Question

3) Please answer questions A and B based on the following information:

The director of transportation of a large company is interested in the usage of her van pool. She considers her routes to be divided into local and non-local. She is particularly interested in learning if there is a difference in the proportion of males and females who use the local routes. She takes a sample of a day's riders and finds the following:

	Male	Female
Local	35	36
Non-Local	33	25

She will use this information to perform a chi-square hypothesis test using a level of significance of 0.05.

A) Please develop the Expected frequencies table based on the above information showing all the relevant calculations.

B) Given that π₁ represents the proportion of males who use local routes and π₂ represents the proportion of females who use local routes; perform the Chi-square test of hypothesis for the following and also show all the relevant calculations.

H₀:π₁ = π₂

H₁:π₁ ≠ π₂

Answer 1

	Male	Female	Total
Local	35	36	71
Non local	33	25	58
Total	68	61	129

Part a)

	Expected Table
	Male	Female	Total
Local	(71 * 68) / 129 = 37.4264	(71 * 61) / = 33.5736	71
Non local	(58 * 68) / 129 = 30.5736	(58 * 61) / = 27.4264	58
Total	68	61	129

part b)

Percentage	Oi	Ei	( Oi - Ei )	( Oi - Ei )2	( Oi - Ei )2 / Ei
	35	37.4264	-2.4264	5.8874	0.1573
	36	33.5736	2.4264	5.8874	0.1754
	33	30.5736	2.4264	5.8874	0.1926
	25	27.4264	-2.4264	5.8874	0.2147
Total	129	129	0	23.5496	0.74

Test Statistic :-
X² = Σ (Oi - Ei )² / Ei
X² = 0.74

Test Criteria
Reject null hypothesis if X² > X2(α, (r-1)(c-1))
Critical value X²(0.05, (2-1) (2-1) = X²(0.05,1) = 3.841 ( From chi square table )
Since, 0.74 < 3.841
Conclusion = Fail to reject null hypothesis

Decision based on P value
P (X² > 0.74) = 0.3897
Reject null hypothesis if P value < α = 0.05
P value = 0.3897 > 0.05, hence we fail to reject null hypothesis
Conclusion = Fail to reject null hypothesis

There is sufficient evidence to conclude that H₀:π₁ = π_2.