Question

A sample of 340 randomly selected people in their 20s were asked about their favorite place to go on a Friday night. The results are given in

the table, together with a person’s gender:

Movies    Sports game Restaurant Shopping Concert    Total
Men 43    50    46    19 42    200
Women 29    21    31    31 28    140
Total    72    71    77    50 70    340

(a) [1] Give the marginal distribution of gender.

(b) [2] Find the conditional distribution of preferred activity for Women.

Test at a 5% level of significance whether the preferred activity depends on the gender.

(c) [1] State the null and alternative hypotheses.

(d) [1] Calculate the test statistic.

(e) [1] Find the corresponding p-value.

(f) [1] Make the decision according to (e).

(g) [1] Give the conclusion by the decision made from (f).

Answer 1

a)

marginal distribution of gender:

Men : 200 / 340 = 0.5882

Women : 140 / 340 = 0.4118

b) Conditional distribution of preferred activity for Women.

	Sports	Game	Restaurant	Shopping	Concert
Women	29 / 140 = 0.2071	21 / 140 = 0.15	31 / 140 = 0.2214	31 / 140 = 0.2214	28 / 140 = 0.20

c)

Observed Frequencies
	Sports	Game	Restaurant	Shopping	Concert	Total
Men	43	50	46	19	42	200
Women	29	21	31	31	28	140
Total	72	71	77	50	70	340
Expected Frequencies
	Sports	Game	Restaurant	Shopping	Concert	Total
Men	72 * 200 / 340 = 42.3529	71 * 200 / 340 = 41.7647	77 * 200 / 340 = 45.2941	50 * 200 / 340 = 29.4118	70 * 200 / 340 = 41.1765	200
Women	72 * 140 / 340 = 29.6471	71 * 140 / 340 = 29.2353	77 * 140 / 340 = 31.7059	50 * 140 / 340 = 20.5882	70 * 140 / 340 = 28.8235	140
Total	72	71	77	50	70	340
	(fo-fe)²/fe
Men	(43 - 42.3529)²/42.3529 = 0.0099	(50 - 41.7647)²/41.7647 = 1.6239	(46 - 45.2941)²/45.2941 = 0.011	(19 - 29.4118)²/29.4118 = 3.6858	(42 - 41.1765)²/41.1765 = 0.0165
Women	(29 - 29.6471)²/29.6471 = 0.0141	(21 - 29.2353)²/29.2353 = 2.3198	(31 - 31.7059)²/31.7059 = 0.0157	(31 - 20.5882)²/20.5882 = 5.2654	(28 - 28.8235)²/28.8235 = 0.0235

Null and Alternative hypothesis:  

Ho: Preferred activity is independent of the gender.

H1: Preferred activity dependent on the gender.  

Test statistic:  

χ² = ∑ ((fo-fe)²/fe) =    12.9855

df = (r-1)(c-1) =    4

p-value = CHISQ.DIST.RT(12.9855, 4) = 0.0113  

Decision:  

p-value < α, Reject the null hypothesis.  

There is enough evidence to conclude that preferred activity depends on the gender.