Question

15. In the original Star Trek, characters were separated by their rank and station on the Enterprise via differently colored uniforms. We wish to determine whether a character dying on the show was independent of their uniform color. The data on the number of characters seen with each color of uniform, along with whether they died, are given below. Alive Dead Blue 1297 Gold 469 Red 215 24 (a) Which test is appropriate? Why? (b) Write the appropriate hypotheses. (c) Make a table of the expected number of each uniform color/death combination under Ho. (d) What is the value of the test statistic? What is/are the associated degree(s) of freedom? (e) Find the p-value, and state your conclusion in the context of the problem at the a= 0.05 level.

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Answer 1

(a) The chi-square test of independence is appropriate because we want to determine if there is a significant relationship between two nominal (categorical) variables.

(b) The hypothesis being tested is:

H₀: A character dying on the show and uniform color are independent

H_a: A character dying on the show and uniform color are not independent

(c)

		Col 1	Col 2	Total
Row 1	Observed	129	7	136
	Expected	123.35	12.65	136.00
	O - E	5.65	-5.65	0.00
	(O - E)² / E	0.26	2.52	2.78
Row 2	Observed	46	9	55
	Expected	49.88	5.12	55.00
	O - E	-3.88	3.88	0.00
	(O - E)² / E	0.30	2.95	3.25
Row 3	Observed	215	24	239
	Expected	216.77	22.23	239.00
	O - E	-1.77	1.77	0.00
	(O - E)² / E	0.01	0.14	0.15
Total	Observed	390	40	430
	Expected	390.00	40.00	430.00
	O - E	0.00	0.00	0.00
	(O - E)² / E	0.58	5.61	6.19
		6.19	chi-square
		2	df
		.0453	p-value

(d) The test statistic is 6.19. The degrees of freedom is 2.

(e) The p-value is 0.0453.

Since the p-value (0.0453) is less than the significance level (0.05), we can reject the null hypothesis.

Therefore, we have sufficient evidence to conclude that a character dying on the show and uniform color are not independent.

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