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

remolatg 2021-02-21 Answered
The Centers for Disease Control reported the percentage of people 18 years of age and older who smoke CDCwebsite,December14,2014. Suppose that a study designed to collect new data on smokers and nonsmokers uses a preliminary estimate of the proportion who smoke of .30.
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.
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)?
c.What is the 95% confidence interval for the proportion of smokers in the population (to 4 decimals)?
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crocolylec
Answered 2021-02-22 Author has 100 answers
Step 1
a.
The given information is as follows:
Given that the preliminary estimate of proportion of smokers among the people of 18 years and older age is 0.30.
That is, p = 0.30.
The confidence level is (1α)=95%.
From this, the level of significance is α=0.05.
Furthermore, it is given that the margin of error is 0.02. That is, E = 0.02.
Step 2
Obtain the critical value for the given situation:
The confidence level is 95%.
The critical value corresponding to the given situation is obtained as Zα2=1.96 from the calculation given below:
1α=10.95
α=0.05
α2=0.052
=0.025
Hence, cumulative area to the left is, Area to the left = 1 - Area to the right
=1-0.025
=0.975
Zα2=1.96[Excelforμla:NORMINV(0.975,0,1)]
Step 3
Obtain the required sample size:
Here, zα2=1.96,p=0.3,E=0.02.
The required sample size is obtained as 2,017 from the calculation given below:
n=p^(1p^)×(Zα2E)2
=(0.3×(10.3))×(1.960.02)2
=0.21×9,604
=2,016.84
2,017(Rounded to nearest whole number)
Step 4
b.
Obtain the point estimate of the proportion of smokers in the population using the sample size obtained in part (a):
It is given that 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 from the calculation given below:
p^=xn
=5202,017
=0.2578(Rounded to 4 decimal places)
Step 5
c.
Obtain the 95% confidence interval for the proportion of smokers in the population:
Here, p=^0.2578,n=2,017andzα2=1.96.
Denote the proportion of smokers in the population as p.
The 95% confidence interval for the proportion of smokers in the population is obtained as 0.2387p0.2769 from the calculation given below:
CI(p^Zα2×p^(1p^)npp^+Zα2×p^(1p^)n
=(0.25781.96×0.2578(10.2578)2,017p0.2578+1.96×0.2578(10.2578)2,017)
=(0.25780.0191p0.2578+0.0191)
=(0.2387p0.2769)
Step 6
Result:
a. The sample size satisfying the given requirement is n = 2,017.
b. The point estimate of the proportion of smokers in the population is 0.2578.
c. The 95% confidence interval for the proportion of smokers in the population is obtained as 0.2387p0.2769.
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