Question

In a certain region, 15% of people over age 50 didn't graduate from high school. we would like to know if this percentage is the same among the 25-30 year age group. use critical values to exactly 3 decimal places.

(a) how many 25-30 year old people should be surveyed in order to estimate the proportion of non-grads to within 5% with 99% confidence? correct: your answer is correct.
(b) suppose we wanted to cut the margin of error to 2%. how many people should be sampled now?
(c) what sample size is required for a margin of error of 8%?

Answer 1

To estimate the proportion of non-grads among the 25-30 year age group, we use the formula for sample size for proportions. For a 99% confidence level and a margin of error of 5%, we need to survey 735 people. If we want to cut the margin of error to 2%, we need to survey 1836 people. For a margin of error of 8%, we need to survey 459 people.

Step-by-step explanation:

To estimate the proportion of non-grads among the 25-30 year age group with a 99% confidence level and a margin of error of 5%, we need to determine the sample size.

Using the formula for sample size for proportions, we have:

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

where n is the sample size, Z is the critical value, p is the estimated proportion, and E is the margin of error.

In this case, the Z-score for a 99% confidence level is approximately 2.576, and the estimated proportion is 15%. Substituting these values into the formula:

n = (2.576^2 * 0.15 * (1-0.15))/(0.05^2) ≈ 734.14

Rounding up to the nearest whole number, we need to survey 735 25-30 year old people in order to estimate the proportion of non-grads to within 5% with 99% confidence.

For a margin of error of 2%, we can use the same formula with a new margin of error value:

n = (2.576^2 * 0.15 * (1-0.15))/(0.02^2) ≈ 1835.35

Rounding up to the nearest whole number, we need to survey 1836 people.

Finally, for a margin of error of 8%, the formula becomes:

n = (2.576^2 * 0.15 * (1-0.15))/(0.08^2) ≈ 458.84

Rounding up to the nearest whole number, we need to survey 459 people.