A Survey of 740 shoppers selected randomly from grocery stores in California asked for their favorite brands of cereal. From the following data, determine whether the cereal companies have the same market share in California as nationwide. Test at the a = .01.
CompanyKelloggsGen. MillsPost Quak. OatsOther
Share35% 20% 12% 7% 17%
KellogsGeneral MillPostQuaker OatsOther
22016310995153

Here we want to test the null hypothesis

Ho: The cereal companies have the same market share in California as nationwide

against the alternative hypothesis

Ha: The cereal companies have different market share in California as nationwide

The test statistic for testing Ho is the Chi-Square statistic

follows a Chi-Square distribution with (5-1) = 4 degrees of freedom, where Oi is the observed frequency and Ei = 740*(1/5) = 148 is the expected frequency for the i-th company.

The calculation of the Chi-Square statistic is given in the following table.

Company / Observed Frequency (Oi) / Expected Frequency (Ei) / (Oi-Ei)^2 / [(Oi-Ei)^2]/Ei
Kelloggs / 220 / 148 / 5184 / 35.0270
Gen. Mills / 163 / 148 / 225 / 1.5203
Post / 109 / 148 / 1521 / 10.2770
Quak. Oats / 95 / 148 / 2809 / 18.9797
Other / 153 / 148 / 25 / 0.1689
Total / 740 / Chi-Square Statistic / 65.9730

Thus, the calculated value of the test statistic is

= 65.9730

Since the level of significance a = 0.01, from Chi-square tables with 4 degrees of freedom we get the critical value as 13.277.

Therefore the critical region of the test is

> 13.277

Here = 65.9730 > 13.277.

Therefore we reject the null hypothesis Ho. So we can conclude that the cereal companies do not have the same market share in California as nationwide.

[P-value approach

The p-value of the test is given by,

p-value = P[>65.9730] = 0.0000

Since the p-value is less than a =0.01, we reject the null hypothesis Ho. So we can conclude that the cereal companies do not have the same market share in California as nationwide.]