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Question 19. Sampling Distribution. A quarterback threw 1 interception in his first game, 2 interceptions in...
Question 19. Sampling Distribution.
A quarterback threw 1 interception in his first game, 2 interceptions in his second game, 5 interceptions in his third game, and then he retired.
Consider the values of 1, 2, and 5 to be a population.
Assume that samples of size 2 are randomly selected (with replacement) from the population.
a. List the 9 different possible samples and find the mean of each sample.
b. What is the mean of the sample means from part a?
c. Is the mean of the sampling distribution from part b equal to the mean of the population of the three listed values? Are those means always equal?
Solutions
Expert Solution
Answer:-
Given That:-
Sampling Distribution:-
A quarterback threw 1 interception in his first game, 2 interceptions in his second game, 5 interceptions in his third game, and then he retired.
Consider the values of 1, 2, and 5 to be a population.
Assume that samples of size 2 are randomly selected (with replacement) from the population.
| Game number | 1 | 2 | 3 | Total |
| Number of interceptions | 1 | 2 | 5 | 8 |
a. List the 9 different possible samples and find the mean of each sample.?
Possible Samples of sample Size 2 :
| Sample | ||
| Sample Number | ||
| 1 | 1 | 1 |
| 2 | 2 | 2 |
| 3 | 5 | 5 |
| 4 | 1 | 2 |
| 5 | 1 | 5 |
| 6 | 2 | 5 |
| 7 | 2 | 1 |
| 8 | 5 | 1 |
| 9 | 5 | 2 |
Sample means
Sample |
||||
| Sample Number | Element 1 | Element 2 | Total | Sample mean |
| 1 | 1 | 1 | 2 | (1+1)/2=2/2=1 |
| 2 | 2 | 2 | 4 | (2+2)/2=4/2=2 |
| 3 | 5 | 5 | 10 | (5+5)/2=10/2=5 |
| 4 | 1 | 2 | 3 | (1+2)/2=3/2=1.5 |
| 5 | 1 | 5 | 6 | (1+5)/2=6/2=3 |
| 6 | 2 | 5 | 7 | (2+5)/2=7/2=3.5 |
| 7 | 2 | 1 | 3 | (2+1)/2=3/2=1.5 |
| 8 | 5 | 1 | 6 | (5+1)/2=6/2=3 |
| 9 | 5 | 2 | 7 | (5+2)/2=7/2=3.5 |
b. What is the mean of the sample means from part a?
Mean of sample means = sum of all sample means / Total number of samples
= (1+2+5+1.5+3+3.5+1.5+3+3.5) / 9 = 24/9 = 8/3
c. Is the mean of the sampling distribution from part b equal to the mean of the population of the three listed values? Are those means always equal?
Population :
| Game number | 1 | 2 | 3 | Total |
| Number of interceptions | 1 | 2 | 5 | 8 |
Mean of the population = (1+2+5) / 3 = 8/3
Mean of the sampling distribution from part b is mean of the sample
means which is equal to 8/3 .
Yes the mean of the sampling distribution from part b : 8/3 is equal to the mean of the population : 8/3
Yes mean of the sampling distribution is always equal to the mean of the population.
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