The Centers for Disease Control reported the percentage of people 18 years of age and older who smoke&nbsp;⟨CDC&nbsp;website,&nbsp;December&nbsp;14,&nbsp;2014⟩. Suppose that a study designed to collect new data on smokers and nonsmokers uses a preliminary estimate of the proportion who smoke of 30.&nbsp;a.) How large a sample should be taken to estimate the proportion of smokers in the population with a margin of error of .02(to the nearest whole number)? Use 95% confidence.&nbsp;b.) Assume that the study uses your sample size recommendation in part(a) and finds 520 smokers. What is the point estimate of the proportion of smokers in the population(to 4 decimals)?&nbsp;c.) What is the 95% confidence interval for the proportion of smokers in the population (to 4 decimals)?

Question

The Centers for Disease Control reported the percentage of people 18 years of age and older who smoke&amp;nbsp;⟨CDC&amp;nbsp;website,&amp;nbsp;December&amp;nbsp;14,&amp;nbsp;2014⟩. Suppose that a study designed to collect new data on smokers and nonsmokers uses a preliminary estimate of the proportion who smoke of 30.&amp;nbsp;a.) How large a sample should be taken to estimate the proportion of smokers in the population with a margin of error of .02(to the nearest whole number)? Use 95% confidence.&amp;nbsp;b.) Assume that the study uses your sample size recommendation in part(a) and finds 520 smokers. What is the point estimate of the proportion of smokers in the population(to 4 decimals)?&amp;nbsp;c.) What is the 95% confidence interval for the proportion of smokers in the population (to 4 decimals)?

crocolylec · Accepted Answer

a.) p = 0.30, the confidence level is&amp;nbsp;(1–α)=95%, the level of significance is&amp;nbsp;α=0.05.&amp;nbsp;So, E = 0.02, the confidence level is 95%.&amp;nbsp;The critical value corresponding&amp;nbsp;Zα2=1.96&amp;nbsp;from the calculation:1−α=1−0.95&amp;nbsp;α=0.05&amp;nbsp;α2=0.052&amp;nbsp;=0.025&amp;nbsp;Area to the left = 1 - Area to the right = 1 - 0.025 = 0.975&amp;nbsp;Zα2=1.96[Excelf&amp;nbsp;or&amp;nbsp;μla:NORMINV(0.975,0,1)]&amp;nbsp;zα2=1.96,p=0.3,E=0.02.&amp;nbsp;The required sample size is obtained as 2,017:&amp;nbsp;n=p^(1−p^)×(Zα2E)2&amp;nbsp;=(0.3×(1−0.3))×(1.960.02)2&amp;nbsp;=0.21×9,604&amp;nbsp;=2,016.84&amp;nbsp;≈2,017&amp;nbsp;(answer)b.) Among a sample of 2,017 people of age 18 years and older, 520 are identified as smokers, the number of people sampled is n = 2,017, the number of smokers among the sampled people is x = 520, the point estimate of the proportion of smokers in the population is obtained as 0.2578:&amp;nbsp;p^=xn&amp;nbsp;=5202,017&amp;nbsp;=0.2578 (answer)c.) &amp;nbsp;p−=^0.2578,n=2,017&amp;nbsp;and&amp;nbsp;zα2=1.96.&amp;nbsp;CI(p^−Zα2×p^(1−p^)n≤p≤p^+Zα2×p^(1−p^)n&amp;nbsp;&amp;nbsp;=(0.2578−1.96×0.2578(1−0.2578)2,017≤p≤0.2578+1.96×0.2578(1−0.2578)2,017)&amp;nbsp;=(0.2578−0.0191≤p≤0.2578+0.0191)&amp;nbsp;=(0.2387≤p≤0.2769)&amp;nbsp; (answer)

The Centers for Disease Control reported the percentage of people 18 years of age and older who smoke <<CDC website, December 14, 2014>>. Suppose that

Answered question

Answer & Explanation

New Questions in High school statistics