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# 10 - STAT E-50 Introduction to Statistics The Chi-Square...

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STAT E-50 - Introduction to Statistics The Chi-Square Test The chi-square test is a nonparametric test that is used to compare experimental results with theoretical models. That is, we will be comparing observed frequencies with expected frequencies . In a hypothesis test, the expected frequencies are those we would expect if the null hypothesis of our test is true. ( ) 2 OE E χ 2 = The formula is: where O represents the observed frequency and E represents the expected frequency. The value of df depends on the type of test you are performing.

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The Chi-Square Distribution The P 2 distribution is: • nonnegative • not symmetrical; it is skewed to the right • distributed to form a family of distributions, with a separate distribution for each different number of degrees of freedom. Page 2
Page 3 The Chi-Square Test for Goodness of Fit The goodness-of-fit test compares the distribution of observed outcomes for a single categorical variable to the expected outcomes predicted by a probability model. This test involves one sample, and one variable. Assumptions and Conditions: Counted data condition Be sure that the data is counts, or frequencies Independence assumption Randomization condition Sample size assumption Expected cell frequency condition: each expected frequency is at least 5

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Page 4 The Chi-square test is one-sided. 0 P 2 (df, α )
Page 5 Automobile insurance is much more expensive for teenage drivers than for older drivers. To justify this cost difference, insurance companies claim that the younger drivers are much more likely to be involved in costly accidents. To test this claim, a researcher obtains information about registered drivers from the Department of Motor Vehicles and selects a sample of 300 accident reports from the police department. The DMV reports the percentage of registered drivers in each age category as reported below. The number of accident reports is also shown. Does this data indicate that accidents occur with the same distribution as the ages of the drivers? H 0 : The distribution of accidents is the same as the distribution of registered drivers. H a : The distribution of accidents is not the same as the distribution of registered drivers. AGE percent of drivers number of accidents (observed) expected O - E () 2 OE 2 E under age 20 16 68 age 20 - 29 28 92 age 30 or older 56 140

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Page 6 Check the conditions: Counted data condition Randomization condition Expected cell frequency condition AGE percent of drivers number of accidents (observed) expected O - E () 2 OE 2 E under age 20 16 68 age 20 - 29 28 92 age 30 or older 56 140
Page 7 Check the conditions: Counted data condition Randomization condition Expected cell frequency condition AGE percent of drivers number of accidents (observed) expected O - E () 2 OE 2 E under age 20 16 68 48 age 20 - 29 28 92 84 age 30 or older 56 140 168 n = 300 300 Note: Σ observed = Σ expected

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Page 8 Check the conditions: Counted data condition Randomization condition Expected cell frequency condition - each expected frequency 5 AGE percent of drivers number of accidents
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10 - STAT E-50 Introduction to Statistics The Chi-Square...

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