A recent study conducted by the government attempts to determine proportion of people who support further increase in cigarette taxes. In this study, 2500 voting age citizens were sampled, and it was found that 1900 of them were in favor of an increase in cigarette taxes. At level α = 0.05, do you believe that 78% of all citizens are in favor of an increase in cigarette taxes.
(A) State the null and alternative hypotheses and compute the p-value.
(B) What is the smallest α you need to reject the null hypothesis.
My work:
(A) H0: μ = 0.78 and Ha: μ < 0.78. I then found the test statistic at level α = 0.05:
Z = (0.76-0.78)/[sqrt(0.78x0.22)/2500] = -2.4096 = -2.41.
I then found the p-value: P(z<-2.41) = 0.0080.
(B) I found Zα=Z0.05= 1.65 for the critical value, but I'm confused as to how to find the smallest α needed to reject the null hypothesis.
Any help is very appreciated!!
When do you reject $H_0$? You reject it when your p-value is less than the stipulated $\alpha$-level. So the computed p-value is precisely the smallest $\alpha$-level that would been needed, because any higher $\alpha$-level would lead to a rejection, and any lower $\alpha$-level would lead you to fail to reject, as then the p-value would have been higher.