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Question:
To earn full credit for this question use your own sheet of paper to solve the problem, showing all steps of your work in order to get full credit. A random sample of 30 male college students was selected, and their heights were measured. The heights (in inches) are given below.
73 68 72 74 6773 69 66 66 68 67 68 73 72 67 71 69 71 70 6769 68 7366 67 66 66 72 72 70
(a) Complete the frequency distribution for the data. Make sure to enter your answers for the relative frequency as decimals, rounded to the nearest tenth.
Height | Frequency | Relative Frequency
66 | |
67 | |
68 | |
69 | |
70 | |
71 | |
72 | |
73 | |
74 | |
(b) Compute the (weighted) sample mean. Make sure to enter your answer rounded to the nearest tenth.
(c) Interpret the sample mean obtained in question (b) in the context of the problem.
Answer:
Height | Frequency | Relative Frequency
66 | 5 | 17
67 | 5 | 17
68 | 4 | 13
69 | 3 | 10
70 | 2 | 7
71 | 2 | 7
72 | 4 | 13
73 | 4 | 13
74 | 1 | 3
Total | 30 | 100
Weighted mean = 69
Explanation:
A random sample of 30 male college students was selected, and their heights were measured. The heights (in inches) are given.
(a) Complete the frequency distribution for the data.
Make sure to enter your answers for the relative frequency as decimals, rounded to the nearest tenth.
Relative frequency is calculated using
Relative frequency = (frequency/30)×100%
Height | Frequency | Relative Frequency
66 | 5 | 17
67 | 5 | 17
68 | 4 | 13
69 | 3 | 10
70 | 2 | 7
71 | 2 | 7
72 | 4 | 13
73 | 4 | 13
74 | 1 | 3
Total | 30 | 100
(b) Compute the (weighted) sample mean. Make sure to enter your answer rounded to the nearest tenth.
Weighted mean = weighted sum/total terms
Weighted sum = 5(66) + 5(67) + 4(68) + 3(69) + 2(70) + 2(71) + 4(72) + 4(73) + 1(74)
Weighted sum = 2080
total terms = 30
So the weighted mean is
Weighted mean = 2080/30
Weighted mean = 69.33
Rounding off to the nearest tenth
Weighted mean = 69
(c) Interpret the sample mean obtained in question (b) in the context of the problem.
If you notice the frequency of students having a height of 69 inches is relatively less as compared to students who has a height less than 69 inches and greater than 69 inches.
This is due to the fact that in weighted mean the contribution of certain data points is different than other data points.
In this case, we have more students having a height of 66 to 68 inches and 72 to 73 inches. Therefore, the mean height is shifted to the middle of these heights that is 69 inches.
Let's find out the median of the heights by arranging the data in ascending order.
66 66 66 66 66 67 67 67 67 67 68 68 68 68 69 | 69 69 70 70 71 71 72 72 72 72 73 73 73 73 74
The median is the 30/2 = 15th value in the data set
The 15th value is 69
So the mean and median both are 69 in this case.