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. Dollars Spent on Federal Projects in Home Districts Party Less than 5 Billion 5 to 10 Billion More than 10 billion Row Total Democratic 9 11 25 45 Republican 11 19 17 47 Column Total 20 30 42 92 (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.) Maple Generated Plot Maple Generated Plot Maple Generated Plot Maple Generated Plot (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: 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: Same 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? Yes No What sampling distribution will you use? binomial uniform normal chi-square Student'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.) p-value > 0.100 0.050 < p-value < 0.100 0.025 < p-value < 0.050 0.010 < p-value < 0.025 0.005 < p-value < 0.010 p-value < 0.005 (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 > α, we fail to reject the null hypothesis. Since the P-value > α, we reject the null hypothesis. Since the P-value ≤ α, we reject the null hypothesis. Since the P-value ≤ α, 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.

Answer 1

(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.

(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?

Level of significance is 0.01.

State the null and alternate hypotheses.

H0: Same 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? Yes No What sampling distribution will you use? binomial uniform normal chi-square Student's t What are the degrees of freedom?

Party		Less than 5 Billion	5 to 10 Billion	More than 10 Billion	Total
Democratic	Observed	9	11	25	45
	Expected	9.783	14.674	20.543	45.00
	O - E	-0.78	-3.67	4.46	0.00
	(O - E)² / E	0.06	0.92	0.97	1.95
Republican	Observed	11	19	17	47
	Expected	10.217	15.326	21.457	47.00
	O - E	0.78	3.67	-4.46	0.00
	(O - E)² / E	0.06	0.88	0.93	1.87
Total	Observed	20	30	42	92
	Expected	20.000	30.000	42.000	92.00
	O - E	0.00	0.00	0.00	0.00
	(O - E)² / E	0.12	1.80	1.89	3.82
		3.815	chi-square
		2	df
		.1484	p-value

Find the value of the chi-square statistic for the sample.

The value of the chi-square statistic for the sample is 3.815.

Are all the expected frequencies greater than 5?

Yes

What sampling distribution will you use?

chi-square

What are the degrees of freedom?

The degrees of freedom is 2.

(c) Find or estimate the P-value of the sample test statistic.

p-value > 0.100

(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 > α, 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 insufficient evidence to conclude that the proportion of spending for Democrats and Republicans within each level of spending is not the same.