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A researcher has examined the influence of gender (Men vs Women) and political orientation (democrat vs republican) on happiness scores He conducts a 2x2 Factorial ...

Question

A researcher has examined the influence of gender (Men vs Women) and political orientation (democrat vs republican) on happiness scores He conducts a 2x2 Factorial ANOVA and reports the following descriptive and inferential statistics. Interpret each result in an English sentence:1. Gender: M= 62, 75,F(1,21)= 5.32,p <.05 2. Political Orientation: DEM= 78, REP 69,F(1,21) = 4.45, p <.05 3. Interaction between gender and political orientation: F(1,21) 7.8,p < .05

A researcher has examined the influence of gender (Men vs Women) and political orientation (democrat vs republican) on happiness scores He conducts a 2x2 Factorial ANOVA and reports the following descriptive and inferential statistics. Interpret each result in an English sentence: 1. Gender: M= 62, 75,F(1,21)= 5.32,p <.05 2. Political Orientation: DEM= 78, REP 69,F(1,21) = 4.45, p <.05 3. Interaction between gender and political orientation: F(1,21) 7.8,p < .05



Answers

Look again at the table of political views for the Intro Stats students in Exercise 24.
a) Find the conditional distributions (percentages) of political views for the females.
b) Find the conditional distributions (percentages) of political views for the males.
c) Make a graphical display that compares the two distributions.
d) Do the variables Politics and Sex appear to be independent? Explain.

For this problem. We have summarized information from a pull of University of Montana students, and they were classified across two variables sex and party affiliation. And we wonder if there is an association between sex and party affiliation. For her part, a were asked, um, if this is a test for homogeneity or independence. So since this is a test for an association of two variables with respect to a single population, this is a test of independence. And for part B were asked what the appropriate hypothesis would be for the no hypothesis would be party affiliation is independent of sex, and so the alternative hypothesis would be that party affiliation is not independent of sex. So that's part B, part A and part B and for part C were asked if the conditions for inference they're satisfied. So remember that we have three main conditions that need to be satisfied. We need to have counted data. We need to show independence of the sampled items, and we need to have an expected the smallest expected cell frequency must be at least five. So we do have counted data in the table given in the question we have counted data so that condition is satisfied with respect to independence were not told. It is a random sample, but we will assume that it was done randomly within this college. Also since party affiliation various from one part of the country to another. We should not seen that this date applies to students in all parts of the country. But we can at least we can, at least in for with respect to students in Montana. So we will say that that condition is met and for expected cell frequency. I have calculated the expected cell frequency for each cell and the smallest I got was 19.2, which is quite a bit bigger than five. So we're good to go there and then in party were asked to calculate the P value for this test. So I use software to do this. I got a chi squared value of 4.851 and a P value of zero point zero Heat 84 for Part E were asked to stay to complete conclusion. So because of the P value being quite large, almost 0.9 ah, we would fail to reject the no life offices. And remember, the no hypothesis is that party affiliation is independent of sex. So therefore, we would say that we do not have sufficient evidence to suggest that there is an association between students sex and party affiliation.

Once again our welcome to a new problem. This time we have physical problems and we're dealing with of Chi Square test or the basic off chi square test. Of course, you know it Exactly. I thought testing, but yeah, the branch off particular where you're going with non pro metric unpermitted has These are based off categorical variables. These are brave off categorical variable drin. And by that you mean that these are variables that qualitative in nature and typical qualitative variables include things like agenda and, of course, things like party aviation. So in the sense off political science, a lot of times when you're dealing with kite quartet or non Parametric test, your intention is to find association or what we call a relationship. And it can be tricky to find relationships when it comes to Christ water. Unless you have, ah, handle on how to, uh, control categorical variables. And one way of doing that is to create a contingency tables, contingency tables. And these tables are such a you have the variable itself are together with the frequency involved in the variable the frequencies involved, uh, in the variables. So when you're looking at associations say between different categorical variables, you will need contingency tables, at least the variables and the frequency off the variables. A few new have a table. Uh, do you have a table? And in a sense, in a sense, the stable hung party affiliation on the road. So we have a Republican. We have Democrats. Yeah, we also have independent. So these are identification Zoff party affiliations. Uh, we're not Not independence, but independent would see that. And then we also have agenda. So by agenda, we have female and you have mail. And so way have one of five. 1 50. Another 1 50 1 15, one of three 1 79. So we're saying there 105. Uh huh. Female Republican, 115 male of republicans. So these are numbers that you see, and our goal in this particular problem is to determine to determine a association between between the party affiliation and agenda agenda is a categorical variable for its body affiliation. The total sample size off this group is 802 adults. Remember, adults can raise stuff in political parties. And so the first in 40 mm. You wanna visualized the frequency. Marginal distribution, A frequency marginal distribution Using this data in 40 relative frequency. Marginal distribution. A relative frequency. Marginal. She distribution. She Hmm. Proportion for portion off. Um, independent registered water proportional independent rate headquarters. And then in what you're looking at, Mm. Party affiliation. Party affiliation by gender. And this is, of course, conditional. It's a conditional distribution. Mm hmm. Because you're setting up gender. You're controlling for gender, and then you commit the body affiliation in what you want to get bag grubs. Hmm. Bathed on gender. E could be a dog in the background, but just know that I'm going to focus on the problem. Uh, and then finally, you want to find associations between agenda and party irrigation? No, no, no. And so, holidays. This happened holidays. Disheartened eso. Those are problems. Those are problems, you know. Just remember that way. Do you have all these variables Dylan with? So, you know, we have a Republican Democrats? No, of course they are independent. Uh huh. One independent. Hmm. You know, we have all of these way female on. Then we have mail, and you wanna write out the frequencies. 15 1, 15 4 Republican males way also have Democrats on 50 one of you. 1 50 1 79. Hmm. Those independent. So the first step. But we're gonna say we need the marginal distribution to marginal frequent. That's simply having an additional column, which you were gonna call the marginal frequency distribution. And we're coming up, you know, we're taking wanna fight. And 1 15 that's going to give us 220 and then 1 51 of three. That's gonna give us, uh, to 56 weeks and then 1 51 79. That's when it was 23. 51 79. That's gonna give us 3 29. Hmm. The total males. So the total total is a grand total of 88 or five. Always crumpled. And then for the mail, we have 5358 nine. That Uh huh. 17. So 17. Well, then nine and then 3. 97 on this side we have. Uh huh. Marginal frequency distribution for female will be five to 45 as imaginable frequency distribution for females in the college. You have the world on the college, so that's the total total for world and columns That gives you the marginal frequency distribution. Learning for deep in the second. But we want to find a relative marginal frequency distribution developing marginal frequency distribution. So, for the marginal frequency distribution were just coming up their roles in the cart. He rolled in the columns. And then Now we're talking about the, uh, relatives. Mm hmm. Large general distribution. Relative imaginable distribution. So we have political party. So we have a Republican. You have Democrats, you have independent. Then you have the marginal distribution. Mm. Then this side, we're also gonna have this agenda. So agenda, we have a female no mail in the end of the earth, marginal distribution. And then you have our relatives roll it to you. Marginal distribution. Hmm. So, looking at the numbers, he got a unified. And then one sister, when you're 50 that's gonna give you a total marginal distribution. 45 Um, maybe hood 1. 15. Dr. Leo, what else could be on 1 70 million and then 3 97 you know, to 20 2, 53 3, 29 and then, uh, eight or 500 toe. So now we want to get the distributions always thing is to get the relatives. Why do you know, uh, distribution? You're gonna take the to marginal distribution divided by, uh, the total candidates total? Mm hmm. I don't wanna call them candidates. Total adult registered adults read step adults Total where you stood, Adult. So in distance. Same to 20 off 85. Mm. 2. 20 off eight or five. He did 20 to 7. Run it off, and then we got to 53 off eight or five. You get 2.31 then each one green off. Pedro. Fire. We get point forward off course that is rounding. But, uh, if you sum up the exact value she stopped the exact values you should get something close to. It's a point. Forward. 031 To serve in, you should get approximately. Well, it's running. So this is This is this is gonna be 0.99 But you approximate that to one point school. So because the rounding elements going on over there And then, of course, But see, Hmm, proportion. If independence proportion off independence, it's gonna be you. Take the total independence so you can see from here 3. 29 and then divided by the grand tour. So we turned around the body in total, um, diffusers. Hmm. Zero point for your eight, which is approximately equal to 0.4. You're looking for one on. Then we wanna get conditional distribution, traditional distribution, uh, which is taking the frequency on the patrol courage. And so what's gonna happen in, you know, we want to get the female and male lessons. So you have a table. Which political party? The Republican, the public can. And then we have democrats, then independent. Mm. Mm hmm. Uh huh. For independence, a Republican feeling. So if you want to get the Republican, he knows what you do is you take the full five inches total female, uh, Republican female. So if we if we go back with saying well Hmm? Construct, we want to get a conditional distribution off. Party affiliation by agenda, party affiliation by agenda. So, Republican female, you have to go through a Republican who see them. You wanna fly and then divide that by, um, the total total that you're looking at. So, for example, in distance, you're gonna take one of five and then divided by the total eight or five. All right, it's one of five, but by 85. So that's the Republican senior 0.13 All right, His 0.23. So all you did was take the Republican female divide by the grand total. You got that? And then female democrats repeat the process 1 50 by 15 by eight or five. That gives you 1 15, 8 or five 0.1 night. It's rounding. Remember, Independent something. 1 50 85 also come to one night and then for male. Same process 1 15, 85. So 1 15 and eight or five. That gives us 0.1 ft and then one of 3 85. And that gives us one of three by eight or five. I'm going to give us a 50.13 Wrong. Finally, we have males who are independence. 3. 29 over eight or five. So 3. 29 over eight or five. Hmm 0.41 Let me put E. So we want to do the graphing. And if you go back, you're seeing that of the graphing conditional distributions. Graphing conditional distributions. So who actually this one is not through 29. You gotta go back to 1 79. I don't want you to be back. So this is 1 79 on. That's gonna change the final help. You want to make sure that go back, Hmm? For 79 that's going to give us just making sure eight or five points to two. And so now we have to graph these numbers onto ah, conditional, uh, distribution work. So the bulletin of your why actually extract you will be, Hmm. We have 33 way have different candidates right here. But we have. Mm. You have Republican, and then you have democrats. And then, of course, we have independent. And then below that, we have female. Thank you. 000 Europe when you're five. 0.10 0.15 We're open to zero were 0.25 the so Republicans female. We go all the way up until zero points. One somewhere. Someone like you, You're gonna continue mhm. Then we have, uh, public turn the mail a little bit higher. Hmm. 0.14 Then Democrats. You have 0.19 on then mail 0.13 which was similar to the other one for male Democrats. There are Democrats. Your points one close to that. Learn independence. You have points 19.22 So this is 0.49 right here. 22. That goes a little bit high. And then finally import f mm. The conclusion Want to come up with the conclusion so mhm. The is on association between between the agenda and party affiliation, whereby we can say we can confidently say the term well, are more likely to be independent unless likely to be Democrats. No. Uh, no. Very clearly to be independent. Hmm. Right. Like to be Democrats. So once again, we had a problem. Six different sections fast the marginal distribution frequency and then relative marginal distribution within the proportion of independence. Tory affiliation by agenda, Conditional distribution, Bagua. And then to take for associations, we had the marginal distributions. We had relative marginal distributions where we took the for specific values total. We took the margins and then divided by the grand total. Um, on then the proportion of independence you could see 1.41 out of the total, uh, conditional distributions by agenda. So Republican female Republican nail hmm. And then the graphene bird work who so is an association between gender and party affiliation. So I hope you enjoy the problems. Feel free to turn any questions or comments and how they want a few days.

So being a Republican or a democrat is a nominal value as well as being independent or all of that, and nominal values cannot be numbers. They aren't. It's like giving yellow and red. Eminem's colors are numbers to identify them, you're going to get some sort of average, but it's not going to be very meaningful. So don't use numbers for something that's supposed to be colors.


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