Question

A researcher wishes to be 95% confident that her estimate of the true proportion of individuals who travel overseas is within 3% of the true proportion. a) Find the sample necessary if, in a prior study, a sample of 200 people showed that 40 traveled overseas last year. b) If no estimate of the sample proportion is available, how large should the sample be?

Answer 1

a) To find the sample size necessary, use the known population proportion, z-score, and margin of error in the formula. b) If no estimate of the sample proportion is available, use the worst-case scenario in the formula to calculate the sample size.

Step-by-step explanation:

a) To find the sample size necessary, we can use the known population proportion from the prior study. The formula to calculate the sample size is:

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

Where n is the sample size, Z is the z-score corresponding to the desired confidence level (in this case, 95% confidence level corresponds to a z-score of 1.96), p is the prior estimated population proportion (0.2), and E is the desired margin of error (0.03).

Using these values:

n = (1.96^2 * 0.2 * (1-0.2)) / 0.03^2

n = 1536

Therefore, the researcher would need a sample size of 1536 people.

b) If no estimate of the sample proportion is available, we can use the worst-case scenario to calculate the sample size. The worst-case scenario is when p = 0.5, which gives the largest sample size. The formula remains the same:

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

Using the values: Z = 1.96, p = 0.5, E = 0.03:

n = (1.96^2 * 0.5 * (1-0.5)) / 0.03^2

n = 1068

Therefore, if no estimate of the sample proportion is available, the researcher would need a sample size of 1068 people.

Answer 2

Answer: a) 683 b) 1067

Step-by-step explanation:

The confidence interval for population proportion is given by :-

`p\pm z_(\alpha/2)\sqrt{(p(1-p))/(n)}`

a) Given : Significance level :
`\alpha=1-0.95=0.05`

Critical value :
`z_(\alpha/2)}=\pm1.96`

Margin of error :
`E=0.03`

Formula to calculate the sample size needed for interval estimate of population proportion :-

`n=p(1-p)((z_(\alpha/2))/(E))^2\\\\=0.2(0.8)((1.96)/(0.03))^2=682.951111111\approx683`

Hence, the required sample size would be 683 .

b) If no estimate of the sample proportion is available then the formula to calculate sample size will be :-

`n=0.25((z_(\alpha/2))/(E))^2\\\\=0.25((1.96)/(0.03))^2=1067.11111111\approx1067`

Hence, the required sample size would be 1067 .