Alright, let's walk through the solution step-by-step.
Given data:
- Sample size [tex]\( n = 43 \)[/tex]
- Population standard deviation [tex]\( \sigma = 11 \)[/tex]
We need to find the standard error of the mean for different population sizes. The standard error of the mean is calculated using different formulas depending on whether the population is considered infinite or finite.
### a. The population size is infinite
When the population size is infinite, the standard error of the mean ([tex]\(SE\)[/tex]) is given by the formula:
[tex]\[
SE = \frac{\sigma}{\sqrt{n}}
\][/tex]
Substituting the given values:
[tex]\[
SE = \frac{11}{\sqrt{43}} \approx 1.68
\][/tex]
Thus, the standard error of the mean for an infinite population is approximately [tex]\( 1.68 \)[/tex].
### b. The population size is [tex]\( N = 50,000 \)[/tex]
For a finite population, we use the finite population correction factor (FPC). The standard error is calculated as:
[tex]\[
SE_{finite} = SE \times \sqrt{\frac{N - n}{N - 1}}
\][/tex]
First, calculate the standard error using the infinite population formula:
[tex]\[
SE = \frac{11}{\sqrt{43}} \approx 1.68
\][/tex]
Then, apply the finite population correction factor:
[tex]\[
\text{FPC} = \sqrt{\frac{50000 - 43}{50000 - 1}} \approx 0.99943
\][/tex]
Now, calculate the finite standard error:
[tex]\[
SE_{finite} = 1.68 \times 0.99943 \approx 1.68
\][/tex]
So, the standard error of the mean when [tex]\( N = 50,000 \)[/tex] is approximately [tex]\( 1.68 \)[/tex].
### c. The population size is [tex]\( N = 5,000 \)[/tex]
Again, we'll use the finite population correction factor:
[tex]\[
\text{FPC} = \sqrt{\frac{5000 - 43}{5000 - 1}} \approx 0.99369
\][/tex]
Now, calculate the finite standard error:
[tex]\[
SE_{finite} = 1.68 \times 0.99369 \approx 1.67
\][/tex]
Thus, the standard error of the mean when [tex]\( N = 5,000 \)[/tex] is approximately [tex]\( 1.67 \)[/tex].
### d. The population size is [tex]\( N = 500 \)[/tex]
Using the finite population correction factor:
[tex]\[
\text{FPC} = \sqrt{\frac{500 - 43}{500 - 1}} \approx 0.95736
\][/tex]
Now, calculate the finite standard error:
[tex]\[
SE_{finite} = 1.68 \times 0.95736 \approx 1.61
\][/tex]
Thus, the standard error of the mean when [tex]\( N = 500 \)[/tex] is approximately [tex]\( 1.61 \)[/tex].
This concludes the detailed step-by-step solution for the given question.
