Question

Between September 25 and 28 of 2014, the Pew Research Center asked 101 randomly selected Americans the following question: On which of these activities does the US government currently spend the most money? Is it Social Security, transportation, foreign aid, or interest on the national debt?
20 people said Social Security (the correct answer), while 4 said transportation, 33 said foreign aid, and 26 said interest. 18 said they don’t know or refused to answer.
Remember for a Bernoulli variable the population mean is p and the population variance is p(1−p).

(a) What proportion of the sample responded that the government spends more on foreign aid than the other activities?

(b) Build a 90% confidence interval around this estimate. Now build a 99% confidence interval. Interpret and explain the difference between them.

(c) Using a two-sided test with a 10 percent significance level, can you reject the null hypothesis that the true proportion is .45? What is the p-value for that test? •

(d) Is is possible that more than 34% of the sample believes the government spends more on foreign aid than Social Security? Why or why not?

Answer 1

The 99% CI is wider than the 90% CI. This is because we want to be more confident of having the estimate within the bounds.

pvalue=2*P(Z<-2..49)=0.0128

Hence as t*<-1.645, we reject H0 and say the true proportion is different from 0.45.

d)Yes its possible, As seen in b), the CI includes 0.34 in the bound. 34% is quite close to the mean as well.