Question

topic: examining data distributions in a box-and-whisker plot.

1. make a box-and-whisker plot for the following test scores.

60, 64, 68, 68, 72, 76, 76, 80, 80, 80, 84, 84, 84, 84, 88, 88, 88, 92, 92, 96, 96, 96, 96, 96, 96, 96, 100, 100
(there is a sketch of a box-and-whisker plot with a number line from 50 to 100)
10 a. how much of the data is represented by the box?
b. how much is represented by each whisker?

1. what does the graph tell you about student success on the test?

use the data from the set section to answer the following questions.

1. make a frequency table with intervals. use an interval of 5.
2. make a histogram of the data using your intervals of 5.

(there is a frequency table with score intervals and frequency, and a sketch of a histogram with a number line from 55 to 105)

Explanation:

Step1: Analyze Question 10a

In a box - and - whisker plot, the box represents the interquartile range (IQR), which contains the middle 50% of the data. This is a fundamental concept in statistics (a subfield of Mathematics) related to data distribution. The box spans from the first quartile ($Q_1$) to the third quartile ($Q_3$), and the data between $Q_1$ and $Q_3$ is 50% of the total data set.

Step2: Analyze Question 10b

Each whisker in a box - and - whisker plot represents the data that is outside the interquartile range but within the minimum and maximum values (excluding outliers, if any). Since the total data is 100% and the box represents 50%, the remaining 50% is split equally between the two whiskers. So each whisker represents 25% of the data.

Step3: Analyze Question 11

To determine what the graph (box - and - whisker plot) tells about student success, we look at the data distribution. First, we can find the minimum, $Q_1$, median, $Q_3$, and maximum. The test scores are: 60, 64, 68, 68, 72, 76, 76, 80, 80, 80, 84, 84, 84, 84, 88, 88, 88, 92, 92, 96, 96, 96, 96, 96, 96, 96, 100, 100.

• The minimum value is 60.
• To find the median (second quartile, $Q_2$), since there are $n = 28$ data points, the median is the average of the 14th and 15th values. The 14th value is 84 and the 15th value is 88, so the median $=\frac{84 + 88}{2}=86$.
• The first quartile ($Q_1$) is the median of the first 14 values. The first 14 values are: 60, 64, 68, 68, 72, 76, 76, 80, 80, 80, 84, 84, 84, 84. The median of these 14 values (average of 7th and 8th values) is $\frac{76+80}{2}=78$.
• The third quartile ($Q_3$) is the median of the last 14 values. The last 14 values are: 88, 88, 88, 92, 92, 96, 96, 96, 96, 96, 96, 96, 100, 100. The median of these 14 values (average of 7th and 8th values) is $\frac{96 + 96}{2}=96$.
• The maximum value is 100.

Most of the data is on the higher end (since the median is 86, and a large number of scores are 80 and above, with a significant number of 96s and 100s). This suggests that a majority of students performed well on the test, with a relatively small proportion of lower scores (the left whisker and the lower part of the box represent a smaller portion of the data).

Step4: Analyze Question 12

To make a frequency table with an interval of 5:

• Interval 60 - 64: Scores are 60, 64. Frequency = 2.
• Interval 65 - 69: Scores are 68, 68. Frequency = 2.
• Interval 70 - 74: Score is 72. Frequency = 1.
• Interval 75 - 79: Scores are 76, 76. Frequency = 2.
• Interval 80 - 84: Scores are 80, 80, 80, 84, 84, 84, 84. Frequency = 7.
• Interval 85 - 89: Scores are 88, 88, 88. Frequency = 3.
• Interval 90 - 94: Scores are 92, 92. Frequency = 2.
• Interval 95 - 100: Scores are 96, 96, 96, 96, 96, 96, 96, 100, 100. Frequency = 9.

Step5: Analyze Question 13

To make a histogram:

• The x - axis will have the intervals: 60 - 64, 65 - 69, 70 - 74, 75 - 79, 80 - 84, 85 - 89, 90 - 94, 95 - 100.
• The y - axis will represent the frequency.
• For each interval, we draw a bar with height equal to the frequency of that interval. For example, the bar for 60 - 64 will have height 2, for 65 - 69 height 2, for 70 - 74 height 1, for 75 - 79 height 2, for 80 - 84 height 7, for 85 - 89 height 3, for 90 - 94 height 2, and for 95 - 100 height 9. The histogram should have no spaces between the bars (since the intervals are continuous) and the width of each bar should be equal (since the interval width is 5 for all intervals).

Answer:

s:
10a. 50% of the data.
10b. 25% of the data.

1. Most students performed well (a large proportion of scores are 80 and above, with many high scores like 96 and 100), and only a small portion of students had relatively low scores (60 - 74 range has a lower frequency).

12.

Score Interval	Frequency
65 - 69	2
70 - 74	1
75 - 79	2
80 - 84	7
85 - 89	3
90 - 94	2
95 - 100	9

1. A histogram with x - axis intervals (60 - 64, 65 - 69, 70 - 74, 75 - 79, 80 - 84, 85 - 89, 90 - 94, 95 - 100) and y - axis as frequency. The bar heights correspond to the frequencies: 2, 2, 1, 2, 7, 3, 2, 9 for the respective intervals.