two hundred fifty students were asked how many hours they spend on studying each night. twenty students responded that they spend less than 1 hour on studying each night. what is the relative frequency of this response?
Answers
Twenty students were asked how many hours they worked per day. Their responses, in hours, are as follows:
5, 6, 3, 3, 2, 4, 7, 5, 2, 3, 5, 6, 5, 4, 4, 3, 5, 2, 5, 3.
The following table lists the different data values in ascending order and their frequencies.
Frequency Table of Student Work Hours
DATA VALUE FREQUENCY
2 3
3 5
4 3
5 6
6 2
7 1
In this research, 3 students studied for 2 hours. 5 students studies for 3 hours.
A frequency is the number of times a value of the data occurs. According to the table, there are three students who work two hours, five students who work three hours, and so on. The sum of the values in the frequency column, 20, represents the total number of students included in the sample.
A relative frequency is the ratio (fraction or proportion) of the number of times a value of the data occurs in the set of all outcomes to the total number of outcomes. To find the relative frequencies, divide each frequency by the total number of students in the sample–in this case, 20. Relative frequencies can be written as fractions, percents, or decimals.
Relative frequency =
frequency of the class
total
Cumulative relative frequency is the accumulation of the previous relative frequencies. To find the cumulative relative frequencies, add all the previous relative frequencies to the relative frequency for the current row, as shown in the table below.
Cumulative relative frequency = sum of previous relative frequencies + current class frequency
Example 1
Frequency Table of Student Work Hours with Relative and Cumulative Relative Frequencies
DATA VALUE FREQUENCY RELATIVE
FREQUENCY
CUMULATIVE RELATIVE
FREQUENCY
2 3
3
20
or 0.15 0.15
3 5
5
20
or 0.25 0.15 + 0.25 = 0.40
4 3
3
20
or 0.15 0.40 + 0.15 = 0.55
5 6
6
20
or 0.30 0.55 + 0.30 = 0.85
6 2
2
20
or 0.10 0.85 + 0.10 = 0.95
7 1
1
20
or 0.05 0.95 + 0.05 = 1.00
The last entry of the cumulative relative frequency column is one, indicating that one hundred percent of the data has been accumulated.
Example 2
We sample the height of 100 soccer players. The result is shown below.
Height (inches) Frequency
59.95 – 61.95 5
61.95 – 63.95 3
63.95 – 65.95 15
65.95 – 67.95 40
67.95 – 69.95 17
69.95 – 71.95 12
71.95 – 73.95 7
73.95 – 75.95 1
Total = 100
Find:
a. the relative frequency for each class.
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b. the percentage for height that is less than 63.95 inches.
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c. the percentage for height that is between 69.95 inches and 73.95 inches.
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In this sample, there are five players whose heights fall within the interval 59.95–61.95 inches, three players whose heights fall within the interval 61.95–63.95 inches, 15 players whose heights fall within the interval 63.95–65.95 inches, 40 players whose heights fall within the interval 65.95–67.95 inches, 17 players whose heights fall within the interval 67.95–69.95 inches, 12 players whose heights fall within the interval 69.95–71.95, seven players whose heights fall within the interval 71.95–73.95, and one player whose heights fall within the interval 73.95–75.95. All heights fall between the endpoints of an interval and not at the endpoints.
Example 3
The table shows the amount, in inches, of annual rainfall in a sample of towns.
Rainfall (inches) Frequency
2.95 – 4.97 6
4.97 – 6.99 7
6.99 – 9.01 15
9.01 – 11.03 8
11.03 – 13.05 9
13.05 – 15.07 5
Find
the relative frequency and cumulative relative frequency for each class.
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the percentage of rainfall that is less than 9.01 inches.
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the percentage of heights that fall between 61.95 and 65.95 inches.