Question

23) In a recent year , the most popular colors for light trucks were white, 29%; black, 17%; silver, 13%; red, 12%; gray, 9%; blue, 8%; and other, 12%. A survey of randomly selected light truck owners, in a particular area revealed the following. At alpha equal to 0.05, do the proportions differ from these stated? Other Red Black Silver Gray Blue White 15 22 33 31 48

Answer 1

Solution:

Level of significance =

g= 0,05

Step 1) State H0 and H1:

H0: The proportions do not differ from stated proportions of colors for light trucks

Vs

H1: The proportions differ from stated proportions of colors for light trucks

Step 2) Find test statistic value:

Chi square test statistic for goodness of fit

AN (=^

Where

O_i = Observed Counts

E_i =Expected Counts =N * % value

Thus we need to make following table

Color	Oi: Observed frequency	Expected % values	Ei: Expected Frequencies	Oi^2/Ei
White	48	29%	53.36	43.178
Black	31	17%	31.28	30.723
Silver	30	13%	23.92	37.625
Red	33	12%	22.08	49.321
Gray	22	9%	16.56	29.227
Blue	15	8%	14.72	15.285
Other	5	12%	22.08	1.132
	N =184			206.492

Thus

AN (=^

x2 = 206.492 – 184

x2 = 22.492

Step 3)

Find Chi-square critical value:

df = k - 1 = 7 -1 = 6

Level of significance = 0.05

00 15 750 X 050 تا من x95 0.000 0.010 (0.072 (1.207 (1.412 0.000 0.020 0.11 0.2017 0.54 0.001 0.004 0.01 0.103 0216 0.32 0.484 0.711 0.831 1.145 T /THEL 0.016 0.211 0384 1.064 1,610 5 20 X100 2.70 4.605 6231 7779 9236 | 10 1001 | 815 88 1L.70 | | 12.592 | | ا- | 0 . 82

Chi-square critical value = 12.592

Step 4) Decision Rule:
Reject null hypothesis H0, if Chi square test statistic > Chi-square critical value = 12.592 , otherwise we fail to reject H0.
Since Chi square test statistic = > Chi-square critical value = 12.592 , we reject null hypothesis H0.

x2 = 22.492

Step 5) Conclusion:

At 0.05 level of significance , data provide sufficient evidence to conclude that: the proportions differ from stated proportions of colors for light trucks