Suppose a random sample of size is selected from a population with . Find the value of the standard error of the mean in each of the following cases (use the finite population correction factor if appropriate). a. The population size is infinite (to decimals). 1.56 b. The population size is N= 50000 is (to decimals). 1.56 c. The population size is N=5000 (to decimals). 1.56 d. The population size is N=500(to decimals). 

To find the standard error of the mean (SEM) for a sample, we use the formula:

\[
\text{SEM} = \frac{\sigma}{\sqrt{n}} \times \text{FPC}
\]

Where:
- \(\sigma\) is the population standard deviation,
- \(n\) is the sample size,
- FPC is the finite population correction factor given by \(\sqrt{\frac{N - n}{N - 1}}\) when the population size \(N\) is finite.

### Given:
- Population standard deviation \(\sigma = 1.56\)
- Sample size \(n\) (not specified, assuming it's the same for all cases)

### a. Population size is infinite (N → ∞)
For an infinite population, the FPC is 1.

\[
\text{SEM}_{\infty} = \frac{1.56}{\sqrt{n}} \times 1 = \frac{1.56}{\sqrt{n}}
\]

### b. Population size is N = 50,000
For \(N = 50,000\):

\[
\text{FPC} = \sqrt{\frac{50000 - n}{50000 - 1}} \approx \sqrt{\frac{50000 - n}{49999}}
\]

\[
\text{SEM}_{50000} = \frac{1.56}{\sqrt{n}} \times \sqrt{\frac{50000 - n}{49999}}
\]

### c. Population size is N = 5,000
For \(N = 5,000\):

\[
\text{FPC} = \sqrt{\frac{5000 - n}{5000 - 1}} \approx \sqrt{\frac{5000 - n}{4999}}
\]

\[
\text{SEM}_{5000} = \frac{1.56}{\sqrt{n}} \times \sqrt{\frac{5000 - n}{4999}}
\]

### d. Population size is N = 500
For \(N = 500\):

\[
\text{FPC} = \sqrt{\frac{500 - n}{500 - 1}} \approx \sqrt{\frac{500 - n}{499}}
\]

\[
\text{SEM}_{500} = \frac{1.56}{\sqrt{n}} \times \sqrt{\frac{500 - n}{499}}
\]

### Summary of SEM Values
To calculate specific values, we need the sample size \(n\). If you provide \(n\), I can compute the exact SEM for each case.

Do you have a specific sample size \(n\) in mind, or would you like to assume a value? 

### Questions for Further Exploration
1. What is the sample size \(n\) you would like to use?
2. How does the finite population correction factor impact the SEM?
3. Can you explain how SEM is different from standard deviation?
4. What are practical applications of SEM in research?
5. Would you like examples of calculating SEM in real-world scenarios?

### Tip
Understanding the relationship between sample size and standard error can help in designing experiments and determining sample sizes needed for desired precision.

Suppose a random sample of size is selected from a population with . Find the value of the standard error of the mean in each of the following cases (use the finite population correction factor if appropriate). a. The population size is infinite (to decimals). 1.56 b. The population size is N= 50000 is (to decimals). 1.56 c. The population size is N=5000 (to decimals). 1.56 d. The population size is N=500(to decimals).

Explore how to calculate the standard error of the mean (SEM) for different population sizes, including infinite populations and finite populations of sizes 50,000, 5,000, and 500. This detailed guide includes the relevant formulas and explanations for better understanding. Suitable for students in Grades 11-12.

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