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In a two-way ANOVA with replications, the test statistic for the interaction test is:MSA a: MSEMSA + MSB b MSEMSB MSEd MS(AB) MSEMST MSE...

Question

In a two-way ANOVA with replications, the test statistic for the interaction test is:MSA a: MSEMSA + MSB b MSEMSB MSEd MS(AB) MSEMST MSE

In a two-way ANOVA with replications, the test statistic for the interaction test is: MSA a: MSE MSA + MSB b MSE MSB MSE d MS(AB) MSE MST MSE



Answers

2. State the hypotheses used in the ANOVA tests.

So there are three hypotheses for a two way in nova and usually just see the null hypothesis. And I'm gonna write down the alternative. It's really just the opposite of what the knoll would be, but I'll write the null hypotheses. I'll just call this h not one, and they're all pretty much the same thing. At least the first two are, it says for the first one, it's there is uh no difference in group means at any level of the first independent variable. That's the first knoll. The second knoll is almost exactly the same. I'm going to write it as a church, not two, and it says there is no difference in groups. In group means at any level of the second independent variable. And then the third null hypothesis is a little bit different, and that talks about effect. So, the way we were there, this when it says the effect of one independent variable does not depend on the effect of the other independent variable or in other words, no interaction effect. Now, usually you'll just see the knoll because the alternative is just the opposite. But just for clarity, I'll go ahead and write the alternative hypothesis, I'm gonna write the whole thing. But basically the first one is saying that there is a difference, there is a difference between or at at least one level of the first independent variable and then the second alternative same thing, there is a difference. But this time for the second independent variable, and then the third hypothesis would again just be the opposite. There is an interaction, there is and interaction effect between the two independent variables. Okay, so again, typically there are three. There will be three, but generally just written as the null hypothesis. So those are the ones to know, you know, for sure.

So you're dealing with a null hypothesis of the main being equal to 82.5. And alternately the mean is less than 82.5. So you're doing a one tailed test and you're using a significance level of 1%. So you're looking at this little picture here, and you're saying also that the distribution is approximately normal, so that the sampling distribution is going to be centered at assuming lee 82.5, and and your sample that you had had 17 items. And the mean of those 17 items, I believe, you said was 70.1, and the standard deviation of those was approximately 20.1. So we want to find what the test statistic is and the test statistic because this is a small sample, and uh and you don't know the population standard deviation, we will be finding a T value and the T value would have 16 degrees of freedom. And your test statistic is to take the X bar minus the mean. You're assuming divided by the sample standard deviation over the square root end. So we'll have the 70.1 minus the mean of 82.5 divided by the standard deviation of the sample over the square root of 17, not 16. And so let me quick type this in my calculator, I have 70.1 minus 82.5, divided by left parenthesis. E 20.1 divided by the square root of 17. And almost have it here and I get the test statistic to be negative 2.536 Okay. And so we know that that value is maybe about down here, that test statistic that we're getting is negative 2.536 And you wanted to know what the critical value was for this situation. And so a couple ways you can look it up in your table and look up for the degrees of freedom 16 and look for the area below it to be 160.1 And actually I think on your table you'd have to look up the 0.99 and find that value which is actually this upper end. I'm going to use my inverse T. Value and the inverse T. Value on my calculator is under second. And distribution and members T. I can put in an area of point on one because I really want to find this value down here that has the area point to one and find out what that corresponding T. Value is. And so on my calculator I'm going to put that area in my degrees of freedom of 16 and then just hit enter and when I do it tells me that that critical T. Value is negative 2.583 and five. And again the back of your book will give you a different value. Probably rounds off only 3000 places and may call this just that value. So are critical T. Value is this value. And so that's our critical thi this is our test statistic and this is the critical T. Value and we can see that they are close but this critical T. Value is a little bit smaller than what our actual T. Value is. So for this instance we would fail to reject the null. You don't ask that question but we would fail to reject the null to reject at a 1% significance level because this value is actually higher not much but it is higher on that and it's not down below this value. So hopefully that helps. There's your test statistic and there's a critical T value.

Okay. Um We're gonna walk through what is called a hypothesis test. Um And so in summary we are told that um kind of like the first drug choice um is heroin. Um For heroin was 60 for speed we were given 40 and for um some other drug of choice was 20 so 100 and 20 were our total count. Um And somewhere it was claimed that that we have been told that typically the proportion um the population proportion um that chose heroin was 60%. The proportion or population proportion who actually chose speed was 30%. And the population proportion that chose other was 10%. And we want to know is does that really actually follow those proportions or is one uh these proportions different? Okay so we're actually going to be using what is called a chi squared um goodness of fit test. And so we would expect we would expect, so these are going to be our expected counts. This is what we observed with 60 40 and 20. What we we would expect is that 60% of these, 120 would choose heritage and 30% of the 120 which would be 36 would choose speed and then 10% of the 120 would choose other. Okay. And so we're going to calculate that chi square statistic and that is always going to be the observed minus its corresponding expected counts square divided by it, it's expected. And then we're just going to some those up. So we would have the observed minus the expected for speed squared over the expected and then we would add it to the observed of the other minus it's expected squared over its expected. Okay. So when we do that we get 7.777 as our chi squared test statistics. Now we're told that and of course there's always um conditions that we should check to make sure that we can actually do this hypothesis test um for chi squared. Um This was a random sample so we need to make sure that some wordings that a random sample and all of our expected counts should be greater than five. Um and that we should have independent in our sample independence in our sample. Um And so if I um and this is a degree of freedom of two and how I find the degree of freedom is I take how many subcategories and subtract one from it. And so for a degree of freedom of two at a type one error equal 2.10 This is also known as my significance level. If I go to a chi squared probability table or chi square distribution table, degree of freedom at two at a 20.1 significance level, I get a chi squared equal to 5.99 nine. Okay, so chi squared distribution is kind of looks like a right skewed normal distribution. Um So if we have a rejection region at 5.999 then the rejection region is going to be that talent and we noticed that 7.77 is in that reject region. So what conclusion can we draw? Well, we would reject, we would reject the null. Mhm. Um And we have sufficient evidence to say that at least one of the claimed proportions is different. Mhm. I hope this helps.

In this problem, we shall be stating the hypothesis. In the novel test. The non hypothesis, hitch not is a new one equals. Mewtwo equals all the way until we get to the last me, you. And this simply means that all the means are equal and that gives us the knowledge. But this is There now are the alternative hypothesis. H one is that at least one meaning is different from the others? Yeah.


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