A population consists of six numbers 4,8,12,16,20,24. Consider all samples of size of
two which can be drawn without replacement from this population.
i. ii. The mean of the sampling distribution of means. The standard deviation of the sampling distribution of means
i. To find the mean of the sampling distribution of means, we first need to find all possible samples of size 2 that can be drawn from the population.
Possible samples:
- {4, 8}
- {4, 12}
- {4, 16}
- {4, 20}
- {4, 24}
- {8, 12}
- {8, 16}
- {8, 20}
- {8, 24}
- {12, 16}
- {12, 20}
- {12, 24}
- {16, 20}
- {16, 24}
- {20, 24}
Now, we find the mean of each sample:
- {4, 8}: Mean = (4+8)/2 = 6
- {4, 12}: Mean = (4+12)/2 = 8
- {4, 16}: Mean = (4+16)/2 = 10
- {4, 20}: Mean = (4+20)/2 = 12
- {4, 24}: Mean = (4+24)/2 = 14
- {8, 12}: Mean = (8+12)/2 = 10
- {8, 16}: Mean = (8+16)/2 = 12
- {8, 20}: Mean = (8+20)/2 = 14
- {8, 24}: Mean = (8+24)/2 = 16
- {12, 16}: Mean = (12+16)/2 = 14
- {12, 20}: Mean = (12+20)/2 = 16
- {12, 24}: Mean = (12+24)/2 = 18
- {16, 20}: Mean = (16+20)/2 = 18
- {16, 24}: Mean = (16+24)/2 = 20
- {20, 24}: Mean = (20+24)/2 = 22
Now, to find the mean of the sampling distribution of means, we take the average of these means:
Mean = (6 + 8 + 10 + 12 + 14 + 10 + 12 + 14 + 16 + 14 + 16 + 18 + 18 + 20 + 22) / 15
Mean = 13
Therefore, the mean of the sampling distribution of means is 13.
ii. To find the standard deviation of the sampling distribution of means, we need to calculate the standard deviation of the sample means:
1. Calculate the squared differences of each sample mean from the mean calculated in part i.
2. Find the average of these squared differences (variance).
3. Take the square root of the variance to find the standard deviation.
Calculations:
1. Calculate the squared differences:
- (6-13)^2 = 49
- (8-13)^2 = 25
- (10-13)^2 = 9
- (12-13)^2 = 1
- (14-13)^2 = 1
- (10-13)^2 = 9
- (12-13)^2 = 1
- (14-13)^2 = 1
- (16-13)^2 = 9
- (14-13)^2 = 1
- (16-13)^2 = 9
- (18-13)^2 = 25
- (18-13)^2 = 25
- (20-13)^2 = 49
- (22-13)^2 = 81
2. Find the average of these squared differences:
Variance = (49 + 25 + 9 + 1 + 1 + 9 + 1 + 1 + 9 + 1 + 9 + 25 + 25 + 49 + 81) / 15
Variance = 18.27
3. Take the square root of the variance to find the standard deviation:
Standard Deviation = √18.27
Standard Deviation ≈ 4.27
Therefore, the standard deviation of the sampling distribution of means is approximately 4.27.