Question

A previous study gave (0.18,0.33) as a confidence interval for p. Find the minimum sample size required to ensure that your estimate ( p ) will be within 0.07 of p with 90% confidence. IMPORTANT: Use a critical value that you found with your calculator (not from a table), and round it to 3 places after the decimal point before you plug it into a formula and perform your calculations. Do not round-off any other intermediate results.

Answer 1

The minimum sample size required to estimate p within 0.07 with 90% confidence involves finding the critical value z₀.₀₅ with a calculator, estimating p and q from given confidence intervals, and applying the sample size formula, rounding up to the nearest whole number.

Step-by-step explanation:

To determine the minimum sample size required to ensure that your estimate of p will be within 0.07 of p with 90% confidence, we can use the formula for required sample size for estimating a population proportion. First, we use a calculator to find the critical value z₀.₀₅ by utilizing the inverse normal function, as the confidence interval is two-tailed. Once the critical value is calculated, round it to three decimal places. We use the average of the provided confidence interval to estimate p and q, where p' = (0.18 + 0.33) / 2 and q' = 1 - p'. Then we apply the formula:

Sample Size Formula

n = ​( (z₀.₀₅) ² · (p'q') ) / E²

where n is the sample size, z is the z-score corresponding to the desired confidence level (0.90 in this case), p' is the estimated population proportion, q' is 1-p', and E is the desired margin of error (0.07).

After calculating the critical value and plugging in the values for p' and q', solve for n and round up to the nearest whole number, as we can't have a fraction of a survey participant.