Question

Based on sales over a six-month period, the five top-selling compact cars are Chevy Cruze, Ford Focus, Hyundai Elantra, Honda Civic, and Toyota Corolla.† Based on total sales, the market shares for these five compact cars were Chevy Cruze 24%, Ford Focus 21%, Hyundai Elantra 20%, Honda Civic 18%, and Toyota Corolla 17%. Suppose a sample of 400 compact car sales in one city showed the following number of vehicles sold.

Chevy Cruze	109
Ford Focus	93
Hyundai Elantra	64
Honda Civic	83
Toyota Corolla	51

Use a goodness of fit test to determine if the sample data indicate that the market shares for the five compact cars in this city are different than the market shares reported by Motor Trend. Use a 0.05 level of significance.

(a) State the null and alternative hypotheses.

A. H₀: The majority of the market shares for the five automobiles in this city differ from the ones reported by Motor Trend.
H_a: The majority of the market shares for the five automobiles in this city are the same as the ones reported by Motor Trend.

B. H₀: The market shares for the five automobiles in this city differ from 0.24, 0.21, 0.20, 0.18, 0.17.
H_a: The market shares for the five automobiles in this city are the same as the above shares.    

C. H₀: The majority of the market shares for the five automobiles in this city are the same as the ones reported by Motor Trend.
H_a: The majority of the market shares for the five automobiles in this city differ from the ones reported by Motor Trend.

D. H₀: The market shares for the five automobiles in this city are 0.24, 0.21, 0.20, 0.18, 0.17.
H_a: The market shares for the five automobiles in this city differ from the above shares.

(b) Find the value of the test statistic. (Round your answer to three decimal places.)

(c) Find the critical value for the test. (Round your answer to three decimal places.)

(d) State your conclusion.

A. Reject H₀. We cannot conclude that the market shares for the five compact cars in this city differ from the market shares reported.

B. Do not reject H₀. We conclude that the market shares for the five compact cars in this city differ from the market shares reported.     

C. Reject H₀. We conclude that the market shares for the five compact cars in this city differ from the market shares reported.

D. Do not reject H₀. We cannot conclude that the market shares for the five compact cars in this city differ from the market shares reported.

Answer 1

a)

D. H₀: The market shares for the five automobiles in this city are 0.24, 0.21, 0.20, 0.18, 0.17.
H_a: The market shares for the five automobiles in this city differ from the above shares.

b)

applying chi square goodness of fit test:

	relative	observed	Expected	residual	Chi square
Category	frequency(p)	Oi	Ei=total*p	R2i=(Oi-Ei)/√Ei	R2i=(Oi-Ei)2/Ei
1	0.24	109	96.00	1.33	1.76
2	0.21	93	84.00	0.98	0.96
3	0.20	64	80.00	-1.79	3.20
4	0.18	83	72.00	1.30	1.68
5	0.17	51	68.00	-2.06	4.25
total	1	400	400		11.86
test statistic X2=		11.855

c)

degree of freedom =categories-1=			4
for 0.05 level and 4 df :crtiical value X2 =			9.488	from excel: chiinv(0.05,4)

d)

since test statistic falls in rejection region we reject null hypothesis

C. Reject H₀. We conclude that the market shares for the five compact cars in this city differ from the market shares reported.