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Ch 15 - Chi-Squared _Categorical Data_

5tofindvariouspercentiles 205

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Unformatted text preview: t discussion is to learn about the chi‐square distribution ‐ what the distribution looks like, some facts, how to use Table A.5 to find various percentiles. 205 The Chi‐Square Distribution General Shape: If we have a chi‐square distribution with df = degrees of freedom, then the ... Mean is equal to df Variance is equal to 2(df) Standard deviation is equal to √[2(df)] These facts will serve as a useful frame of reference for making decision. Table A.5 provides some upper‐tail percentiles for chi‐square distributions. Try It! Consider the 2 (4) distribution. a. What is the mean for this distribution? ___4____ b. What is the median for this distribution? ___3.36_______ c. How likely would it be to get a value of 4 or even larger? Draw a picture to help show it. Area is between 0.25 and 0.50. d. How likely would it be to get a value of 10.3 or even larger? Draw a picture to help show it. Area is between 0.025 and 0.05. This is how bounds for a p‐value will be found 206 The BIG IDEA The data consists of observed counts. We compute expected counts under the H0 ‐ these counts are what we would expect (on average) if the corresponding H0 were true. Compare the observed and expected counts using the X 2 test statistic. The statistic will be a measure of how close the observed counts are to the expected counts under H0. If this distance is large, we have support for the alternative Ha. With this in mind, we turn to our first chi‐square test of goodness of fit. We will derive the methodology for the test through an example. An overall summary of the test will be presented at the end. Test of Goodness of Fit: Helps us assess if a particular discrete model is a good fitting model for a discrete characteristic, based on a random sample from the population. Goodness of Fit Test Scenario: We have one population of interest, say all cars exiting a toll road that has four booths at the exit. Question: Are the four booths used equally often? Data: 1 random sample of 100 cars, we...
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