Question

Suppose a random sample of 10,000 individuals is asked to identify their favorite brand of soap among ten choices. The following results from the survey are obtained:

Observed
Brand Frequency

A             1200
B             900
C             850
D             1160
E                1020
F               975
G            1100
H            980
I              1035
J               780

Test the hypothesis that the preferences for each brand are equal (or uniform). Test this at the 0.05 level.

Answer 1

For uniform distribution expected proportion = 1/10 = 0.1

The following table is obtained:

Categories	Observed	Expected	(fo-fe)2/fe
A	1200	10000*0.1=1000	(1200-1000)2/1000 = 40
B	900	10000*0.1=1000	(900-1000)2/1000 = 10
C	850	10000*0.1=1000	(850-1000)2/1000 = 22.5
D	1160	10000*0.1=1000	(1160-1000)2/1000 = 25.6
E	1020	10000*0.1=1000	(1020-1000)2/1000 = 0.4
F	975	10000*0.1=1000	(975-1000)2/1000 = 0.625
G	1100	10000*0.1=1000	(1100-1000)2/1000 = 10
H	980	10000*0.1=1000	(980-1000)2/1000 = 0.4
I	1035	10000*0.10=1000	(1035-1000)2/1000 = 1.225
J	780	10000*0.1=1000	(780-1000)2/1000 = 48.4
Sum =	10000	10000	159.15

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

H_0: p_1 = 0.1, p_2 = 0.1, p_3 = 0.1, p_4 = 0.1, p_5 = 0.1, p_6 = 0.1, p_7 = 0.1, p_8 = 0.1, p_9 = 0.10, p_10 = 0.1

H_a​: Some of the population proportions differ from the values stated in the null hypothesis

This corresponds to a Chi-Square test for Goodness of Fit.

(2) Rejection Region

Based on the information provided, the significance level is α = 0.05, the number of degrees of freedom is df = 10 - 1 = 9.

(3) Test Statistics

The Chi-Squared statistic is computed as follows:

= 40+10+22.5+25.6+0.4+0.625+10+0.4+1.225+48.4

= 159.15

(4) Decision about the null hypothesis

Since it is observed that = 159.15 > = 16.919, it is then concluded that the null hypothesis is rejected.

(5) Conclusion

It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that some of the population proportions differ from those stated in the null hypothesis, at the α = 0.05 significance level.