Question

question 4
researchers at gallup were interested in whether there was a decrease in the proportion of nonretirees (in other words, people who are not yet retired) who did not believe that the u.s. social security program will be able to pay them a retirement benefit when they retire. in the summer of 2010, the gallup poll found that 60% of nonretirees thought that social security would not be able to pay a benefit. five years later in the summer of 2015, gallup asked the same question to 1,282 nonretirees and found that 51% of respondents did not think that social security would be able to pay a retirement benefit by the time that they retire.
we would like to test the hypothesis that in 2015 there is a lower proportion of nonretirees who do not think that social security will be able to pay them a benefit. a z - test for the population proportion showed that z = - 4.29, p < 0.001.
which of the following are the appropriate hypotheses in this case?
a. ○h₀: p = 0.51 vs. hₐ: p < 0.51
b. ○h₀: p = 0.51 vs. hₐ: p > 0.51
c. ○h₀: p < 0.60 vs. hₐ: p = 0.60
d. ○h₀: p = 0.60 vs. hₐ: p < 0.60
e. ○h₀: p = 0.60 vs. hₐ: p ≠ 0.60

Explanation:

In hypothesis testing for a proportion, the null hypothesis \(H_0\) is a statement of no change. Here, the 2010 proportion is \(p = 0.60\) (the baseline proportion). The alternative hypothesis \(H_a\) is what we are testing for. Since we want to test if there is a decrease (lower proportion) in 2015 compared to 2010, \(H_a\) is \(p<0.60\).

Answer:

D. \(H_0: p = 0.60\) vs. \(H_a: p < 0.60\)