Two random samples were drawn from members of the U.S. Congress. One sample was

Question

Two random samples were drawn from members of the U.S. Congress. One sample was taken from members who are Democrats and the other from members who are Republicans. For each sample, the number of dollars spent on federal projects in each congressperson's home district was recorded.

(i) Make a cluster bar graph showing the percentages of Congress members from each party who spent each designated amount in their respective home districts. (In the graphs, blue represents Democrats and red represents Republicans.)

(ii) Use a 1% level of significance to test whether congressional members of each political party spent designated amounts in the same proportions.
(a) What is the level of significance?


State the null and alternate hypotheses.

H0: Different proportion of Democrats and Republicans within each spending level.
H1: Same proportion of Democrats and Republicans within each spending level.H0: Same proportion of Democrats and Republicans within each spending level.
H1: Same proportion of Democrats and Republicans within each spending level.    H0: Same proportion of Democrats and Republicans within each spending level.
H1: Different proportion of Democrats and Republicans within each spending level.H0: Different proportion of Democrats and Republicans within each spending level.
H1: Different proportion of Democrats and Republicans within each spending level.

(b) Find the value of the chi-square statistic for the sample. (Round the expected frequencies to at least three decimal places. Round the test statistic to three decimal places.)


Are all the expected frequencies greater than 5?

YesNo    

What sampling distribution will you use?

binomialchi-square    normaluniformStudent's t

What are the degrees of freedom?


(c) Find or estimate the P-value of the sample test statistic. (Round your answer to three decimal places.)


(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis of independence?

Since the P-value > , we fail to reject the null hypothesis.Since the P-value > , we reject the null hypothesis.    Since the P-value , we reject the null hypothesis.Since the P-value , we fail to reject the null hypothesis.

(e) Interpret your conclusion in the context of the application.

At the 1% level of significance, there is sufficient evidence to conclude that the proportion of spending for Democrats and Republicans within each level of spending is not the same.At the 1% level of significance, there is insufficient evidence to conclude that the proportion of spending for Democrats and Republicans within each level of spending is not the same.   

Dollars Spent on Federal Projects
in Home Districts Party Less than
5 Billion 5 to 10
Billion More than
10 billion Row Total Democratic 10 15 20 45 Republican 13 22 12 47 Column Total 23 37 32 92 60 50 20 Percentage 30 12 20 10 0 10 billion

Explanation / Answer

top right graph is right

(ii)

a) level of significance =0.01

H0: Same proportion of Democrats and Republicans within each spending level.
H1: Different proportion of Democrats and Republicans within each spending level.

b)

test statistic X2 =3.6739

expected frequencies greater than 5 : Yes

sampling distribution will you use--normaluniform

degrees of freedom---2

c) P-value =0.1593

d) Since the P-value > , we fail to reject the null hypothesis

e)At the 1% level of significance, there is insufficient evidence to conclude that the proportion of spending for Democrats and Republicans within each level of spending is not the same.

Observed O <5 billion 5-10 billion >10 billion Total democratic 10 15 20 45 republic 13 22 12 47 Total 23 37 32 92 Expected E=rowtotal*column total/grand total <5 billion 5-10 billion >10 billion Total democratic 11.250 18.098 15.652 45 republic 11.750 18.902 16.348 47 Total 23 37 32 92 chi square =(O-E)^2/E <5 billion 5-10 billion >10 billion Total democratic 0.139 0.5303 1.2077 1.877 republic 0.133 0.508 1.156 1.797 Total 0.272 1.038 2.364 3.6739

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