Question

at 95% confidence, how large a sample should be taken to obtain a margin of error of 0.04 for the estimation of a population proportion? assume that past data are not available for developing a planning value for p*. (round your answer up to the nearest whole number.)

Answer 1

To achieve a 95% confidence level with a margin of error of 0.04 for an unknown population proportion, a sample size of 601 is required after rounding up to the nearest whole number.

Step-by-step explanation:

When calculating the sample size required to estimate a population proportion with a specified margin of error, we can use the following formula for the sample size (n):

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

where Z is the z-score corresponding to the desired confidence level, p* is the estimated population proportion, and E is the margin of error.

In this case, since we do not have prior data to estimate the population proportion, we use p* = 0.5 for the most conservative sample size calculation, because 0.5 maximizes the product p*(1 - p*). For a 95% confidence interval, the z-score is approximately 1.96. The desired margin of error E is 0.04.

Substituting these values into the formula:

n = (1.96^2 * 0.5 * (1 - 0.5)) / 0.04^2,

n = (3.8416 * 0.25) / 0.0016,

n = 0.9604 / 0.0016,

n = 600.25.

Therefore, after rounding up to the nearest whole number, the minimum sample size required is 601.

Answer 2

To estimate a population proportion with a margin of error of 0.04 and a confidence level of 95%, a sample size of 601 is needed.

Step-by-step explanation:

To determine the sample size needed to estimate a population proportion with a margin of error of 0.04 and a confidence level of 95%, we can use the formula:

`Sample size = (Z-score)^2 * (p-hat * (1-p-hat)) / (E^2)`

Where:

• Z-score = 1.96 for a 95% confidence level
• p-hat is the estimated proportion (0.5 for a conservative estimate)
• E is the margin of error (0.04)

Plugging in these values, we get:

Sample size
`= (1.96)^2 * (0.5 * (1-0.5)) / (0.04^2)`

= 600.25

Rounding up to the nearest whole number, the sample size needed is 601.