The article "Most Customers OK with New Bulbs" (USA Today, Feb. 18,2011 )
describes a survey of 1,016 randomly selected adult Americans. Each person in
the sample was asked if they have replaced standard light bulbs in their home
with the more energy efficient compact fluorescent (CFL) bulbs. Suppose you
want to use the survey data to determine if there is evidence that more than
\(70 \%\) of adult Americans have replaced standard bulbs with CFL bulbs. Let
\(p\) denote the proportion of all adult Americans who have replaced standard
bulbs with CFL bulbs.
a. Describe the shape, center, and spread of the sampling distribution of
\(\hat{p}\) for random samples of size 1,016 if the null hypothesis \(H_{0}:
p=0.70\) is true.
b. Would you be surprised to observe a sample proportion as large as
\(\hat{p}=0.72\) for a sample of size 1,016 if the null hypothesis \(H_{0}:
p=0.70\) were true? Explain why or why not.
c. Would you be surprised to observe a sample proportion as large as
\(\hat{p}=0.75\) for a sample of size 1,016 if the null hypothesis \(H_{0}:
p=0.70\) were true? Explain why or why not.
d. The actual sample proportion observed in the study was \(\hat{p}=0.71\).
Based on this sample proportion, is there convincing evidence that more than
\(70 \%\) have replaced standard bulbs with CFL bulbs, or is this sample
proportion consistent with what you would expect to see when the null
hypothesis is true? Support your answer with a probability calculation.
