Question

12. In a survey of 645 males ages​ 18-64, 394 say they have gone to the dentist in the past year. Construct​ 90% and​ 95% confidence intervals for the population proportion. Interpret the results and compare the widths of the confidence intervals. If​ convenient, use technology to construct the confidence intervals.

The​ 90% confidence interval for the population proportion p is ​( nothing​, nothing​). ​(Round to three decimal places as​ needed.)

The​ 95% confidence interval for the population proportion p is ​( nothing​, nothing​). ​(Round to three decimal places as​ needed.)

Interpret your results of both confidence intervals.

A. With the given​ confidence, it can be said that the sample proportion of males ages​ 18-64 who say they have gone to the dentist in the past year is between the endpoints of the given confidence interval. B. With the given​ confidence, it can be said that the population proportion of males ages​ 18-64 who say they have gone to the dentist in the past year is between the endpoints of the given confidence interval. C. With the given​ confidence, it can be said that the population proportion of males ages​ 18-64 who say they have gone to the dentist in the past year is not between the endpoints of the given confidence interval.

Which interval is​ wider? The​ 90% confidence interval The​ 95% confidence interval

Answer 1

From the given data sample no.of male under the age group 18 - 64 selected at random = n= 645

No. of male under the age group 18- 64 in the sample visited the dentist in the past one year= x=394

sample proportion of male under the age group 18-64 visited the dentist in the past one yeat p= x/n =394/645 = 0.6109

sample proportion of male under the age 18-64 didnt visit the dentist in the past one year q=1-p = 0.3891

1.a) 90% confidence interval for the population proportion of male under the age 18-64 visited dentist in the past one year is p -1.645x(pxq /n) P p + 1.645x(pxq / n)

0.6109 - 1.645 x0.019197 P 0.6109 + 1.645x 0.019197

0.5793 P 0.6425

90 % chance that the proporton of male under the age 18-64 visited dentist in the past one year is (0.579 , 0.6425) i.e it lies in between 57.9% to 64.25%

b) 95% confidence limits for the proportion of male under the age group 18- 64 visited the dentist in the last one year is given as

p -1.96x(pxq /n) P p + 1.96x(pxq / n)

0.6109 - 1.96 x0.019197 P 0.6109 + 1.96x 0.019197

0.5732 P 0.6485

95% confidence limits for the proportion of male under the age 18-64 visited the dentist in the last one year is (0.5732 , 0.6485) i.e it lies in between 57.32% to 64.85%

comparison of width of 1a and 1b

we observe that width of 95% confidence interval is more wider than that of 90% confidence interval

2.a) 90% confidence limits for the population proportion of male under the age 18-64 didnot visit the dentist in the last one year

q -1.645x(pxq /n) Q q + 1.645x(pxq / n)

0.3891 - 1.645 x0.019197 P 0.3891 + 1.645x 0.019197

0.3575 P 0.42067

90% confidence limits for the population proportion of male under the age 18-64 didnot visit the dentist in the last one year is ( 0.3575, 0.42067) i.e it lies in between 35.75% to 42.067%

b) 95 % confidence interval for the population proportion of male under the age 18-64 didnot visit the dentist in the last one year is given as

q -1.96x(pxq /n) Q q+ 1.96x(pxq / n)

0.3891 - 1.96 x0.019197 Q 0.3891+ 1.96x 0.019197

0.35147 Q 0.42672

95% confidence limits for the population proportion of male under the age group 18-64 didnot visit the dentist in the last one year is ( 0.35147, 0.42672) i.e it lies in between 35.147% to 42.67%

comparison between 2a) and 2b)

   We observe that width of 95% confidence limits is more wider than that of 90% confidence limits     

conclusion

A. sample proportion of male under the age 18-64 visit the dentist in the last one year is 61.09%

sample proportion of male in the age group 18-64 didnot visit the dentist in the last one year is 38.91%

B. 90% confidence limits for the population proportion of male under 18-64 visit the dentist in the last one year is (57.93%, 64.25%)

similarly 95% confidence limits for the same is (57.32%, 64.85)

95% confidence limits are more wider than 90% limits

C. 90% confidence interval for the population proportion of male in the age group 18-64 did not visit the dentist in the last one year is (35.75% , 42.067%)

similarly 95% confidence interval for the same is (35.14% ,42.67%)

In this case also we observe that 95% confidence interval is more wider than that of 90% confidence interval

D. 95% confidence interval is wider than 90% confidence interval in both the cases