In Exercises 29–36, answer the given questions, which are related to percentages.

Chillax USA Today reported results from a Research Now for Keurig survey in which 1458 men and 1543 women were asked this: “In a typical week, how often can you kick back and relax?”

a. Among the women, 19% responded with “rarely, if ever.” What is the exact value that is 19% of the number of women surveyed?

b. Could the result from part (a) be the actual number of women who responded with “rarely, if ever”? Why or why not?

c. What is the actual number of women who responded with “rarely, if ever”?

d. Among the men who responded, 219 responded with “rarely, if ever.” What is the percentage of men who responded with “rarely, if ever.”?

e. Consider the question that the subjects were asked. Is that question clear and unambiguous so that all respondents will interpret the question the same way? How might the survey be improved?

There are 1458 men and 1543 women in the survey.

The question of the survey is, “How often can you kick back and relax?”

a.

Of 1543 women, 19% responded, “Rarely, if ever.”

The percentage symbol implies the value is divided by 100.

Here,19% of the total women surveyed are:

$$\begin{gathered} 19\%\times 1543=\frac{19}{100}\times 1543 \\ =293.17 \end{gathered}$$

Thus, the exact value is 293.17.

b.

The actual count of women who responded “rarely, if ever” is in decimal form. And the expected count is in the whole number.

Thus, the actual count cannot be 293.17.

c.

To estimate the closest actual count of responses, convert the decimal value to a whole number.

In this case, the rounded-off value is 293.

Thus, the actual number of the proposed responses is expected to be close to 293.

d.

Of 1458 men, 219 responded, “Rarely, if ever.”

The fraction of men who gave this response is$\frac{219}{1458}$.

The percentage is obtained by multiplying thefraction by 100.

Therefore,

$\frac{219}{1458}\times 100\approx 15.0206\%$

Thus, the percentage of men who responded is 15.0206%, approximately.

e.

The question of the survey is, “In a typical week,how often can you kick back and relax?”

Each respondent may interpret the question differently as it is asked in qualitative categories, one of them being “rarely, if ever.”

In such a case, the qualitative approach may lead to different interpretations.

The survey can be improved by asking responses in the quantitative format, that is, asking the number of days in a week.