Question

For each of the situations ( #5a-e below) construct the 95% confidence interval estimate to the population. Assume that the results are based on a representative sample of Canadians. Express the estimate in a sentence.

a) The average age of the 925 respondents was 37.3 years with a standard deviation of 2.9.
b) Of the 799 respondents questioned, 126 identified themselves as atheist.
c) Of the 1220 respondents for whom we have information, 222 said that they had
been divorced at least once.
d) Four hundred and thirty two of the 668 respondents questioned said that they
are concerned about the environment.
e) The average number of children for the 945 respondents for which we have
information was 1.67 with a standard deviation of .94.

Answer 1

Answer on Question #49069 – Math – Statistics and Probability

For each of the situations (#5a-e below) construct the 95% confidence interval estimate to the population. Assume that the results are based on a representative sample of Canadians. Express the estimate in a sentence.

a) The average age of the 925 respondents was 37.3 years with a standard deviation of 2.9.

b) Of the 799 respondents questioned, 126 identified themselves as atheist.

c) Of the 1220 respondents for whom we have information, 222 said that they had been divorced at least once.

d) Four hundred and thirty two of the 668 respondents questioned said that they are concerned about the environment.

e) The average number of children for the 945 respondents for which we have information was 1.67 with a standard deviation of .94.

Solution

a) The 95% confidence interval of the mean is

$$\begin{array}{l} \left(\mu - \frac{z_{1-\alpha}}{2} \frac{\sigma}{\sqrt{n}}; \mu + \frac{z_{1-\alpha}}{2} \frac{\sigma}{\sqrt{n}}\right) = \left(37.3 - z_{0.025} \frac{2.9}{\sqrt{925}}; 37.3 + z_{0.025} \frac{2.9}{\sqrt{925}}\right) \\ = \left(37.3 - 1.96 \frac{2.9}{\sqrt{925}}; 37.3 + 1.96 \frac{2.9}{\sqrt{925}}\right) = (37.11; 37.49). \end{array}$$

With 95% confidence level, the mean age of the respondents lies in the interval (37.11; 37.49) years.

b) The 95% confidence interval of the population proportion is

$$\begin{array}{l} \left(p - \frac{z_{1-\alpha}}{2} \sqrt{\frac{p(1-p)}{n}}; p - \frac{z_{1-\alpha}}{2} \sqrt{\frac{p(1-p)}{n}}\right) \\ = \left(\frac{126}{799} - z_{0.025} \sqrt{\frac{\frac{126}{799} \left(1 - \frac{126}{799}\right)}{799}}; \frac{126}{799} + z_{0.025} \sqrt{\frac{\frac{126}{799} \left(1 - \frac{126}{799}\right)}{799}}\right) \\ = \left(0.158 - 1.96 \sqrt{\frac{0.158(1 - 0.158)}{799}}; 0.158 + 1.96 \sqrt{\frac{0.158(1 - 0.158)}{799}}\right) \\ = (0.133; 0.183). \end{array}$$

With 95% confidence level, the proportion of the respondents who identified themselves as atheist lies in the interval (0.133; 0.183).

d) The 95% confidence interval of the population proportion is

$$\begin{array}{l} \left(p - z _ {\frac {1 - \alpha}{2}} \sqrt {\frac {p (1 - p)}{n}}; p - z _ {\frac {1 - \alpha}{2}} \sqrt {\frac {p (1 - p)}{n}}\right) \\ = \left(\frac {432}{668} - z _ {0.025} \sqrt {\frac {\frac {432}{668} \left(1 - \frac {432}{668}\right)}{668}}; \frac {432}{668} + z _ {0.025} \sqrt {\frac {\frac {432}{668} \left(1 - \frac {432}{668}\right)}{668}}\right) \\ = \left(0.647 - 1.96 \sqrt {\frac {0.647 (1 - 0.647)}{668}}; 0.647 + 1.96 \sqrt {\frac {0.647 (1 - 0.647)}{668}}\right) \\ = (0.611; 0.683). \end{array}$$

With 95% confidence level, the proportion of the respondents who had been divorced at least once lies in the interval (0.611; 0.683).

c) The 95% confidence interval of the population proportion is

$$\begin{array}{l} \left(p - z _ {\frac {1 - \alpha}{2}} \sqrt {\frac {p (1 - p)}{n}}; p - z _ {\frac {1 - \alpha}{2}} \sqrt {\frac {p (1 - p)}{n}}\right) \\ = \left(\frac {222}{1220} - z _ {0.025} \sqrt {\frac {\frac {222}{1220} \left(1 - \frac {222}{1220}\right)}{1220}}; \frac {222}{1220} + z _ {0.025} \sqrt {\frac {\frac {222}{1220} \left(1 - \frac {222}{1220}\right)}{1220}}\right) \\ = \left(0.182 - 1.96 \sqrt {\frac {0.182 (1 - 0.182)}{1220}}; 0.182 + 1.96 \sqrt {\frac {0.182 (1 - 0.182)}{1220}}\right) \\ = (0.160; 0.204). \end{array}$$

With 95% confidence level, the proportion of the respondents who are concerned about the environment lies in the interval (0.160; 0.204).

e) The 95% confidence interval of the mean is

$$\begin{array}{l} \left(\mu - z _ {\frac {1 - \alpha}{2}} \frac {\sigma}{\sqrt {n}}; \mu + z _ {\frac {1 - \alpha}{2}} \frac {\sigma}{\sqrt {n}}\right) = \left(1.67 - z _ {0.025} \frac {0.94}{\sqrt {945}}; 1.67 + z _ {0.025} \frac {0.94}{\sqrt {945}}\right) \\ = \left(1.67 - 1.96 \frac {0.94}{\sqrt {945}}; 1.67 + 1.96 \frac {0.94}{\sqrt {945}}\right) = (1.61; 1.73). \end{array}$$

With 95% confidence level, the average number of children for the 945 respondents for which we have information lies in the interval (1.61; 1.73).

www.AssignmentExpert.com