Question

Determining Sample Size. In Exercises 31–38, use the given data to find the minimum sample size required to estimate a population proportion or percentage.

IOS MarketshareYou plan to develop a new iOS social gaming app that you believe will surpass the success of Angry Birds and Facebook combined. In forecasting revenue, you need to estimate the percentage of all smartphone and tablet devices that use the iOS operating system versus android and other operating systems. How many smartphones and tablets must be surveyed in order to be 99%. Confident that your estimate is in error by no more than two percentage points?

a.Assume that nothing is known about the percentage of portable devices using the iOS operating system.

b.Assume that a recent survey suggests that about 43% of smartphone and tablets are using the iOS operating system (based on data from NetMarketShare).

c. Does the additional survey information from part (b) have much of an effect on the sample size that is required?

Short answer

a. The required sample size is equal to 4147.

b. The required sample size is equal to 4065.

c. The knowledge of the sample proportions does not have a significant effect on the sample size because the two sample sizes are approximately equal,

Step-by-step solution

Given information

The percentage of portable devices that use the iOS operating system needs to be estimated

The sample size needs to be determined.

The following values are known:

The margin of error is equal to 0.02.

The level of confidence is equal to 99%.

Finding the sample size when the sample proportion is not known

a.

Let $\hat{p}$ denote the sample proportion of smartphone and tablet devices that use the iOS operating system.

Let $\hat{q}$ denote the sample proportion of smartphone and tablet devices that do not use the iOS operating system.

Here, nothing is known about the sample proportions.

The formula for finding the sample size is as follows:

$n=\frac{{\left[z_{\frac{\alpha }{2}}\right]}^{2}0.25}{E^2}$

The confidence level is equal to 99%. Thus, the level of significance is equal to 0.01.

The value of $z_{\frac{\alpha }{2}}$ for $\alpha =0.01$ from the standard normal table is equal to 2.5758.

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

$$\begin{gathered} n=\frac{{\left(2.5758\right)}^2\times 0.25}{{\left(0.02\right)}^2} \\ =4146.72 \\ \approx 4147 \end{gathered}$$

Hence, the required sample size is equal to 4147.

Finding the sample size when the sample proportion is known

b.

The value of $\hat{p}$ is given to be equal to:

$$\begin{gathered} \hat{p}=43\% \\ =\frac{43}{100} \\ =0.43 \end{gathered}$$

Thus, the value of is computed below:

$$\begin{gathered} \hat{q}=1-\hat{p} \\ =1-0.43 \\ =0.57 \end{gathered}$$

The formula for finding the sample size is as follows:

$n=\frac{{\left[z_{\frac{\alpha }{2}}\right]}^2\hat{p}\hat{q}}{E^2}$

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

$$\begin{gathered} n=\frac{{\left(2.5758\right)}^2\times 0.43\times 0.57}{{\left(0.02\right)}^2} \\ =4065.44 \\ \approx 4065 \end{gathered}$$

Hence, the required sample size is equal to 4065.

Comparison

c.

Since the two sample sizes are approximately equal, the knowledge of the sample proportions does not have a significant effect on the sample size.