Sampling Distribution of ๐ฬ
3. A Suppose a random sample of size 80 is selected from a population with s = 8. Find the value of the standard error of the mean in each of the following cases (use the finite population correction factor if appropriate).
1. The population size is infinite.
2. The population size is N = 50,000.
3. The population size is N = 5000.
4. The population size is N = 500.
To find the value of the standard error of the mean in each of the given cases, we can use the formula:
Standard Error of the Mean (SE) = Population Standard Deviation (s) / Square Root of Sample Size (n)
1. When the population size is infinite:
Here, you are given that the population standard deviation (s) is 8 and the sample size (n) is 80. In this case, you don't need to consider any finite population correction factor since the population size is infinite.
Using the formula, the standard error of the mean:
SE = 8 / โ80 โ 0.8944
2. When the population size is N = 50,000:
Similar to the previous case, you are given that the population standard deviation (s) is 8 and the sample size (n) is 80. However, in this case, we need to consider the finite population correction factor since the population size is finite.
Finite Population Correction Factor (FPC) = โ(N - n) / โ(N - 1)
where N is the population size and n is the sample size.
Substituting the values:
FPC = โ(50000 - 80) / โ(50000 - 1) = โ49840 / โ49999 โ 0.9998
The standard error of the mean (SE) with finite population correction:
SE = s / โn * FPC
SE = 8 / โ80 * 0.9998 โ 0.8944 * 0.9998 โ 0.8939
3. When the population size is N = 5000:
Using the same formula for the finite population correction factor as before, and substituting the values:
FPC = โ(5000 - 80) / โ(5000 - 1) = โ4920 / โ4999 โ 0.9898
The standard error of the mean with finite population correction:
SE = s / โn * FPC
SE = 8 / โ80 * 0.9898 โ 0.8944 * 0.9898 โ 0.8856
4. When the population size is N = 500:
Again, using the formula for the finite population correction factor and substituting the values:
FPC = โ(500 - 80) / โ(500 - 1) = โ420 / โ499 โ 0.9586
The standard error of the mean with finite population correction:
SE = s / โn * FPC
SE = 8 / โ80 * 0.9586 โ 0.8944 * 0.9586 โ 0.8579
In summary, the standard error of the mean varies depending on the population size and whether a finite population correction factor is applied.