Question

(a) Find the size of each of two samples (assume that they are of equal size) needed to estimate the difference between the proportions of boys and girls under 10 years old who are afraid of spiders. Assume the "worst case" scenario for the value of both sample proportions. We want a 98% confidence level and for the error to be smaller than 0.02.

b) Again find the sample size required, as in part (a), but with the knowledge that a similar student last year found that the proportion of boys afraid of spiders is 0.45 and the proportion of girls afraid of spiders was 0.58.

Answer 1

(a) The sample sizes are 6787.

(b) The sample sizes are 6666.

Explanation:

(a)

The information provided is:

Confidence level = 98%

MOE = 0.02

n₁ = n₂ = n

`\hat p_(1) = \hat p_(2) = \hat p = 0.50\ (\text{Assume})`

Compute the sample sizes as follows:

`MOE=z_(\alpha/2)*\sqrt{(2*\hat p(1-\hat p))/(n)`

`n=(2*\hat p(1-\hat p)* (z_(\alpha/2))^(2))/(MOE^(2))`

`=(2*0.50(1-0.50)* (2.33)^(2))/(0.02^(2))\\\\=6786.125\\\\\approx 6787`

Thus, the sample sizes are 6787.

(b)

Now it is provided that:

`\hat p_(1)=0.45\\\hat p_(2)=0.58`

Compute the sample size as follows:

`MOE=z_(\alpha/2)*\sqrt{(\hat p_(1)(1-\hat p_(1))+\hat p_(2)(1-\hat p_(2)))/(n)`

`n=((z_(\alpha/2))^(2)* [\hat p_(1)(1-\hat p_(1))+\hat p_(2)(1-\hat p_(2))])/(MOE^(2))`

`=(2.33^(2)* [0.45(1-0.45)+0.58(1-0.58)])/(0.02^(2))\\\\=6665.331975\\\\\approx 6666`

Thus, the sample sizes are 6666.