a researcher wishes to be 95% confident that her estimate of the true proportion of individuals who travel…

a researcher wishes to be 95% confident that her estimate of the true proportion of individuals who travel overseas is within 4% of the true proportion. find the sample necessary if in a prior study, a sample of 200 people showed that 40 traveled overseas last year. 385 if no estimate of the sample proportion is available, how large should the sample be? 200

a researcher wishes to be 95% confident that her estimate of the true proportion of individuals who travel overseas is within 4% of the true proportion. find the sample necessary if in a prior study, a sample of 200 people showed that 40 traveled overseas last year. 385 if no estimate of the sample proportion is available, how large should the sample be? 200

Answer

Explanation:

Step1: Determine the formula for sample size when no prior estimate is available

When no estimate of the sample proportion (p) is available, we use (p = 0.5) (this gives the maximum value of (p(1 - p))). The formula for sample size (n) in estimating a proportion is (n=\frac{z^{2}_{\alpha/2}\times p(1 - p)}{E^{2}})

For a 95% confidence level, (\alpha=1 - 0.95=0.05), and (\alpha/2=0.025). The (z)-value (z_{\alpha/2}=z_{0.025}). From the standard normal table, (z_{0.025} = 1.96). The margin of error (E = 0.04)

Step2: Substitute the values into the formula

Substitute (z_{\alpha/2}=1.96), (p = 0.5), (1-p=0.5) and (E = 0.04) into the formula:

[ \begin{align*} n&=\frac{(1.96)^{2}\times0.5\times0.5}{(0.04)^{2}}\ &=\frac{3.8416\times0.25}{0.0016}\ &=\frac{0.9604}{0.0016}\ &=600.25 \end{align*} ]

Since the sample size (n) must be an integer, we round up.

Answer:

(n = 601)