Confidence Interval

Question

In a survey of 1003 people, 59% said that they have never hesitated to give a handshake because they had a fear of germs. The survey results were reported in USA Today, and the survey was conducted by Wakefield Research for Purell, a supplier of had sanitizer products. 

 

a.) Construct a 95% confidence interval estimate of the proportion of people in the population who have never hesitated to give a handshake because of a fear of germs.

 

b.) Is there anything about the survey that might make the results questionable?

 

c.) If an independent pollster wanted to conduct another survey to confirm or refute the results, how many people must be surveyed? Assume that we want 90% confidence that the sample percentage is within 2.5 percentage points of the true population proportion. 

Answer 1

Part a) The formula to build a confidence interval for a proportion is:

𝑝̂ –𝑧_𝛼/2⋅√(𝑝̂ (1−𝑝̂) < 𝑝 < 𝑝̂ +𝑧_𝛼/2⋅√[𝑝̂ (1−𝑝̂ )]. The problem tells us that n=1003, and the proportion we're being

n n

asked to find is the proportion of people in the population who have never hesitated to give a handshake because of a fear of germs, and we're told in the problem that proportion is 59%. so 59% (0.59), is our 𝑝̂. This means 1−𝑝̂ =1-0.59=0.41, Since the confidence level 𝐶𝐿=0.95, and we know 𝛼=1–𝐶𝐿 = 1–0.95 =0.05. So, 𝛼/2=0.025. When you look this up in the tables, you'll see that 𝑧_𝛼/2=𝑧0.025=1.96. Now that we have all the variables figured out, you can just plug them into the equation to calculate the 95% Confidence Interval. It's worth noting that the

𝑧_𝛼/√2⋅√𝑝̂ (1−𝑝.). is also called the margin of error.

n

Part b) The fact that the survey was conducted by Wakefield Research for Purell, a supplier of hand sanitizer products, would question how they choose the people that they surveyed,

Part c) To calculate the sample size, you can use the same equation as we did in Part a), but be aware that a couple of parameters have changed. First, we only need a 90% CI, so you'll need to look up a different table value to use in the calculation, but the method for how to find that table value follows what we did in Part a). Additionally, this time we're told what the width of the CI needs to be (+{/-2.5%), you'll need to find what 2.5% is in terms of p (true population proportion). In general, when one needs to find a sample size, you need to do a little research to find a good estimate of the parameters you want to estimate from your study, in this case, I would use the 59% as my estimate for the population proportion. Once again, I'm not allowed to do the work for you, but you have the equation & values to plug in it, so you should be able to finish the work from here.