What’s in a Word? Part II In a recent survey conducted by the ...

Question

What’s in a Word? Part II In a recent survey conducted by the Pew Research Center, a random sample of adults 18 years of age or older living in the continental United States was asked their reaction to the word capitalism. In addition, the individuals were asked to disclose which political party they most associate with. Results of the survey are given in the table below.

a. Does the evidence suggest individuals within each political affiliation react differently to the word capitalism? Use the alpha = 0.05 level of significance.

Table showing positive and negative reactions to the word capitalism by political affiliation: Democrat, Independent, Republican.

a. Does the evidence suggest individuals within each political affiliation react differently to the word capitalism? Use the alpha = 0.05 level of significance.

Accepted Answer

Hello. In this video, we are told that a company surveys employees about their preferred work schedule and their departments. The results are given in the following table. At a 0.05 significance level, is there evidence to suggest that the preferred work schedule is related to the department? Now, the wording here is a little tricky, but we're trying to find if there is evidence that the preferred work schedule is related to the department, which the keyword here is related. So, what this means is that if we want to find the evidence where two things are related to each other, then what we want to do is we want to use a G2 test for independence. So, let's first set up the null and alternate hypothesis for the G score test. Now, for the no hypothesis, we are going to claim that the preferred work schedules are independent. So there is going to be independence. Now, what about the alternate hypothesis? Well, if the no hypothesis is claiming that there's independence, then the alternate hypothesis states the opposites in which they are dependent. OK, so now let's go ahead and look at the given information. Now, the only thing that's really given to us here is the is the significance level in which we are working with a significance level of 0.05. But what we need to do is we need to go ahead and check to make sure that all the expected values of the columns for the G score test are greater than or equal to 5. Now, for the expected value, expected value is calculated as the following. Expected value is the row total. Minus the column total. And this will all be divided. By the grand total of everything. So let's go ahead and take the sum of each row and each column, and the sum of all the cells that are given to us within the table. Now, the sum of the first row for sales in the morning, afternoon, night is going to be 40. The sum for engineering is going to be 50. And the sum for HR's row is going to be 30. Now, for the columns, the sum of the morning column is going to be 54. The sum of the afternoon column is going to be 37. And the sum of the night column will be 29. Now, if we take the grand sum, the grand sum of all 9 of the cells together is going to be 120. All right, so now that we have these totals, what we need to do is we need to go ahead and create a table for the estimation, estimated rows. So let's go ahead and do that real quick. So again, we have sales. Engineering And we have HR. And we have 3 columns to to work with. There's the morning shift. The evening shift, or I'm sorry, afternoon shift. And then there is the night shift. And what we need to do is we need to go ahead and calculate the estimated value or the estimated frequency for each of the cells. So, starting with the first cell, for sales, that is going to be the row total minus the column total divided by the grand total. So that is going to be 40 minus 54, I'm sorry, 40. Multiplied by 54, because I made an error on the estimated frequency. It's not a difference, it is a product instead. So it's the row total multiplied by the column total. So it's gonna be 40 multiplied by 54, and then divided by 120, and that is going to give us an estimated frequency of 18.00. And if we repeat this process for the next two cells, we get 12.33. 9.67. For engineering, we get 22.50. 15.42. And 12.08. And for HR we get 13.50. 9.25 and 7.25. So, remember, the reason why we did this is because we want to make sure that all cells are greater than or equal to 5. So do we see any values here that are less than 5? Well, the answer is no, which means that we can go ahead and apply the chi square test for independence. Now what we need to do next is we need to go ahead and calculate our test statistic for G squared. The test statistic is defined as G 2 equal to the sum. Of the observed values minus the estimated frequency squared, divided by the estimated frequencies. So, let's go ahead and organize this data into a table. So what I've done here is I went ahead and I organized all the cell values with respect to the expected frequencies. So, for example, if I wanted to take the first element of the sum, I would take the difference of the expected or the observed value, which is 18, minus the expected, which is 18 as well, and that gives me a total of 0. Then we will square that value and then divide it by the expected value, and we need to repeat that for all the values that were given to us in the table. So that is going to give us a total sum of 0.4077, which is going to be our C square test statistic. Now that we have our G square test statistic, what we need to go ahead and do next is we need to find the critical value. So, the first thing we need for the critical value is the degree of freedom. Now keep in mind that the degree of freedom for a table is going to be the amount of rows minus 1, multiplied by the amount of columns minus 1. So here, that is going to be 3 minus 1, multiplied by 3 minus 1, and that is going to give us a total product of 4. We have a degree of freedom of 4, and so now what we need to do is we need to go ahead and use a G score table in order to determine what the G square critical value is gonna be with respect to a degree of freedom of 4 and a significance level of 0.05. Now, here, that is going to give us a total value or a G2 critical value. of 9.488. Now, if we compare the two, the G2 critical value is larger than the calculated G square test statistic that we found. And so what this means is that we fail to reject the null hypothesis. And so what this means is that there is insufficient evidence to suggest that the preferred work schedule is related to the department. And so with that being said, the correct solution to this problem is going to be option A. So I hope this video helps you in understanding how to approach this problem, and we will go ahead and see you all in the next video.

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