Chapter 6.4, Problem 30PS

Solution

(a)

To find: It needs to be determined the number of students that were surveyed.

The number of students surveyed were 787.

Given information: A Gallup Youth Survey reported that 23% of students surveyed or about 181 students and it is given that math is their favorite subject in school.

Calculation: The total number of students are assumed as n and 23% of total number of students are equal to 181 so one gets as:

  $\begin{array}{c}23\%n=181\\ \frac{23}{100}\cdot n=181\\ n=181\cdot \frac{100}{23}\\ n\approx 787\end{array}$

Hence the number of students surveyed were 787.

(b)

To find: It needs to be determined the margin of error for the survey.

The margin of error for the survey is $\pm 3.6\%$ .

Given information: A Gallup Youth Survey reported that 23% of students surveyed or about 181 students and it is given that math is their favorite subject in school.

Formula used: The formula that is used for evaluating the margin of error is $\pm \frac{1}{\sqrt{n}}$ .

Calculation: The number of students is 787 so one has $n=787$, thus the margin of error is evaluated as:

  $\begin{array}{c}\pm E=\pm \frac{1}{\sqrt{n}}\\ =\pm \frac{1}{\sqrt{787}}\\ =\pm 0.036\\ =\pm 3.6\%\end{array}$

Hence the margin of error for the survey is $\pm 3.6\%$ .

(c)

To find: It needs to be determined an interval that is likely to contain the exact percent of all the students who would say that math is their favorite subject.

The interval will be $\pm 19.4\%$ to $\pm 26.6\%$ .

Given information: A Gallup Youth Survey reported that 23% of students surveyed or about 181 students and it is given that math is their favorite subject in school.

Explanation: As 23% of students were surveyed then the required interval is with limits as:

  $14\%\pm 3.6\%=19.4\%\text{ to 26}\text{.6\%}$

Hence the interval will be $\pm 19.4\%$ to $\pm 26.6\%$ .