West SouthEast 4 7 8 3 4 4 4 5 6 10 9 7 7 4 3 2 1 2 3 9 MidWest 3 2 4 2 8 3 5 6 2 4 3 3 9 2 3 1 2 1 2 2 1 1 3 2 NewEngland 2 2 8 9 10 12 1 9 9 3 6 5 6 4 4 4 4 2 2 1 1 3 3 4 8 7 3 5 6 2 2 1 4 3 3 5 10 10 9 11 Wyoming Colorado 54.4 50.5 48.3 72.2 71.5 63.3 75.2 64.4 66.3 59.5 66.6 39.7 70.4 44.2 40.5 44.8 44.5 52.5 49.9 55.9 62.6 55.9 55.9 63.3 69.4 39.8 73.3 44.9 71.4 Arizona 60.6 59.9 57.6 50.3 49.7 69.6 70.3 55.9 58.5 61.3 65.2 60.1 49.4 Utah Montana 44.8 62.6 51.5 55.4 55.8 43.4 63.4 42.9 69.5 49.9 72.2 55.8 75.1 70.2 45.8 69.3 49.6 49.8 53.8 39.4 55.2 44.8 68.3 42.4 66.7 56.4 STAT 2050/2070 Statistics Lab Report 4 Guidelines Chi-Square Test for Independence Consider the data given in problem 11 on page 215. We wish to determine if there is an association between the two variables: political party affiliation and the view on background checks. We state the hypotheses first: 0 : the variables are independent (not associated) 1 : the variables are dependent (associated) We use Vassar's College statistics calculator available at: http://vassarstats.net/newcs.html We need to enter the data as follows: Click on 2 for the number of rows, and click on 3 for the number of columns. Then enter the values as shown in problem 11, page 215. Next, click \"Calculate\" to obtain the following results: In this example, df = (r - 1)(c - 1) = 1*2 = 2 (see page 198). So, the critical value (see page 293) is 5.991 (assuming the significance level is 0.05). Since the test statistic is 58.92 (see the figure above), and the p-value is less than 0.0001, we reject the null hypothesis, and conclude that the two variables are dependent. In other words, the political party affiliation does have an influence on the view on background checks. Moreover, since the p-value is very small, the observed differences (see the frequencies) are highly significant. This means that we have very strong evidence in support of the alternative hypothesis. One-Way ANOVA We consider the data in Education by Region. We will compare the means of the four groups. The hypotheses are as follows: 0 : 1 : , . . , We use http://vassarstats.net/; click on ANOVA in the main menu to the left. Then select oneway ANOVA and enter the number of groups (samples). In our data set, there are four groups, so we enter 4: Be sure to select \"Independent Samples.\" Now copy and paste the data from the Excel spreadsheet: Next, click on \"Calculate\" to obtain the following results: Observe that the p-value is 0.037647, and the test statistic is 2.95. Therefore, we reject the null hypothesis, and conclude that at least one mean is significantly different from all the other means. If you look at the means, you will see that the second group's (Southeast) mean is the lowest, while all the others are around 5. We conclude that the average number of years of education beyond high school is significantly lower in the Southeast, and this is why the null hypothesis is rejected. Now the p-value is less than 0.05, but is relatively close to 0.05. Therefore, the observed differences between the means are only moderately significant. In other words, the evidence against the null hypothesis is moderately strong