Question

In a certain region, 20% 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 7% with 95% confidence?

(b) Suppose we wanted to cut the margin of error to 4%. How many people should be sampled now?

(c) What sample size is required for a margin of error of 5%?

Answer 1

a)

The following information is provided,
Significance Level, α = 0.05, Margin of Error, E = 0.07

The provided estimate of proportion p is, p = 0.2
The critical value for significance level, α = 0.05 is 1.96.

The following formula is used to compute the minimum sample size required to estimate the population proportion p within the required margin of error:
n >= p*(1-p)*(zc/E)^2
n = 0.2*(1 - 0.2)*(1.96/0.07)^2
n = 125.44

Therefore, the sample size needed to satisfy the condition n >= 125.44 and it must be an integer number, we conclude that the minimum required sample size is n = 126
Ans : Sample size, n = 126 or 125

b)

The following information is provided,
Significance Level, α = 0.05, Margin of Error, E = 0.04

The provided estimate of proportion p is, p = 0.2
The critical value for significance level, α = 0.05 is 1.96.

The following formula is used to compute the minimum sample size required to estimate the population proportion p within the required margin of error:
n >= p*(1-p)*(zc/E)^2
n = 0.2*(1 - 0.2)*(1.96/0.04)^2
n = 384.16

Therefore, the sample size needed to satisfy the condition n >= 384.16 and it must be an integer number, we conclude that the minimum required sample size is n = 385
Ans : Sample size, n = 385 or 384

c)
The following information is provided,
Significance Level, α = 0.05, Margin of Error, E = 0.05

The provided estimate of proportion p is, p = 0.2
The critical value for significance level, α = 0.05 is 1.96.

The following formula is used to compute the minimum sample size required to estimate the population proportion p within the required margin of error:
n >= p*(1-p)*(zc/E)^2
n = 0.2*(1 - 0.2)*(1.96/0.05)^2
n = 245.86

Therefore, the sample size needed to satisfy the condition n >= 245.86 and it must be an integer number, we conclude that the minimum required sample size is n = 246
Ans : Sample size, n = 246