vertical line segment drawn through the box at 6.5, and a horizontal line segment extending from 2 to 10 that bisects the box. All values are approximate.
B.
0
2
4
6
8
10
Ratings

A boxplot titled Ratings is plotted above a horizontal scale from 0 to 10 in increments of 1. The boxplot consists of a box that extends from 4.5 to 8.5, a vertical line segment drawn through the box at 5.5, and a horizontal line segment extending from 1 to 10 that bisects the box. All values are approximate.
C.
0
2
4
6
8
10
Ratings

A boxplot titled Ratings is plotted above a horizontal scale from 0 to 10 in increments of 1. The boxplot consists of a box that extends from 5.5 to 9.5, a vertical line segment drawn through the box at 6.5, and a horizontal line segment extending from 2 to 10 that bisects the box. All values are approximate.
D.
0
2
4
6
8
10
Ratings

1 answer

To construct the boxplot and identify the 5-number summary for the given speed dating ratings, we will first determine the five numbers: minimum, first quartile (Q1), median (Q2), third quartile (Q3), and maximum.

Given Data:

2.0, 3.0, 3.5, 4.5, 5.5, 5.5, 6.5, 6.5, 6.5, 6.5, 6.5, 7.5, 8.5, 9.5, 9.5, 9.5, 10.0, 10.0, 10.0, 10.0

Step 1: Sort the Data

The ratings are already sorted in ascending order.

Step 2: Identify the 5-number summary

  1. Minimum: The smallest value = 2.0
  2. Maximum: The largest value = 10.0
  3. Median (Q2): The middle value. For n = 20 (even), the median is the average of the 10th and 11th values.
    • 10th value = 6.5
    • 11th value = 6.5
    • Median \( (Q2) = \frac{6.5 + 6.5}{2} = 6.5 \)
  4. First Quartile (Q1): The median of the first half (first 10 values).
    • First half: 2.0, 3.0, 3.5, 4.5, 5.5, 5.5, 6.5, 6.5, 6.5, 6.5
    • Median of this set = average of the 5th and 6th values (5.5 and 5.5)
    • First Quartile \( (Q1) = \frac{5.5 + 5.5}{2} = 5.5 \)
  5. Third Quartile (Q3): The median of the second half (last 10 values).
    • Second half: 6.5, 6.5, 7.5, 8.5, 9.5, 9.5, 9.5, 10.0, 10.0, 10.0
    • Median of this set = average of the 5th and 6th values (9.5 and 9.5)
    • Third Quartile \( (Q3) = \frac{9.5 + 9.5}{2} = 9.5 \)

5-number summary:

  • Minimum: 2.0
  • Q1: 5.5
  • Median (Q2): 6.5
  • Q3: 9.5
  • Maximum: 10.0

Thus, the 5-number summary is: 2.0, 5.5, 6.5, 9.5, 10.0

Part 1

The 5-number summary is:

  • Minimum: 2.0
  • Q1: 5.5
  • Median: 6.5
  • Q3: 9.5
  • Maximum: 10.0

Part 2: Constructing the Box Plot

  • The boxplot will have:
    • The box stretching from Q1 (5.5) to Q3 (9.5)
    • The median line inside the box at 6.5
    • "Whiskers" extending from the minimum (2.0) to the maximum (10.0)

Identify the correct boxplot:

Let's analyze the options:

  • Option A: Box from 4.5 to 9.75 is incorrect; whiskers don't match.
  • Option B: Box from 4.5 to 8.5 is incorrect; whiskers don't match.
  • Option C: Box from 5.5 to 9.5, median at 6.5, and extends from 2 to 10 are correct.
  • Option D: No details available here, but if it's not option C, it's likely incorrect.

Conclusion:

The correct answer for the 5-number summary is: 2.0, 5.5, 6.5, 9.5, 10.0

The best boxplot that represents the data is Option C.