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a.
To construct: A percentile graph for the given data.
a.
![Check Mark](/static/check-mark.png)
Answer to Problem 3.3.23RE
The percentile graph for the given data is as follows,
Explanation of Solution
Given info:
The data represents the salaries (in millions of dollars) for 29 NFL teams for the 1999 2000 season.
Class limits | Frequency |
39.9-42.8 | 2 |
42.9-45.8 | 2 |
45.9-48.8 | 5 |
48.9-51.8 | 5 |
51.8-54.8 | 12 |
54.9-57.8 | 3 |
Calculation:
The cumulative frequency for the distribution is calculated and tabulated below,
Class limits | Frequency | Cumulative frequency |
39.9-42.8 | 2 | 2 |
42.9-45.8 | 2 |
|
45.9-48.8 | 5 |
|
48.9-51.8 | 5 |
|
51.8-54.8 | 12 |
|
54.9-57.8 | 3 |
|
The formula to calculate the cumulative percentage is as follows,
For the cumulative frequency 2:
Substitute cumulative frequency as 2 and n as 29 in the formula,
Similarly for remaining values the cumulative percent are tabulated below
Class limit | Frequency |
Cumulative frequency | Cumulative percent |
39.9-42.8 | 2 | 2 | 6.89 |
42.9-45.8 | 2 | 4 |
|
45.9-48.8 | 5 | 9 |
|
48.9-51.8 | 5 | 14 |
|
51.9-54.8 | 12 | 26 |
|
54.9-57.8 | 3 | 29 |
|
|
Software procedure:
Step-by-step software procedure to draw ogive curve using MINITAB software is as follows:
- Choose Graph >
Scatter plot . - Choose With Connect Line, and then click OK.
- In Y variables, enter the Cumulative Percent.
- In X variables enter the Lower class limits.
- In Data view, select Symbols and Connect line under Data display.
- In Data view, select Smoother and enter 0 for Degree of smoothing and 1 for Number of steps under Lowness.
- Click OK
- To modify the interval settings, double click on the horizontal axis of the graph. Then, select Labels > Specified. In this box, enter the values for the cutpoints of the bin intervals (39.9, 42.9, 45.9, 48.9, 51.9, 54.9).
b.
The values that correspond to the 35th, 65th and 85th percentiles.
b.
![Check Mark](/static/check-mark.png)
Answer to Problem 3.3.23RE
The values corresponding to 35th, 65th and 85th percentile are 50, 53 and 55.
Explanation of Solution
Calculation:
From part (a) the cumulative percent table is as follows,
Class limit | Frequency |
Cumulative frequency |
Cumulative percent |
39.9-42.8 | 2 | 2 | 6.89 |
42.9-45.8 | 2 | 4 | 13.79 |
45.9-48.8 | 5 | 9 | 31.03 |
48.9-51.8 | 5 | 14 | 73.68 |
51.9-54.8 | 12 | 26 | 89.65 |
54.9-57.8 | 3 | 29 | 100 |
|
For 35th percentile:
The 35th percentile location is calculated below,
Substitute n as 29 and m as 35 in the formula,
Here the 10th observation corresponds to the cumulative frequency 14 which falls in the class 48.9-51.8.
The formula to calculate the percentile for the grouped data is given below,
Where,
- l, the lower limit of the class.
- h, the width of class.
- f, the frequency of the class.
- p, the percentiles rank.
- n is the total number
- c is the preceding cumulative frequency
Substitute 35 for m, 48.9 for
Thus, the 35th percentile of the data is approximately 50.
For 65th percentile:
The 65th percentile location is calculated below,
Substitute n as 29 and m as 65 in the formula,
Here the 19th observation corresponds to the cumulative frequency 26 which falls in the class 51.9-54.8.
Substitute 65 for m, 51.9 for
Thus, the 65th percentile of the data is approximately 53.
For 85th percentile:
The 85th percentile location is calculated below,
Substitute n as 29 and m as 85 in the formula,
Here the 25th observation corresponds to the cumulative frequency 26 which falls in the class 51.9-54.8.
Substitute 85 for m, 51.9 for
Thus, the 85th percentile of the data is approximately 55.
c.
The percentile rank of values 44, 48 and 54.
c.
![Check Mark](/static/check-mark.png)
Answer to Problem 3.3.23RE
The percentile rank for values 44, 48 and 54 are 10th, 26th and 78th respectively.
Explanation of Solution
Calculation:
Class limit | Frequency |
Cumulative frequency |
39.9-42.8 | 2 | 2 |
42.9-45.8 | 2 | 4 |
45.9-48.8 | 5 | 9 |
48.9-51.8 | 5 | 14 |
51.9-54.8 | 12 | 26 |
54.9-57.8 | 3 | 29 |
|
For the value 44:
Here, the value 44 falls in the interval 42.9-45.8.
Substitute 44 for
Thus, the percentile rank for the value 44 is 10th percentile.
For the value 48:
Here, the value 48 falls in the interval 45.9-48.8
Substitute 48 for
Thus, the percentile rank for the value 48 is 26th percentile.
For the value 48:
Here, the value 48 falls in the interval 51.9-54.8
Substitute 54 for
Thus, the percentile rank for the value 54 is 78th percentile
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Chapter 3 Solutions
Elementary Statistics: A Step By Step Approach
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