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Exercise 9.5 Take the linear modelYi = XiiB1trzibz+ei E(xiei) =0where both xli and xzi are q x 1. Show how to test the hypotheses Ho : B1 = 82 against H : B1 # 82:...

Question

Exercise 9.5 Take the linear modelYi = XiiB1trzibz+ei E(xiei) =0where both xli and xzi are q x 1. Show how to test the hypotheses Ho : B1 = 82 against H : B1 # 82:

Exercise 9.5 Take the linear model Yi = XiiB1trzibz+ei E(xiei) =0 where both xli and xzi are q x 1. Show how to test the hypotheses Ho : B1 = 82 against H : B1 # 82:



Answers

Use a computer or calculator to complete the hypothesis test $H_{o}: \mu=52, H_{a}: \mu<52, \alpha=0.01$ using the data: $$\begin{array}{rrrrrrrr}45 & 47 & 46 & 58 & 59 & 49 & 46 & 54 & 53 & 52 & 47 & 41\end{array}$$

So here is a given information. Our null hypothesis is that our population mean is greater than or equal to 80 are alternative hypothesis is that our population means less than 80. We have a sample of 100 a population standard deviation of 12 and we are comparing R P values to an Alfa value of 0.1 So the first thing we have to do is compute the test statistic for each of the, um the sample means that we're given, and the 1st 1 we're going to start off with is a sample mean of 78.5. So because our end is, um 100 it is large enough that we can do a Z test that so will compute ese test statistic. And to do that, we will take the sample mean minus the population mean over the population standard deviation divided by the square root of the sample size. And for us, that is 78.5, minus 80 over 12. Divided by the square root of 100 78.5, minus 80 is negative, 1.5 12 divided by the square root of 100 is 1.2 and negative 1.5 over 1.2 is equal to negative 1.25 So we end up with a Z score of negative 1.25 now, because our Z score is negative 1.25 We want to compare that, um, we want to compute a probability value a p value so that we can compare it. Thio, our Alfa of 0.1 So what is this P value? Tell us. Um, it tells us the area under our Z score here, Z equals 1.25 Our easy score to left of our Z score under the area of the curve. Okay, so we're finding this area right here. What is the probability that Z is less than or equal to negative 1.25 Okay, um, and we get a P value of 0.106 Okay, So because we have a p value of 0.106 we're going to compare it to an Alfa. 0.10 point 106 is much greater than 0.1. So we phil to reject the no. Okay, So now if we go to part B. Ah, we're doing the same thing with a sample mean of 77. So our Z score is equal to 77 minus 80 over our population. Standard deviation of 12. Divided by our sample size of 100. This is equal to three. Or start negative. Three over one point two, which is equal to negative. 2.5. Sorry. Yeah, negative. Two point five. Let me just check. I am doing this right? Yeah, it is. Ah, negative. 2.5. Um, so we have a Z value of negative 2.5. So we draw our normal distribution here with a Z value of negative 2.5. We're finding the area to the left of negative 2.5. And underneath this curve, probability that sea is less than negative 2.5. So that's our p value. So our P value is equal to 0.6 Um, and now we're going to compare our p value of 0.6 to our Alpha of 0.1 and 0.6 is less than 0.1 So we have sufficient evidence to reject the no hypothesis. Okay. At an Alfa uh, 0.1 and P value of 0.6 All right, now we're going to do the same thing for part C Z c. Sorry. We're going to compute a Z value, which is equal to our sample. Mean, which is 75.5 minus our population. Mean, which is 80 over our population center deviation, which is 12 divided by our population size, our sample size, which is 100 and we're going to get a value of negative 3.75 Okay, so now we're going to draw our normal distribution a Z value of negative 3.75 If we have Z equals zero, here would be somewhere here. No, 3.75 We're looking at the area to the left of negative 3.75 and under the curb. This is equal to the probability of ze being less than negative. 3.75 We calculate that value. We get, um, a alpha value. All right. Of approximately 0.1 All right. Okay. So with a P value that this is part. See, with a P value of 0.1 we're going to compare that to an Alfa value of 0.1 And this P value is much less than 0.1 So we have sufficient evidence. Two reject the no hypothesis with AP value of 0.1 Um, we can write about approximately 0.1 as in Alpha of 0.1 Okay, not for the last part of the problem for D. We're going to do the same thing we've been doing. We're going to take find ese test statistic. Um, which is equal to our sample mean of 81 minus our population mean of 80 divided by our population. Standard deviation, which is 12 divided by our population size, which is 100. And this is equal to one divided by 1.2, which is equal to about 0.83 Sorry. Exactly zero point. About 0.83 Um, if we draw our our normal distribution, we have Z equals zero here, and this is where you have to be careful because our Z test statistic is positive. Ours e value ends up being here, but we're still going to find the area to the left and underneath the curve. Now you can see that this Z test statistic, this probability here, probably a Z, is less than or equal to 0.83 is pretty big. All right is pretty, pretty big. Um, so we will most likely end up not rejecting, failing to reject the no hypothesis. But once we get this value, our probability that Z is less than zero point 83 is equal to 0.797 And if we compare our p value of 0.7972 are Alfa of 0.1 we get that 0.797 is much, much, much greater than 0.1 So we fail to reject the no, um, with key value of 0.797 at an Alfa uh, 0.1 Hope that help


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