Sampling Theory

Step 1: Understanding the Concept:
This question is about calculating the required sample size for estimating a population proportion. The notation, though slightly unconventional, maps to the standard formula for sample size calculation where the population size (N) is large enough to be considered infinite. The term is likely a typo in the OCR and should be . This value of is used to get the maximum possible sample size when no prior estimate of the proportion is available.

Step 2: Key Formula or Approach:
The formula for the sample size ( ) required to estimate a population proportion with a given margin of error is: Where: - is the Z-score corresponding to the desired confidence level (here represented by ).
- is the estimated population proportion.
- is the desired margin of error.

Step 3: Detailed Explanation:
Let's interpret the given values based on the standard formula: - Margin of error, .
- Z-score, . A Z-score of 2 corresponds to approximately a 95.45% confidence level.
- Estimated proportion, . Using maximizes the product , ensuring a sufficiently large sample size for any true proportion.
Now, substitute these values into the formula: Since ,
Step 4: Final Answer:
The required sample size is 400.

Step 1: Understanding the Concept:
The standard error of the estimate of the population mean is the standard deviation of the sampling distribution of the sample mean, . For Simple Random Sampling Without Replacement (SRSWOR) from a finite population, the formula includes a finite population correction (FPC) factor.

Step 2: Key Formula or Approach:
The standard error (SE) of the sample mean is given by: Where: - is the population size. - is the sample size. - is the population standard deviation.

Step 3: Detailed Explanation:
We are given the following values: - Population size, . - Sample size, . - Population standard deviation, . First, calculate the finite population correction (FPC) factor: Next, calculate the variance of the sample mean: Finally, calculate the standard error by taking the square root of the variance: This value is closest to 1.95.
Step 4: Final Answer:
The standard error of the estimate of the population mean is approximately 1.95.