To solve this problem, we need to calculate the standard error of the mean ($SE$) in each case using the formula:

$SE = \frac{\sigma}{\sqrt{n}}$

where:

• $\sigma = 12$ (the population standard deviation),
• $n = 53$ (sample size).

If the population size ($N$) is finite, we need to apply the finite population correction factor (FPC), which adjusts the standard error as follows:

$SE = \frac{\sigma}{\sqrt{n}} \times \sqrt{\frac{N - n}{N - 1}}$

Let's calculate each case:

(a) Population size is infinite:

Since the population is considered infinite, we don’t apply the FPC, so:

$SE = \frac{12}{\sqrt{53}}$

(b) Population size $N = 50,000$:

Applying the FPC:

$SE = \frac{12}{\sqrt{53}} \times \sqrt{\frac{50000 - 53}{50000 - 1}}$

(c) Population size $N = 5000$:

Applying the FPC:

$SE = \frac{12}{\sqrt{53}} \times \sqrt{\frac{5000 - 53}{5000 - 1}}$

(d) Population size $N = 500$:

Applying the FPC:

$SE = \frac{12}{\sqrt{53}} \times \sqrt{\frac{500 - 53}{500 - 1}}$

I’ll calculate each value to two decimal places.import math

Given values

sigma = 12 n = 53

Function to calculate standard error with FPC

def calculate_se(n, sigma, N=None): se = sigma / math.sqrt(n) if N is not None: # apply finite population correction if N is provided fpc = math.sqrt((N - n) / (N - 1)) se *= fpc return round(se, 2)

Calculations

se_infinite = calculate_se(n, sigma) # Case a: infinite population se_50000 = calculate_se(n, sigma, 50000) # Case b: N = 50,000 se_5000 = calculate_se(n, sigma, 5000) # Case c: N = 5000 se_500 = calculate_se(n, sigma, 500) # Case d: N = 500

se_infinite, se_50000, se_5000, se_500(1.65, 1.65, 1.64, 1.56)(1.65, 1.65, 1.64, 1.56)successHere are the standard error calculations for each case, rounded to two decimal places:

• (a) Population size is infinite: $SE = 1.65$
• (b) Population size $N = 50,000$: $SE = 1.65$
• (c) Population size $N = 5000$: $SE = 1.64$
• (d) Population size $N = 500$: $SE = 1.56$

Would you like further details on any part of this calculation or have any other questions?

Here are some related questions to deepen your understanding:

1. What effect does the finite population correction factor have on the standard error?
2. How would the standard error change if the sample size increased?
3. Why do we assume the population size is "infinite" in some statistical contexts?
4. How is the finite population correction factor derived?
5. In what scenarios would we ignore the finite population correction factor entirely?

Tip: The finite population correction factor has a more significant impact when the sample size is a large portion of the population size.