## Related questions with answers

Use the given data to find the minimum sample size required to estimate a population proportion or percentage. An investor is considering funding of a new video game. She wants to know the worldwide percentage of people who play video games, so a survey is being planned. How many people must be surveyed in order to be 90% confident that the estimated percentage is within three percentage points of the true population percentage? Assume that about 16% of people play video games (based on a report by Spil Games).

Solution

VerifiedGiven:

$c=90\%=0.90$

$E=3\%=0.03$

$\hat{p}=16\%=0.16$

Formula sample size:

$\hat{p}\text{ known: }n=\dfrac{[z_{\alpha/2}]^2\hat{p}\hat{q}}{E^2}=\dfrac{[z_{\alpha/2}]^2\hat{p}(1-\hat{p})}{E^2}$

$\hat{p}\text{ unknown: }n=\dfrac{[z_{\alpha/2}]^2 0.25}{E^2}$

For confidence level $1-\alpha=0.90$, determine $z_{\alpha/2}=z_{0.05}$ using using the normal probability table in the appendix (look up 0.05 in the table, the z-score is then the found z-score with opposite sign):

$z_{\alpha/2}=1.645$

$\hat{p}$ is known, then the sample size is (round up to the nearest integer!):

$n=\dfrac{[z_{\alpha/2}]^2 \hat{p}(1-\hat{p})}{E^2}=\dfrac{1.645^2\times 0.16(1-0.16)}{0.03^2}\approx 405$

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