Testing Goodness of Fit

Definitions / Notation  

Properties of Chi-Square Distribution Not symmetric, it is skewed to the right Values of the chi-square statistic are always greater than or equal to 0 (never negative)

Finding Chi-Square Statistic Determine degrees of freedom (DF = n – 1) Determine the area to the right of the desired value (may have to divide α into tails in certain cases, assume right tail only for this topic) Using DF and “Area to the Right”, look up the chi-square statistic in Table A.4 Critical Values for the χ2 Distribution

Expected Frequencies If the probabilities specified by Ho are p1, p2, …, and the total number of trials is n, the expected frequencies are E1 = np1, E2 = np2, and so on

Hypothesis for Goodness-of-Fit Tests The null hypothesis always specifies a probability for each category. The alternate hypothesis says that some or all of these probabilities differ from the true probabilities of the categories

Classical Approach (By Hand)  

Classical Approach (TI-83/84) Write down a shortened version of claim Come up with null and alternate hypothesis (Ho has a probability for each category while H1 says that some or all of the actual probabilities differ from those specified in Ho) See if claim matches Ho or H1 The picture is always a right tail so put α into the right tail Place observed frequencies into L1 and expected frequencies (np for each category) into L2 Find critical values (Table A.4 using α and DF = k – 1 where k is the number of categories) Find test statistic (χ2GOF-Test) If test statistic falls in tail, Reject Ho. If test statistic falls in main body, Accept Ho. Determine the claim based on step 3

P-Value Approach (TI-84 Plus) Write down a shortened version of claim Come up with null and alternate hypothesis (Ho has a probability for each category while H1 says that some or all of the actual probabilities differ from those specified in Ho) See if claim matches Ho or H1 Find p-value (X2GOF-Test) If p-value is less than α, Reject Ho. If p-value is greater than α, Accept Ho. Determine the claim based on step 3

1. Claim Following are observed frequencies. The null hypothesis is Ho: p1 = 0.25, p2 = 0.20, p3 = 0.50, p4 = 0.05 1 2 3 4 Category Observed 200 150 350 20

Check It Out In the 1950’s-1960’s there was a tendency to grade on a “true curve”: 10% got A’s, 20% got B’s, 40% got C’s, 20% were D’s, and 10% were F’s. Applying this flawed model, an instructor has the following distribution: 94 A’s, 19 B’s, 11 C’s, 2 D’s, and 14 F’s. Does their distribution differ from the “true curve”? Is this instructor inflating their grades?