Question

A researcher wishes to​ estimate, with 99​% ​confidence, the population proportion of adults who think the president of their country can control the price of gasoline. Her estimate must be accurate within 2​% of the true proportion. ​a) No preliminary estimate is available. Find the minimum sample size needed. ​b) Find the minimum sample size​ needed, using a prior study that found that 38​% of the respondents said they think their president can control the price of gasoline. ​c) Compare the results from parts​ (a) and​ (b). ​(a) What is the minimum sample size needed assuming that no prior information is​ available?

Answer 1

Answer:

a. n=4148

b. n=3909

c. The sample size is smaller if a known proportion from prior study is used. The difference in sample sizes is 239

Step-by-step explanation:

a. For sample where no preliminary estimate is given, the minimum sample size is calculated using the formula:

$n=p(1-p)(\frac{z}{ME})^2$

Where:

• $ME=$Margin of error
• $p=$ is the assumed proportion

#Let p=0.5, substitute in the formula to solve for n:

$n=0.5(1-0.5)\times (2.576/0.02)^2\\\\=4147.36\approx 4148$

Hence, the minimum sample size is 4148

b. If given a preliminary estimate p=0.38, we use the same formula but substitute p with the given value:

$n=p(1-p)(z/ME)^2\\\\=0.38(1-0.38)(2.576/0.02)^2\\\\=3908.47\approx3909$

Hence, the minimum sample size is 3909

c. Comparing the sample sizes from a and b:

$n_{0.5}>n_{0.38}\\\\n_{0.5}-n_{0.38}=4148-3909=239$

Hence, the actual sample size is smaller for a known proportion from prior a prior study.