Question

In Exercises 1–5, use the following survey results: Randomly selected subjects were asked if they were aware that the Earth has lost half of its wildlife population during the past 50 years. Among 1121 women, 23% said that they were aware. Among 1084 men, 26% said that they were aware (based on data from a Harris poll).

Biodiversity Find the values of \({x_1}\)(the number of women who were aware of the statement), \({x_2}\) (the number of men who were aware of the statement),\({\hat p_1}\), \({\hat p_2}\)and the pooled proportion \(\bar p\)obtained when testing the claim given in Exercise 1.

Short answer

The value of \({x_1}\) is equal to 258.

The value of \({x_2}\) is equal to 282.

The value of \({\hat p_1}\) is equal to 0.23.

The value of \({\hat p_2}\) is equal to 0.26.

The value of \(\bar p\) is equal to 0.245.

Step-by-step solution

Given information

In a sample of 1121 women, 23% said that they were aware of the fact that the Earth has lost half of its wildlife population during the past 50 years. In another sample of 1084 men, 26% said that they were aware that the Earth had lost half of its wildlife population during the past 50 years.

Values of \({x_1}\) and \({x_2}\)

It is given that\({x_1}\)is the number of women who were about the fact.

Thus, the value of\({x_1}\)is computed below:

\(\begin{array}{c}{x_1} = 23\% \;{\rm{of}}\;1121\\ = \frac{{23}}{{100}} \times 1121\\ \approx 258\end{array}\)

Thus, the value of\({x_1}\)is equal to 258.

Similarly,

It is given that\({x_2}\)is the number of men who were about the fact.

Thus, the value of\({x_2}\)is computed below:

\(\begin{array}{c}{x_2} = 26\% \;{\rm{of}}\;1084\\ = \frac{{26}}{{100}} \times 1084\\ \approx 282\end{array}\)

Thus, the value of \({x_2}\) is equal to 282.

Values of \({\hat p_1}\) and \({\hat p_2}\)

Let\({\hat p_1}\)denote the sample proportion of women who were about the stated fact.

It is given that out of 1121 women, 23% said that they were aware of the fact.

Thus,\({\hat p_1}\)is equal to:

\(\begin{array}{c}{{\hat p}_1} = 23\% \\ = \frac{{23}}{{100}}\\ = 0.23\end{array}\)

Therefore, the value of\({\hat p_1}\)is equal to 0.23.

Let\({\hat p_2}\)denote the sample proportion of men who were about the stated fact.

It is given that out of 1084 men, 26% said that they were aware of the fact.

Thus,\({\hat p_2}\)is equal to:

\(\begin{array}{c}{{\hat p}_1} = 26\% \\ = \frac{{26}}{{100}}\\ = 0.26\end{array}\)

Therefore, the value of \({\hat p_2}\) is equal to 0.26.

Value of \(\bar p\)

Let\({n_1}\)denote the sample size corresponding to women.

\({n_1}\)is equal to 1121.

Let\({n_2}\)denote the sample size corresponding to men.

\({n_2}\)is equal to 1084.

The value of the pooled proportion is equal to:

\(\begin{array}{c}\bar p = \frac{{\left( {{x_1} + {x_2}} \right)}}{{\left( {{n_1} + {n_2}} \right)}}\\ = \frac{{\left( {258 + 282} \right)}}{{\left( {1121 + 1084} \right)}}\\ = 0.245\end{array}\)

Thus, the value of \(\bar p\) is equal to 0.245.