Question

An automobile manufacturer ships one model of its economy car in just four colours. the manufacturer’s contract with its dealers states that the colours white, red, blue, and yellows are in the proportions 9:3:3:1. A sample of 400 contains 231 white, 70 red, 79 blue, and 20 yellow cars.

(a) Estimate the proportion of each colour in the manufacturer’s shipments.

(b) Test the hypothesis that the proportions are in the ratio 9:3:3:1. Let α =0.05

Answer 1

a)

Total cars = 400

The proportion for a car = Number of cars of that type/Total number of cars

Colour	Observed values, O	Observed Proportion
White	231	231/400 = 0.58
Red	70	70/400 = 0.18
Blue	79	79/400 = 0.20
Yellow	20	20/400 = 0.05
Total	400	1

b)

Null Hypothesis Ho: The proportions are in the ratio 9:3:3:1

Alternate Hypothesis, Ha: The proportions do no follow the given ratio

We use the Chi-Square test to test this hypothesis

n=400
Colour	Observed, O	Observed Proportion	Expected Proportion	Expected Value, E	(O-E)2/E
White	231	0.58	0.56	225	0.16
Red	70	0.18	0.19	75	0.333333
Blue	79	0.20	0.19	75	0.213333
Yellow	20	0.05	0.06	25	1
Total	400	1	1	400	1.706667

Chi-square value = 1.706667

degree of freedom = No. of colours -1 = 4 -1 = 3

p-value for chi-sqaure = 1.706667 and 3 degree of freedom is 0.635

Since p-value > alpha (0.05), we fail to reject Null and conclude that the proportions are in the ratio 9:3:3:1