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5_ To test the hypothesis Ho p = 4 against the alternative hypothesis Hi p > 4 in N(p,4) population, a random sample of size 25 is taken_ If the critical region ...

Question

5_ To test the hypothesis Ho p = 4 against the alternative hypothesis Hi p > 4 in N(p,4) population, a random sample of size 25 is taken_ If the critical region is {1 1 > 5} where € is the sample mean then determine the probability of a type I error_6_ To test the hypothesis Ho p = 6 against the alternative H : p > 6 in N(p, 4) population, a sample of size 25 is taken: The critical region is {1 1 > 6.5}. If the mean is 7 then determine the probability of a type II error_

5_ To test the hypothesis Ho p = 4 against the alternative hypothesis Hi p > 4 in N(p,4) population, a random sample of size 25 is taken_ If the critical region is {1 1 > 5} where € is the sample mean then determine the probability of a type I error_ 6_ To test the hypothesis Ho p = 6 against the alternative H : p > 6 in N(p, 4) population, a sample of size 25 is taken: The critical region is {1 1 > 6.5}. If the mean is 7 then determine the probability of a type II error_



Answers

A hypothesis will be used to test that a population mean equals 5 against the alternative that the population mean is less than 5 with unknown variance. What is the critical value for the test statistic $T_{0}$ for the following significance levels? (a) $\alpha=0.01$ and $n=20$ (b) $\alpha=0.05$ and $n=12$ (c) $\alpha=0.10$ and $n=15$

To this lesson in this lesson we'll find their critical values for the T. Distribution using enough A level and the sample size. So what this would have at T. Isn't a two tailed tests because we have no you call to in the Now in the alternative hypothesis we have the not hypothesis us. The mean the meaning cause too seven. And we have the alternative hypothesis. The me no equal to seven. So we have to tell test means to divide the affront to and we have degrees of freedom Which is in -1. So this particular one we have T. That is zero point 005. Okay. And the degree of freedom of 19. So this correspondences To a 19 here. 10 0 point 005. Which is also here for 12 and equivalent for two tails. Okay, so that goes to this. Okay, This is the two tail And this is equivalent for the 12. All right. So that is 2.861. So two point 681 2.861 two point 861. Unless you cut the 2nd 1 where we have and after a zero 05 and An end of 12. Okay. Okay. So in the one we can convert it to until where we have enough for two. Mhm. The group of freedom of And -1. Just recalled to T. You have joe point joe 25. Then we have 11 as the degrees of freedom. So until we are looking at their point Gerald 25 and two tails. You are looking at zero point general five. Okay. So we have 11 here and this is where you goes. So in one till we are looking at 0.5 as re in two tails You are looking at their .5 corresponds to one tail of 0.0 25. Okay, so that is 2.205. Mostly we have this table with Yes one till Yeah, so if it had been 12 and we have actually two tails, you are going to divide the effort into to buy this to avoid house one till correspondent with the two tails. Okay, so just the column that we need, That's the rule that we need. So 2.201 2.201, last but not least. We have enough to Which is 0.1 uh and the sample size which is 15. So we have in one in one tool we have are found to But in two tails we have 0.1 the same thing Than in -1. So this corresponds to if you are using The one with just one tool would have 0.05. That is their .1 over two than 14° of freedom. So we are looking at 0.05 from one tail. We're looking at 0.1 from two tails Still with a degree of freedom was 14. So 0.1 here, two tails. 0.0512 In the 14 this year. So it goes this way and it comes that way. So emits 1.761, 1.761 1.761. Okay, so these are the various critical values. Transfer time. This is the end of the lesson.

Mhm. We got another hypothesis here and our alternative that is less than five. So that's how we're gonna write it. Uh We're gonna be looking at a left tail test since it's less than the mean, Our means, always going to be in the middle here. And what we're trying to calculate is the probability that this is true. That's R. P. Value. What's the probability that are non hypothesis is true. And the point of doing it at a significance level of one is to be able to Be certain within 1% probability that are null hypothesis is wrong. So for a left tail test, which we're looking at right here, since that's beyond our mean, which is in the middle of this bell curve, we're just simply going to be looking at the something called the feet are See score or significance level. So it's gonna be our feet of 1%. And when we look it up in the back of the book here, Table three 1% is going to get us a critical value of -2.33. Same thing with the 5%. Although these numbers are gonna be the same in magnitude as the last problem. So where is 5% in table three, That's gonna get us AZ score of -164. And then finally, for our 5% or 10 Or is 10% 10% probability get us as a Z critical value. That is going to be negative 1.29. Yeah. Yeah

Everyone. So today we're gonna be doing a similar question to the previous video. But we're searching for the population standard deviation rather than looking for the sample me. And so I'm just gonna write out the information we're given in the question. So we're given. I bought. Sees fictional M U is equal to 16. H a mu is less than 16 were given that it's a sample of 50. So and it's 50 we're told that Expo is 14 Onda. We are given a P value off. No point, not 17 Now the first thing we need to do is convert this P value into a said start value. I would do this using the inverse normal, which I talked a little bit about when I answered question 19 So I'd look back to that if you want a little bit of an idea what's going on with this calculation? But you basically put it into calculator and you get it out is 2.120 what, So now we gotta set style value, and we also have a handy said star formula so we can use this formula here, and the values that we've got available to us to isolate the value of Sigma. So if I write this set style value here on the left hand side, right, 2.1 to 1 is equal to And then we know what experience. Because we were told it was 14 over here. And we know what Mu is because you were told what Mu is, that's 16. Now we're looking for Sigma and we've been told that and it's 50. And so we've got Sigma. Now, this is a little bit harder than just isolating the sample mean because this is in the center, off a off double decker fraction. Essentially. And so what we want to do is to start off with. I'm just gonna I'm gonna move this fraction here to the left hand side. This might seem counterintuitive, cause surely we want to get this on this side, but it's more helpful to get rid of this fraction this larger fraction in the first place than it is to do anything else first. So what you're then have is 2.1 to 1. We'll have, uh, sigma over 50 here and then 14 minus 60 is minus two. We can do that in our heads. And so then we can up. We can move this to the other side here by dividing, if you can believe me, that's a division symbol. Should try that somewhere else by dividing. That's it then what we have is Sigma over 15 built right? All of this is right now, my dio sigma over great 50 is equal to, uh to over 2.1201 And so then we can get rid of the bottom of this direction by just multiplying over here, just as we would normally, if this was a normal fraction, it might help you toe sort of visualize what I mean. I haven't put these brackets on here. So what we have is just too normal fractions. I could multiply both by route 50 to isolate sick my hair. So then we have Sigma is to route 50 over 2.1201 on. Then you just put the anti calculator and you get out is 6.6706 And that is our standard deviation in this question and again a little bit harder than the previous one. Looking at the sample mean? But we've been given all the information we need. Remembering how to convert for a P value to Azad style value on then just rearranging this equation until you can plug it into your calculator on. There you have it.


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