Question

10.9 Exercise: Approval Rating Let p denote the proportion of Americans who approve President Trump. In a poll of December, 2018, an estimated approval rating for President Trump was 48% among 3,018 registered voters. Treat the sample as a simple random sample (i.e., a fair representation of all registered voters in U.S.). Is this statistical evidence that the population approval rating is less than 50% of all registered voters in the nation? Use a = 0.05 for the significance level. a. State Ho and H in terms of p. b. Is the sample size large enough to perform hypothesis testing for p? c. State the decision rule. d. Calculate the standard error. e. Calculate the Z statistic f. Calculate the p-value. g. Draw the standard normal distribution, and shade the area which represents the p-value. h. Is Ho rejected in favor of H/? i. State the conclusion in the context of the problem. (Try not to use the symbols Ho, H1, and p in your sentence.)

Answer 1

a) Hypothesis: $H_0:p\geq 0.50$ Vs  

b) Since , n=3018 > 30

Therefore , the sample size is large enough

c) The critical value is ,

$Z_{\alpha}=Z_{0.05}=-1.64$

Decision Rule : Reject Ho if $Z_{test}<-1.64$

OR

Reject Ho if p-value < $\alpha=0.05$

d) The standard error is ,

$SE=\sqrt{\frac{P_0(1-P_0)}{n}}=\sqrt{\frac{0.50(1-0.50)}{3018}}=0.0091$

e) The Z statistic is ,

$Z_{stat}=\frac{p-P_0}{SE}=\frac{0.48-0.50}{0.0091}=-2.20$

f) p-value = $P(Z>|Z_{stat}|)=P(Z>2.20)=1-P(Z\leq 2.20)=1-\Phi(2.20)$

$=1-0.9861=0.0139$ ; From standard normal probability table

g)

P-value - P 2=0 Estat

h) Here , p-value=0.0139<$\alpha=0.05$

Therefore , reject Ho.

i) Conclusion : Hence , there is sufficient evidence to support the claim that the population approval rating is less than 50% of all resistered voters in the nation.