Answer to Question #188919 in Statistics and Probability for kel

Question

1. In March 2014, the Gallup Organization found in a survey that 42% of U.S. adults think global warming is greatly exaggerated. The number of adults polled was 1,048.
2. Can we say that less than half (significantly more than 50% at an alpha level of .05, using a two-tailed alpha test – this means .025 in each tail) of U.S. adults believe global warming is greatly exaggerated?  Here the p for your H0 (null hypothesis) is p=.50. is the p you should use in your standard error (SE) formula.
3. This same survey finds that 68% Republicans believe global warming is greatly exaggerated compared to 18% Democrats. The survey is that same as above - administered to 1,048 adults, 400 of whom were Republicans and 450 of whom were Democrats (the rest were independents)
4. Using information from question #1, with hypothesis testing can we say that fewer Democrats think global warming is exaggerated than Republicans? Why or why not? Use an alpha level of .01 this time with a two-tailed test (99% confidence).

Answer 1

$(1) n=1048,p=0.42$

(2) Let $H_o: p<0.5$ , Half of the U.S. adults believe that global warming is greatly exaggerated

and $H_a: p>0.5,$ Not half of the US adults believe that global warming is greatly exaggerated.

Here, \hat{p}=0.5,

Test statistics-

$z=\dfrac{0.5-0.42}{\sqrt{\frac{0.42(1-0.42)}{1048}}}=\dfrac{0.08\times 32.37}{0.4935}=5.2474$

$\text{P-value-} \\ p(z<5.2474)=1$

As $p-value>\alpha i.e. 0.05$ , So Null hypothesis is accepted. There is enough evidence to support the claim that Half of the US adults believe that global warming is greatly exaggerated.

(3) p_1=0.68,p_2=0.18,n_1=400,n_2=450

(4) Let $H_o: p_2'<p_1',$  fewer Democrats think global warming is exaggerated than Republicans

and $H_a:p_2'>p_1',$ The claim is wrong.

Test statistics-

$z=\dfrac{p_1-p_2}{\sqrt{\dfrac{p_1(1-p_1)}{n_1}+\dfrac{p_2(1-p_2)}{n_2}}}\\[9pt]=\dfrac{0.68-0.18}{\sqrt{\dfrac{0.68(1-0.68)}{400}+\dfrac{0.18(1-0.18)}{450}}}$

$\\[9pt]=\dfrac{0.5}{0.029}=32.804$

$P(value) =P(z>32.804)=1$

As $p-value>\alpha,$ so Null hypothesis is accepted, There is enough evidence to support the claim that fewer Democrats think global warming is exaggerated than Republicans