Question

the test scores from mrs. gray’s math class are shown below. 66, 71, 72, 78, 82, 85, 85, 86, 86, 89, 91, 92 (with some crossed - out numbers). a. construct a box plot to display these data. a number line with marks at 65, 70, 75, 80, 85, 90, 95, 100 is shown. the data set 20, 36, 52, 56, 24, 16, 40, 4, 28 represents the number of books purchased by nine book club members in a year. construct a box plot for these data on the number line below. a number line with marks at 0, 20, 40, 60 is shown.

Answer:

For Mrs. Gray's math class test scores:

Step 1: Organize the data

First, we need to have the complete and ordered data. Let's list out the scores clearly. From the given, the scores are: 66, 71, 72, 78, 82, 85, 85, 86, 86, 89, 89, 91, 92. Wait, let's count the number of data points. Let's re - list them properly. Let's assume the crossed - out numbers are maybe errors or duplicates, but the visible scores are: 66, 71, 72, 78, 82, 85, 85, 86, 86, 89, 89, 91, 92. Wait, actually, let's check the count. Let's list all the scores as per the problem: 66, 71, 72, 78, 82, 85, 85, 86, 86, 89, 89, 91, 92. Wait, maybe I miscounted. Let's see: the first row has 66, 71, 72, 78 (4 numbers), then 82, 85 (2), then 85, 86, 86 (3), then 89, 91, 92 (3). Wait, 4 + 2+3 + 3=12? Wait, maybe the original data is: 66, 71, 72, 78, 82, 85, 85, 86, 86, 89, 89, 91, 92? Wait, no, maybe the correct ordered data is: Let's sort them: 66, 71, 72, 78, 82, 85, 85, 86, 86, 89, 89, 91, 92. Wait, the number of data points \(n = 13\) (odd).

Step 2: Find the median (Q2)

The median is the middle value. For \(n = 13\), the position of the median is \(\frac{n + 1}{2}=\frac{13+1}{2}=7\)th term. The 7th term in the ordered data (66, 71, 72, 78, 82, 85, 85, 86, 86, 89, 89, 91, 92) is 85.

Step 3: Find the lower quartile (Q1)

The lower half of the data is the first 6 terms: 66, 71, 72, 78, 82, 85. The median of the lower half (Q1) is the average of the 3rd and 4th terms. The 3rd term is 72 and the 4th term is 78. So \(Q1=\frac{72 + 78}{2}=\frac{150}{2}=75\).

Step 4: Find the upper quartile (Q3)

The upper half of the data is the last 6 terms: 86, 86, 89, 89, 91, 92. The median of the upper half (Q3) is the average of the 3rd and 4th terms. The 3rd term is 89 and the 4th term is 89. So \(Q3=\frac{89+89}{2}=89\).

Step 5: Find the minimum and maximum

The minimum value is 66 and the maximum value is 92.

Step 6: Construct the box - plot

• Draw a number line from 65 to 100 (as given).
• Plot the minimum (66), Q1 (75), median (85), Q3 (89), and maximum (92).
• Draw a box from Q1 to Q3, draw a vertical line inside the box at the median, and draw whiskers from the box to the minimum and maximum.

For the book club data (20, 36, 52, 56, 24, 16, 40, 4, 28):

Step 1: Order the data

First, we sort the data set: 4, 16, 20, 24, 28, 36, 40, 52, 56. The number of data points \(n = 9\) (odd).

Step 2: Find the median (Q2)

The position of the median is \(\frac{n + 1}{2}=\frac{9 + 1}{2}=5\)th term. The 5th term in the ordered data (4, 16, 20, 24, 28, 36, 40, 52, 56) is 28.

Step 3: Find the lower quartile (Q1)

The lower half of the data is the first 4 terms: 4, 16, 20, 24. The median of the lower half (Q1) is the average of the 2nd and 3rd terms. The 2nd term is 16 and the 3rd term is 20. So \(Q1=\frac{16+20}{2}=\frac{36}{2}=18\).

Step 4: Find the upper quartile (Q3)

The upper half of the data is the last 4 terms: 36, 40, 52, 56. The median of the upper half (Q3) is the average of the 2nd and 3rd terms. The 2nd term is 40 and the 3rd term is 52. So \(Q3=\frac{40 + 52}{2}=\frac{92}{2}=46\).

Step 5: Find the minimum and maximum

The minimum value is 4 and the maximum value is 56.

Step 6: Construct the box - plot

• Draw a number line from 0 to 60 (as given).
• Plot the minimum (4), Q1 (18), median (28), Q3 (46), and maximum (56).
• Draw a box from Q1 to Q3, draw a vertical line inside the box at the median, and draw whiskers from the box to the minimum and maximum.

Final Answer (Box - Plot Construction Summary)

For Mrs. Gray's math class:

• Min: 66, Q1: 75, Median: 85, Q3: 89, Max: 92. Plot on the number line 65 - 100.

For the book club:

• Min: 4, Q1: 18, Median: 28, Q3: 46, Max: 56. Plot on the number line 0 - 60.

(Note: To actually draw the box - plots, you would mark the points on the respective number lines, draw the box between Q1 and Q3, the line at the median, and the whiskers to min and max.)