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 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)?
( , )
In order to solve this problem, part a is necessary to lead to the answers in the following problems. So I don't want to worry about problem b and c until I have a figured out. What I have so far is that to find a sample size, I should set up the equation as such:
[(1.96)^2(#)^2]/(.02)^2. I could easily solve this problem but I just don't know how to solve for the missing value.
To solve for the sample size (#) in the equation [(1.96)^2(#)^2]/(.02)^2, we can rearrange the equation to isolate (#). Here's how:
1. Start with the equation: [(1.96)^2(#)^2]/(.02)^2 = ?
2. Rearrange the equation to solve for (#): (#)^2 = [(1.96)^2(.02)^2]/1
3. Simplify the right side of the equation: (#)^2 = (1.96)^2(.02)^2
4. Take the square root of both sides of the equation to solve for (#): # = √[(1.96)^2(.02)^2]
5. Use a calculator to compute the right side of the equation: # ≈ √[3.8416 * 0.0004]
6. Calculate the square root: # ≈ √0.00153664
7. Calculate the approximate value of # to the nearest whole number: # ≈ 0.0392
Since we cannot have a fraction of a sample, we round up the value to the nearest whole number. Therefore, the sample size needed is approximately 1.
Now that we have the sample size recommendation, we can move on to parts (b) and (c) of the problem.
To solve for the missing value in the equation [(1.96)^2(#)^2]/(.02)^2, you need to rearrange the equation to solve for the sample size (#). Here's the step-by-step process:
1. Start with the equation: [(1.96)^2(#)^2]/(.02)^2
2. Multiply both sides of the equation by (.02)^2: [(1.96)^2(#)^2] = (.02)^2 * [(1.96)^2(#)^2]
3. Divide both sides of the equation by (.02)^2: [(1.96)^2(#)^2] / (.02)^2 = [(1.96)^2(#)^2) / (.02)^2 * (.02)^2
4. Take the square root of both sides of the equation: √{[(1.96)^2(#)^2]/(.02)^2} = √{[(1.96)^2(#)^2) / (.02)^2 * (.02)^2}
5. Simplify the equation: √[(1.96)^2(#)^2] = 1.96 * # = √[(1.96)^2 *(#)^2]
6. Divide both sides of the equation by 1.96: √[(1.96)^2 * (#)^2]/1.96 = # * 1.96/1.96
7. Simplify the equation: √((1.96)^2 * (#)^2)/1.96 = #
8. Simplify further: (1.96 * #)/1.96 = #
9. Cancel out the 1.96 terms: # = #
Therefore, the missing value in the equation is (#). This means that the equation as written represents the sample size needed to estimate the proportion of smokers in the population with a margin of error of .02.