Question

A paper described a survey of 500 undergraduate students at a state university in the southwestern region of the United States. Each student in the sample was classified according to class standing (freshman, sophomore, junior, or senior) and body art category (body piercings only, tattoos only, both tattoos and body piercings, no body art).

Use the data in the accompanying table to determine if there is evidence that there is an association between class standing and response to the body art question. Assume that it is reasonable to regard the sample of students as representative of the students at this university. Use

α = 0.01.

	Body Piercings Only	Tattoos Only	Both Body Piercing and Tattoos	No Body Art
Freshman	63	7	15	86
Sophomore	45	11	10	64
Junior	20	9	7	44
Senior	21	17	25	56

State the null and alternative hypotheses.

H₀: Class standing and body art response are not independent.
H_a: Class standing and body art response are independent. H₀: The response proportions are not all the same for all class standings.
H_a: The response proportions are the same for all class standings.     H₀: Class standing and body art response are independent.
H_a: Class standing and body art response are not independent. H₀: The response proportions are the same for all class standings.
H_a: The response proportions are not all the same for all class standings.

Calculate the test statistic. (Round your answer to two decimal places.)
χ² =

What is the P-value for the test? (Round your answer to four decimal places.)
P-value =

What can you conclude?

Reject H₀. There is convincing evidence to conclude that the proportions falling into the body art response categories are not all the same for the four class standings. Reject H₀. There is convincing evidence to conclude that there is an association between class standing and response to the body art question.     Do not reject H₀. There is not enough evidence to conclude that the proportions falling into the body art response categories are not all the same for the four class standings. Do not reject H₀. There is not enough evidence to conclude that there is an association between class standing and response to the body art question.

You may need to use the appropriate table in Appendix A to answer this question.

Answer 1

Expected Values Column 1 Column 2 Column 3 Column 4 Total 250x171 500 149x171 500 44x171 500 57x171 500 =50,958 171 =15.048 - 19.494 - 85.5 44x130 500 149x130 500 57x130 500 250x130 500 38.74 11.44 - 14.82 130 44x80 500 250x80 500 149x80 500 23.84 57x80 9.12 7.04 40 80 44x119 500 57x119 500 149x119 500 250x119 500 =35.462 - 13.566 =10.472 119 = 59.5 Total 149 44 57 250 500

Expected Values Column 1 Column 4 Column 2 Column 3 Total 44x171 500 57x171 500 250x171 500 149x171 500 = 19.494 =50.958 = 15.048 = 85.5 171 149x130 500 44x130 500 57x130 500 250x130 500 = 11.44 38.74 14.82 = 65 130 250x 80 500 44x80 500 57x 80 500 149x80 500 -23.84 = 7.04 = 9.12 40 80 44x119 500 149x119 500 57x119 500 250x119 500 - 35.462 = 10.472 = 13.566 =59.5 119 Total 149 44 57 250 500

Column 3 Squared Distances Column 1 Column 2 Column 4 (63-50.958) 50,958 (15-19.494) 19.494 (86-85.5)2 85.5 (7-15.048)2 15.048 = 4.304 - 2.846 1.036 0.003 (64-65) (45-38.74) 38.74 (10-14.82)2 14.82 (11-11.44) 11.44 - 1.012 - 1.568 0.015 =0.017 65 (9-7.04)2 7.04 (7-9.12)2 9.12 (44-40) 40 (20-23.84)2 23,84 0.493 0,619 0.546 = 0.4 (21-35.462) 35.462 (17-10.472) 10.472 (56-59.5) 59.5 (25-13.566) 13.566 5.898 9.637 0.206 =4.069

Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

H0 : The two variables are independent

Ha : The two variables are dependent

Rejection Region

Based on the information provided, the significance level is α=0.01 , the number of degrees of freedom is df=(4−1)×(4−1)=9, so then the rejection region for this test is R={χ^2 :χ2>21.666}.

Test Statistics

The Chi-Squared statistic is computed as follows:

2.846+1.012+0.619+5.898+4.304+0.017+0.546+4.069+1.036+1.568+0.493+9.637+0.003+0.015+0.4+0.206=32

Decision about the null hypothesis

Since it is observed that χ^2 =32.668>χc^2​=21.666, it is then concluded that the null hypothesis is rejected.

Conclusion

It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that the two variables are dependent, at the 0.01 significance level.

The corresponding p-value for the test is p=Pr(χ^2>32.668)= 0.0002

​ Reject H0. There is convincing evidence to conclude that there is an association between class standing and response to the body art question.