Goodness of Fit Test

7.1.1 Goodness of Fit Test

The test below is for a single categorical variable. The null hypothesis and alternative hypothesis for the goodness of fit test are

H0 : π1 = π1,0,π2 = π2,0,...,πk = πk,0, and

HA : At least one of the probabilities does not equal its hypothesized value.

The π_1,0,π_2,0,…,π_k,0 represent the hypothesized probabilities of each category. The Chi-Squared test statistic is:

∑ 2 χ2 = [Oi---Ei]-, Ei

where O_i is the observed frequency and E_i = nπ_i,0 is the expected frequency. The test statistic follows, approximately, a Chi-squared distribution with k - 1 degrees of freedom.

For example, imagine that it is believed that for mobile phones black is chosen 70% of the time, then pink at 20% and others at 10%. This would be the null hypothesis.

H0 : πblack = .7,πpink = .2 and πothers = .1

The researcher decides to reject the null hypothesis at an alpha level of 0.05. A random sample of 100 people that have mobile phones are asked the color of their phone. From the survey, 75 people have black phones, 10 people pink, and 15 another color. Thus

(75 - 70)2 (10 - 20 )2 (15 - 10)2 χ2 = ---------- + ---------- + ----------= 7.857 70 20 10

which has approximately a Chi-squared distribution with (3-1) degrees of freedom. Thus since 7.857 > 5.991 the researcher decides to reject the null hypothesis and believes that the percent of black, pink and other phones are not the percent previously believed.

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