A) An insurance company wants to estimate the proportion of people unsatisfied with their new telephone help service. A survey of 200 callers revealed 45 were unsatisfied with the service. Construct by hand a 95% confidence interval for the proportion of unsatisfied customers.

B) If the company wanted to estimate the sample proportion to within 2% of the population proportion, what sample size would be necessary assuming no prior estimate of the proportion is known?

A) To construct a confidence interval for the proportion of unsatisfied customers, we can use the formula:

Confidence Interval = Sample Proportion ± (Z * Standard Error)

Where:
- Sample Proportion = Number of unsatisfied customers / Total number of callers
- Z = Z-score corresponding to the desired confidence level (in this case, 95% confidence level)
- Standard Error = Square root of [(Sample Proportion * (1 - Sample Proportion)) / Sample Size]

Let's calculate the values step by step:

1. Calculate the Sample Proportion:
Sample Proportion = Number of unsatisfied customers / Total number of callers
Sample Proportion = 45 / 200
Sample Proportion = 0.225

2. Determine the Z-score:
To calculate the Z-score for a 95% confidence level, you can refer to a Z-table or use a statistical calculator. The Z-score for a 95% confidence level is approximately 1.96.

3. Calculate the Standard Error:
Standard Error = Square root of [(Sample Proportion * (1 - Sample Proportion)) / Sample Size]
Standard Error = Square root of [(0.225 * (1 - 0.225)) / 200]
Standard Error ≈ 0.0357

4. Construct the Confidence Interval:
Confidence Interval = Sample Proportion ± (Z * Standard Error)
Confidence Interval = 0.225 ± (1.96 * 0.0357)
Confidence Interval ≈ 0.225 ± 0.0701

Therefore, the 95% confidence interval for the proportion of unsatisfied customers is approximately 0.155 to 0.295.

B) To determine the necessary sample size to estimate the sample proportion within 2% of the population proportion, we can use the formula:

Sample Size = [Z^2 * (P * (1 - P))] / (E^2)

Where:
- Z = Z-score corresponding to the desired confidence level (you can choose a Z-score corresponding to a particular confidence level, such as 95%)
- P = Estimated population proportion (since no prior estimate is known, we can assume 0.5)
- E = Maximum acceptable margin of error (within 2% of the population proportion)

Let's calculate the sample size:

1. Determine the Z-score:
You need to choose a desired confidence level and find the corresponding Z-score. Let's assume a 95% confidence level, which corresponds to a Z-score of approximately 1.96.

2. Calculate the Sample Size:
Sample Size = [Z^2 * (P * (1 - P))] / (E^2)
Sample Size = [1.96^2 * (0.5 * (1 - 0.5))] / (0.02^2)
Sample Size ≈ 2401

Therefore, a sample size of approximately 2401 would be necessary to estimate the sample proportion within 2% of the population proportion, assuming no prior estimate is known.

A) To construct a confidence interval for the proportion of unsatisfied customers, we can use the formula:

CI = p-hat ± z * sqrt((p-hat * (1-p-hat))/n)

where:
- p-hat is the sample proportion of unsatisfied customers (45/200 = 0.225)
- z is the z-score corresponding to the desired confidence level (for a 95% confidence level, z = 1.96)
- n is the sample size (200)

Let's substitute these values into the formula:

CI = 0.225 ± 1.96 * sqrt((0.225 * (1-0.225))/200)

Calculating the values:

CI = 0.225 ± 0.051

Therefore, the 95% confidence interval for the proportion of unsatisfied customers is (0.174, 0.276).

B) To determine the sample size required to estimate the proportion to within 2% of the population proportion, without any prior estimate, we can use the formula:

n = (z^2 * p * (1-p)) / E^2

where:
- z is the z-score corresponding to the desired confidence level (for a 95% confidence level, z = 1.96)
- p is the estimated proportion (unknown in this case)
- E is the desired margin of error (2% or 0.02)

Since we don't have any prior estimate of the proportion, we can assume a worst-case scenario of p = 0.5 (maximum variability).

Let's substitute these values into the formula and solve for n:

n = (1.96^2 * 0.5 * (1-0.5)) / (0.02^2)

Calculating the values:

n ≈ 9604

Therefore, a sample size of approximately 9604 would be necessary to estimate the sample proportion to within 2% of the population proportion, assuming no prior estimate is known.