The following table contains the number of successes and...

The following table contains the number of successes and failures for three categories of a variable. Test whether the proportions are equal for each category at the \(\alpha = 0.1\) level of significance. Failures Successes Category 1 Category 2 Category 3 74 79 58 68 76 78 Click the icon to view the Chi-Square table of critical values. B. \(H_0\): The categories of the variable and success and failure are independent. \(H_1\): The categories of the variable and success and failure are dependent. C. \(H_0\): The categories of the variable and success and failure are dependent. \(H_1\): The categories of the variable and success and failure are independent. D. \(H_0: \mu_1 = E_1 \text{ and } \mu_2 = E_2 \text{ and } \mu_3 = E_3\) \(H_1\): At least one mean is different from what is expected. Compute the value of the chi-square test statistic. \(\chi^2_0 = \) (Round to three decimal places as needed.)

Transcript

00:01 So we have this table here which contains the number of failures and successes for three different categories and we want to test whether the proportions are equal for each category at the alpha of 0 .1 level of significance.

00:18 So what that means is we're going to get a p -value and if the p -value falls below this alpha we're going to reach the null hypothesis.

00:23 So the null hypothesis is that the proportions are all equal, p1 equal to p2 equals to p3.

00:28 The alternative is that at least one of the proportions is different from the others.

00:32 And so here are the p -hat values, the estimates, and i did the successes.

00:37 So i looked at the successes, so s over n, and what i have here i totaled the columns, total for cat1, cat2, and cat3, 88, 140, 141, and the number of successes divided by the total number there.

00:53 And as you can see they're not all equal but the question is are they different enough to say that there's some difference among them, some combination that's different.

01:01 If we're looking at 1, this is close to 3 but 2 is different from quite a bit lower than 1 so we're going to see is that difference significant.

01:10 And for this we're going to do a chi -squared test for, or a chi -squared goodness of fit test because we're saying hey, is this the fit, are all the proportions equal, is that a good fit.

01:27 So let's get our expected values because that's going to tell us what we'd expect based on what we observed.

01:33 So the expected values are found by taking the total in the row multiplied by the total in the column divided by the total in the sample.

01:41 So those values are here and i tallied across the rows too and here are the expected values and a little bit of i think about the geometry here.

01:51 So we observed for example 40 failures in category 1, we'd expect 48 and the way i got that are 48 .17.

01:59 I got that number by taking the total in the row, 202, multiplied by the total in the column, 88, divided by the total in the sample, 369.

02:07 And you do that same calculation five more times and these are the expected values.

02:14 And you can see that they're not exactly what we get, right? we observed 40 cat 1 failures but we'd expect 48...

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