91Ó°ÊÓ

The percentage of people who play video games is to be estimated.

The sample size needs to be determined.

The following values are known:

The margin of error is equal to 0.03.

The confidence level is equal to 90%.

a.

Let \(\hat p\) denote the sample proportion of people who play video games.

Let \(\hat q\) denote the sample proportion of people who do not play video games.

Here, nothing is known about the sample proportions.

The formula for finding the sample size is as follows:

\(n = \frac{{{{\left( {{z_{{\alpha \mathord{\left/

{\vphantom {\alpha 2}} \right.

\kern-\nulldelimiterspace} 2}}}} \right)}^2}0.25}}{{{E^2}}}\)

The confidence level is equal to 90%. Thus, the level of significance is equal to 0.10.

The value of \({z_{\frac{\alpha }{2}}}\) for \(\alpha = 0.10\) from the standard normal table is equal to 1.645.

Substituting the required values, the following value of the sample size is obtained:

\(\begin{array}{c}n = \frac{{{{\left( {1.645} \right)}^2} \times 0.25}}{{{{\left( {0.03} \right)}^2}}}\\ = 751.67\\ \approx 752\end{array}\)

Hence, the required sample size is equal to 752.

b.

The value of \(\hat p\) is given to be equal to:

\(\begin{array}{c}\hat p = 16\% \\ = \frac{{16}}{{100}}\\ = 0.16\end{array}\)

Thus, the value of is computed below:

\(\begin{array}{c}\hat q = 1 - \hat p\\ = 1 - 0.16\\ = 0.84\end{array}\)

The formula for finding the sample size is as follows:

\(n = \frac{{{{\left( {{z_{{\alpha \mathord{\left/

{\vphantom {\alpha 2}} \right.

\kern-\nulldelimiterspace} 2}}}} \right)}^2}\hat p\hat q}}{{{E^2}}}\)

Substituting the required values, the following value of the sample size is obtained:

\(\begin{array}{c}n = \frac{{{{\left( {1.645} \right)}^2} \times 0.16 \times 0.84}}{{{{\left( {0.03} \right)}^2}}}\\ = 404.10\\ \approx 404\end{array}\)

Hence, the required sample size is equal to 404.

c.

A sample consisting of people that are easily approachable or the people that you know is a convenience sample and is not a simple random sample.

For estimating the population proportion of people who play video games, a simple random sample that represents the entire population needs to be considered.

Thus, a convenience sample consisting of people you know should not be taken to reach accurate conclusions.