Question

ComputersIn order to better plan for student resources, the chairperson of the mathematics department at Broward College wants to estimate the percentage of students who own a computer. If we want to estimate that percentage based on survey results, how many students must we survey in order to be 90% confident that we are within four percentage points of the population percentage? Assume that the number of students is very large.

Short answer

The number of students that must be surveyed to be 90% confident is 423.

Step-by-step solution

Given information

The level of confidence is 90%.

The margin of error is $E=0.04$.

Compute the sample size

The level of confidence is 90%, which implies that the level of significance is 0.10.

From the Z table, the critical value at 0.10 level of significance is 1.645.

Since no prior information on the sample proportion $\hat{p}$is given, the number of students that must be surveyed to be 90% confident is computed as follows:

role="math" localid="1648195763160" $$\begin{gathered} \mathit{n}\mathbf{=}\frac{{\left(z_{\frac{\alpha }{2}}\right)}^{\mathbf{2}}\mathbf{0}\mathbf{.}\mathbf{25}}{{\mathbf{E}}^{\mathbf{2}}} \\ =\frac{{1.645}^2\times 0.25}{{0.04}^2} \\ =422.8 \\ \approx 423 \end{gathered}$$

Therefore, the number of students that must be surveyed to be 90% confident is 423.