Question

Find the minimum sample size you should use to assure that your estimate of pÌ will be within the required margin of error around the population p. Assume the Margin of error is 0.08 with a confidence level of 95%. From a prior study, pÌ is estimated to be 0.2.

Answer 1

The minimum sample size required is 97.

Explanation:

The (1 - α)% confidence interval for the population proportion is:

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

The margin of error for this confidence interval is:

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

The information provided is:

`\hat p=0.20\\MOE=0.08\\\text{Confidence level}=95\%`

The critical value of z for 95% confidence level is, z = 1.96.

Compute the minimum sample size required as follows:

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

`n=[(z_(\alpha/2)\cdot√(\hat p(1-\hat p)))/(MOE)]^(2)\\\\=[(1.96* √(0.20(1-0.20)))/(0.08)]^(2)\\\\=96.04\\\\\approx 97`

Thus, the minimum sample size required is 97.