Question
In: Statistics and Probability
A survey asked a group of medical doctors how many children they had fathered. The results...
A survey asked a group of medical doctors how many children they had fathered. The results are summarized by this ungrouped frequency distribution.
Number of Children 0 1 2 3 4 6
Number of Doctors 19 25 12 18 6 5
(a) Calculate the sample mean for the number of children the doctors had fathered. (Give your answer correct to two decimal places.)
(b) Calculate the variance for the number of children the doctors had fathered.
(c) Calculate the standard deviation for the number of children the doctors had fathered.
Solutions
Expert Solution
Given that a survey asked a group of medical doctors how many children they had fathered.
The results are summarized by an ungrouped frequency distribution.
| No. | Number of Children | Number of Doctors |
| 1 | 0 | 19 |
| 2 | 1 | 25 |
| 3 | 2 | 12 |
| 4 | 3 | 18 |
| 5 | 4 | 6 |
| 6 | 6 | 5 |
(a) Here we need to calculate the sample mean for the number of children the doctors had fathered.
The sample mean is given by,
The table of calculations is provided below,
| No. | Number of Childern(xi) | Number of Doctors(fi) | xi*fi |
| 1 | 0 | 19 | 0 |
| 2 | 1 | 25 | 25 |
| 3 | 2 | 12 | 24 |
| 4 | 3 | 18 | 54 |
| 5 | 4 | 6 | 24 |
| 6 | 6 | 5 | 30 |
| Total | 85 | 157 | |
Hence the sample mean for the number of children the doctors had fathered is 1.85
(b) Here we need to calculate the variance for the number of children the doctors had fathered.
The sample variance is given by,
The table of calculations is provided below,
| No. | Number of Childern(xi) | Number of Doctors(fi) | xi*fi | ((xi-x̄)^2)*fi |
| 1 | 0 | 19 | 0 | 64.821 |
| 2 | 1 | 25 | 25 | 17.9378 |
| 3 | 2 | 12 | 24 | 0.2807 |
| 4 | 3 | 18 | 54 | 23.9269 |
| 5 | 4 | 6 | 24 | 27.8109 |
| 6 | 6 | 5 | 30 | 86.2346 |
| Total | 85 | 157 | 221.0119 | |
Hence the variance for the number of children the doctors had fathered is 2.6311
(c) Here we need to calculate the standard deviation for the number of children the doctors had fathered.
The sample standard deviation is given by,
The table of calculations is provided below,
| No. | Number of Childern(xi) | Number of Doctors(fi) | xi*fi | ((xi-x̄)^2)*fi |
| 1 | 0 | 19 | 0 | 64.821 |
| 2 | 1 | 25 | 25 | 17.9378 |
| 3 | 2 | 12 | 24 | 0.2807 |
| 4 | 3 | 18 | 54 | 23.9269 |
| 5 | 4 | 6 | 24 | 27.8109 |
| 6 | 6 | 5 | 30 | 86.2346 |
| Total | 85 | 157 | 221.0119 | |
Hence the standard deviation for the number of children the doctors had fathered 1.6221
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