Ch 15 - Chi-Squared _Categorical Data_

# 45 since we reject h0 for large values of x 2 and we

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Unformatted text preview: erved X 2 test statistic value = 1.44 df = 4 – 1 = 3 . Sketch distribution to find bounds… p‐value is > 0.50. Are the results statistically significant at the 5% significance level? NO Conclusion at a 5% level: It appears that .... the 4 booths are used equally often Aside: Using our frame of reference for chi‐square distributions. Recall that if we have a chi‐square distribution with df degrees of freedom, then the mean is equal to df , and the standard deviation is equal to 2(df ) So, if H0 were true, we would expect the X 2 test statistic to be about 3 give or take about sqrt(2*3) = 2.45 . Since we reject H0 for large values of X 2 , and we only observed a value of 1.44 , even less than expected under H0, we certainly do not have enough evidence to reject H0. Goodness of Fit Test Summary Assume: We have 1 random sample of size n . We measure one discrete response X that has k possible outcomes Test: H0: A specified discrete model for X p1 p10 , p 2 p 20 , , pk pk 0 Ha: The probabilities are not as specified in the null hypothesis. Test Statistic: X 2 observed - expected 2 expected where expected E i np i 0 If H0 is true, then X 2 has a 2 distribution with ( k 1) degrees of freedom, where k is the number of categories. The necessary conditions are: at least 80% of the expected counts are greater than 5 and none are less than 1. Be aware of the sample size (pg 598). 209 Try It! Crossbreeding Peas For a genetics experiment in the cross breeding of peas, Mendel obtained the following data in a sample from the second generation of seeds resulting from crossing yellow round peas and green wrinkled peas. n = 556 Yellow Round Yellow Wrinkled Green Round Green Wrinkled 315 101 108 32 312.75 104.25 104.25 34.75 556(9/16) = 312.75, etc. Do these data support the theory that these four types should occur with probabilities 9/16, 3/16, 3/16, and 1/16 respectively? Use = 0.01. H 0 : p1 9/16 , p 2 3/16 , p3 3/16 , p 4 1/16 . X2 315 312.752 101 104.252 108 104.252 32 34.75...
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