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 When using the given sample data to construct a 95% confidence interval estimate of the difference between the two population proportions, the result of (-0.0659, 0.00591) is obtained from technology.

a. Express that confidence interval in a format that uses the symbol <.

b. What feature of the confidence interval is a basis for deciding whether there is a significant difference between the proportion of women aware of the statement and the proportion of men who are aware?

Short answer

a.The 95% confidence interval can be expressed as \( - 0.0659 < \left( {{p_1} - {p_2}} \right) < 0.00591\).

b. The presence of 0 in the confidence interval is the feature that is required to conclude whether there is a significant difference between the proportion of women who are aware of the statement and the proportion of men who are aware.

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.

The 95% confidence interval for the difference between the two population proportions is (-0.0659, 0.00591).

Alternate expression of the confidence interval

a.

In general, the confidence interval for the difference between the two population proportions can be expressed as follows:

\(\left( {{{\hat p}_1} - {{\hat p}_2}} \right) - E < \left( {{p_1} - {p_2}} \right) < \left( {{{\hat p}_1} - {{\hat p}_2}} \right) + E\)

The confidence interval is equal to (-0.0659,0.00591).

Here, the value of -0.0659 (lower limit) is comparable to the expression\(\left( {{{\hat p}_1} - {{\hat p}_2}} \right) - E\).

Similarly, the value of 0.00591 (upper limit) is comparable to the expression \(\left( {{{\hat p}_1} - {{\hat p}_2}} \right) + E\).

Therefore, the expression of the confidence interval in terms of the symbol “<” is given below:

\( - 0.0659 < \left( {{p_1} - {p_2}} \right) < 0.00591\)

Interpretation of the Confidence Interval

b.

The confidence interval equal to (-0.0659, 0.00591) contains the value 0.

This suggests that the difference between the two population proportions can be equal to 0.

In simple words, the value of the population proportion of women who were aware of the statement can be equal to the population proportion of men who were aware of the statement.

Thus, there is not sufficient evidence to conclude that there is a significant difference in the proportion of women who were aware of the statement and the proportion of men who were aware of the statement.