Suppose a random sample of size 40 is selected from a population with = 9. 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 2 decimals).
b. The population size is N = 50,000 (to 2 decimals).
c. The population size is N = 5000 (to 2 decimals).
d. The population size is N = 500 (to 2 decimals).
Some notes on finite population correction factor:
If the population is small and the sample is large (more than 5% of the small population), use the finite population correction factor.
For standard error of the mean, use:
sd/√n
If you need to adjust for the finite population correction factor, use:
sd/√n * √(N-n)/N-1)
N = number in population
n = number in sample
Therefore:
a. Use the formula sd/√n (population is infinite)
b. Is the sample size of 40 more than 5% of the population? No, it isn't; your answer will be the same as part a).
c. Is the sample size of 40 more than 5% of the population? No, it isn't; your answer will be the same as part a) and b).
d. Is the sample size of 40 more than 5% of the population? Yes, it is, so use the finite population correction factor.
I hope this will help get you started.
To find the value of the standard error of the mean in each of these cases, we need to use the formula for the standard error. The formula for the standard error of the mean is:
Standard error = σ / √n
Where σ represents the population standard deviation and n represents the sample size.
a. When the population size is infinite:
In this case, there is no need for the finite population correction factor since the population is considered infinite. Thus, the standard error formula becomes:
Standard error = σ / √n
b. When the population size is N = 50,000:
When the population size is large but finite, we need to use the finite population correction factor. The finite population correction factor adjusts the calculation of the standard error to account for the finite population. The formula for the standard error with the finite population correction factor is:
Standard error = σ / √n * √(N - n) / (N - 1)
Where N represents the population size.
c. When the population size is N = 5000:
Similarly to case b, we need to use the finite population correction factor. So the formula becomes:
Standard error = σ / √n * √(N - n) / (N - 1)
d. When the population size is N = 500:
Again, we need to use the finite population correction factor. So the formula becomes:
Standard error = σ / √n * √(N - n) / (N - 1)
To find the specific values of the standard error in each case, we need the population standard deviation (σ) and the sample size (n). Since they are not provided in your question, you would need to obtain them from the given data or the context of the problem. Once you have these values, plug them into the appropriate formula to calculate the standard error for each case.