Ch 15 - Chi-Squared _Categorical Data_


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Unformatted text preview: tistic Our measure of how close the observed counts are to what we expect under the null hypothesis. 21 33.352 36 19.97 2 30 23.682 48 35.652 26 32.032 19 25.322 X2 33.35 29.97 23.68 35.65 32.03 25.32 14.5 Do you think a value of X 2 14.5 is large enough to reject H0? The next step is to find the p‐value, the probability of getting an X 2 test statistic value as large or larger than the one we observed, assuming H0 is true. To do this we need to know the distribution of the X 2 test statistic under the null hypothesis. If H0 is true, then X 2 has the 2 distribution with degrees of freedom = (r‐1)(c‐1) 217 Aside: Using our frame of reference for chi‐square distributions. If H0 were true, we would expect the X 2 test statistic to be about 2 give or take about sqrt(2*2) = 2 . About how many standard deviations is the observed X 2 value of 14.5 from the expected value under H0? What do you think the decision will be? (14.5 – 2)/2 = 6.25 about 6 standard deviations above the expected value under H0. Find the p‐value for our factory worker example: Observed X 2 test statistic value = 14.5 df = 2 Find the p‐value and use it to determine if the results are statistically significant at the 1% significance level. Sketch the distribution to show the bounds are: p‐value < 0.001 So the results are statistically significant at the 1% level. Conclusion at a 1% level: It appears that .... there is an association between smoking and hypertension for the population of factory workers represented by this sample. Test of Independence Summary Assume: We have 1 random sample of size n . We measure 2 discrete responses: X which has r possible outcomes and Y which has c possible outcomes. Test: H0: The two variables X and Y are independent for the population. Test Statistic: X 2 observed - expected2 expected (row total)(column total) where expected Total n If H0 is true, then X 2 has a 2 distribution with ( r 1)(c 1) degrees of freedom. The necessary conditions are: at least 80% of the expected counts are greater than 5 and none are less than 1. 218 Relationship between Age Group and Appearance Satisfaction Are you satisfied with your overall appearance? A random sample of 150 women were surveyed. Their answer to this question (Yes or No) was recorded along with their age category (1 = under 30, 2 = 30 to 50, and 3 = over 50). SPSS was used to generate the following output from the data. Are You Satisfied? * Age Group Crosstabulation Count Are You Satisfied? Yes No Total under 30 38 10 48 Age Group 30 to 50 30 29 59 over 50 34 9 43 Total 102 48 150 Chi-Square Tests Pearson Chi-Square Likelihood Ratio Linear-by-Linear Association N of Valid Cases Value 13.149 13.039 .018 df 2 2 Asymp. Sig. .001 .001 1 .893 150 a. Give the name of the test to be used for assessing if there is a relationship between age group and appearance satisfaction. __ Chi‐squared test of independence ______________________ b. Assuming there is no relationship between age group and app...
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