The accompanying data represent the homework scores for material on Polynomial and Rational Functions for a random sample of students in a college algebra course. Complete parts (a) through (i).
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(a) Construct a relative frequency distribution with a lower class limit of the first class equal to 30 and a class width of 10.
(Type integers or decimals. Do not round.)
College Algebra Homework Scores
36 66 78 83 91
48 69 79 85 91
55 72 79 85 92
59 76 79 86 92
63 77 79 86 94
65 77 79 87 96
65 78 82 89 97
66 78 82 89 98
Answer (2)
A relative frequency distribution was constructed using class intervals of 30-39, 40-49, 50-59, 60-69, 70-79, 80-89, and 90-99. Frequencies were counted for each interval, then relative frequencies were calculated by dividing each frequency by the total number of scores. The results are displayed in a table format for easy interpretation.
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Determine the frequency of each class interval: 30-39 (1), 40-49 (1), 50-59 (2), 60-69 (6), 70-79 (12), 80-89 (10), 90-99 (8).
Calculate the relative frequency for each class by dividing the frequency by the total number of data points (40).
The relative frequencies are: 30-39 (0.025), 40-49 (0.025), 50-59 (0.05), 60-69 (0.15), 70-79 (0.3), 80-89 (0.25), 90-99 (0.2).
The relative frequency distribution is constructed with the calculated relative frequencies for each class interval. See table in step 4
Explanation
Understanding the Problem We are given a set of homework scores and asked to construct a relative frequency distribution. The first class should have a lower limit of 30, and the class width should be 10. This means our classes will be 30-39, 40-49, 50-59, 60-69, 70-79, 80-89, and 90-99. We need to count how many scores fall into each class and then calculate the relative frequency for each class by dividing the frequency by the total number of scores.
Calculating Frequencies First, let's determine the frequency of each class:
30-39: The scores in this range are 36. So, the frequency is 1.
40-49: The scores in this range are 48. So, the frequency is 1.
50-59: The scores in this range are 55, 59. So, the frequency is 2.
60-69: The scores in this range are 66, 69, 63, 65, 65, 66. So, the frequency is 6.
70-79: The scores in this range are 78, 79, 72, 76, 77, 77, 78, 78, 79, 79, 79, 79. So, the frequency is 12.
80-89: The scores in this range are 83, 85, 85, 86, 86, 87, 82, 82, 89, 89. So, the frequency is 10.
90-99: The scores in this range are 91, 91, 92, 92, 94, 96, 97, 98. So, the frequency is 8.
Calculating Relative Frequencies Next, we calculate the relative frequency for each class. The total number of scores is 40.
30-39: Relative Frequency = 40 1 = 0.025
40-49: Relative Frequency = 40 1 = 0.025
50-59: Relative Frequency = 40 2 = 0.05
60-69: Relative Frequency = 40 6 = 0.15
70-79: Relative Frequency = 40 12 = 0.3
80-89: Relative Frequency = 40 10 = 0.25
90-99: Relative Frequency = 40 8 = 0.2
Presenting the Distribution Finally, we present the relative frequency distribution:
Class
Frequency
Relative Frequency
30-39
1
0.025
40-49
1
0.025
50-59
2
0.05
60-69
6
0.15
70-79
12
0.3
80-89
10
0.25
90-99
8
0.2
Examples
Understanding relative frequency distributions is crucial in many real-world scenarios. For instance, in market research, businesses use these distributions to analyze customer demographics and purchasing patterns. Imagine a store owner wants to understand the age distribution of their customers. By creating a relative frequency distribution, they can quickly see what percentage of their customers falls into different age groups (e.g., 20-30, 31-40, etc.). This information helps them tailor their marketing strategies and product offerings to better suit their customer base, ultimately increasing sales and customer satisfaction. Similarly, in healthcare, relative frequency distributions can help analyze the prevalence of certain diseases within different populations.
