Chapter 11.1, Problem 9E

To determine

Verify the facts.

Answer to Problem 9E

No, it is not possible.

Explanation of Solution

Given information:

If a survey with a random sample of $2500$ students is conducted, is it possible that only $20\%$ of students could respond that they text during class?

Calculation:

Sample size of random survey,

  $\begin{array}{l}n=2500\\ \text{Margin of error}=\pm \frac{1}{\sqrt{n}}\left(100\right)\\ =\pm \frac{1}{\sqrt{2500}}\left(100\right)\\ =\pm \frac{1}{50}\left(100\right)\\ =\pm 2\end{array}$

If only $20\%$ of students could respond that they text during class, the sample size be,

  $\begin{array}{l}=2500\times \frac{20}{100}\\ =500\\ \text{Margin of error}=\pm \frac{1}{\sqrt{n}}\left(100\right)\\ =\pm \frac{1}{\sqrt{500}}\left(100\right)\\ =\pm \frac{1}{22.36}\times \left(100\right)\\ =\pm 4.472\end{array}$

As the sample size decreases the margin of error inceases,

Hence, it is not possible that only $20\%$ of students could respond that they text during class, because the range of $20\%$ of students lies in $\pm 4.472$ which is beyond the range of original sample size $\pm 2$ .