Certainly, let's go step by step to find the standard error of the mean in the given cases:
### Given Data:
- Sample size, [tex]\( n = 44 \)[/tex]
- Population standard deviation, [tex]\( \sigma = 11 \)[/tex]
### Case a: The Population Size is Infinite
For an infinite population, the finite population correction factor is not needed. The standard error of the mean is calculated using the formula:
[tex]\[ \text{Standard Error} = \frac{\sigma}{\sqrt{n}} \][/tex]
Substituting the given values:
[tex]\[ \text{Standard Error} = \frac{11}{\sqrt{44}} \][/tex]
Rounded to 2 decimals, the standard error is:
[tex]\[ \boxed{1.66} \][/tex]
### Case b: The Population Size is [tex]\( N = 50,000 \)[/tex]
When the population size is very large (50,000 in this case), it approximates an infinite population, so the finite population correction factor is essentially 1. The standard error remains:
[tex]\[ \text{Standard Error} = \frac{11}{\sqrt{44}} \][/tex]
Rounded to 2 decimals, the standard error is:
[tex]\[ \boxed{1.66} \][/tex]
### Case c: The Population Size is [tex]\( N = 5000 \)[/tex]
For a finite population, we need to apply the finite population correction factor. The formula for the finite population correction factor (FPC) is:
[tex]\[ \text{FPC} = \sqrt{\frac{N - n}{N - 1}} \][/tex]
Substituting [tex]\( N = 5000 \)[/tex]:
[tex]\[ \text{FPC} = \sqrt{\frac{5000 - 44}{5000 - 1}} \][/tex]
The standard error with the finite population correction is:
[tex]\[ \text{Standard Error} = \left( \frac{11}{\sqrt{44}} \right) \times \text{FPC} \][/tex]
[tex]\[ \boxed{1.65} \][/tex]
### Case d: The Population Size is [tex]\( N = 500 \)[/tex]
Similarly, for [tex]\( N = 500 \)[/tex]:
[tex]\[ \text{FPC} = \sqrt{\frac{500 - 44}{500 - 1}} \][/tex]
The standard error with the finite population correction is:
[tex]\[ \text{Standard Error} = \left( \frac{11}{\sqrt{44}} \right) \times \text{FPC} \][/tex]
Rounded to 2 decimals, the standard error is:
[tex]\[ \boxed{1.59} \][/tex]
So the final answers are:
- For an infinite population: [tex]\( \boxed{1.66} \)[/tex]
- For [tex]\( N = 50,000 \)[/tex]: [tex]\( \boxed{1.66} \)[/tex]
- For [tex]\( N = 5000 \)[/tex]: [tex]\( \boxed{1.65} \)[/tex]
- For [tex]\( N = 500 \)[/tex]: [tex]\( \boxed{1.59} \)[/tex]
