A population consists of six numbers 4,8,12,16,20,24. Consider all samples of size two which can be drawn without replacement from this population.
i. The mean of the sampling distribution of means
can be calculated by finding the mean of all possible sample means.
First, we list all possible samples of size two that can be drawn without replacement from the population {4,8,12,16,20,24}:
- Sample 1: (4,8)
- Sample 2: (4,12)
- Sample 3: (4,16)
- Sample 4: (4,20)
- Sample 5: (4,24)
- Sample 6: (8,12)
- Sample 7: (8,16)
- Sample 8: (8,20)
- Sample 9: (8,24)
- Sample 10: (12,16)
- Sample 11: (12,20)
- Sample 12: (12,24)
- Sample 13: (16,20)
- Sample 14: (16,24)
- Sample 15: (20,24)
Next, we calculate the mean of each sample:
- Sample 1 mean: (4+8)/2 = 6
- Sample 2 mean: (4+12)/2 = 8
- Sample 3 mean: (4+16)/2 = 10
- Sample 4 mean: (4+20)/2 = 12
- Sample 5 mean: (4+24)/2 = 14
- Sample 6 mean: (8+12)/2 = 10
- Sample 7 mean: (8+16)/2 = 12
- Sample 8 mean: (8+20)/2 = 14
- Sample 9 mean: (8+24)/2 = 16
- Sample 10 mean: (12+16)/2 = 14
- Sample 11 mean: (12+20)/2 = 16
- Sample 12 mean: (12+24)/2 = 18
- Sample 13 mean: (16+20)/2 = 18
- Sample 14 mean: (16+24)/2 = 20
- Sample 15 mean: (20+24)/2 = 22
Finally, we calculate the mean of all sample means:
(6+8+10+12+14+10+12+14+16+14+16+18+18+20+22) / 15 = 14
Therefore, the mean of the sampling distribution of means is 14.