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Consider the following hypotheses test HO: pZ 80 Ha: / < 80A sample of 121 provided sample mean of 77.3. The population standard deviation is known to be 16.5 , ...

Question

Consider the following hypotheses test HO: pZ 80 Ha: / < 80A sample of 121 provided sample mean of 77.3. The population standard deviation is known to be 16.5 , Determine the p-value: and using 063 lest the above hypotheses_(Hint: uSe the 2 distribution table t0 findOur p-value il you do nol have the TI-84]

Consider the following hypotheses test HO: pZ 80 Ha: / < 80 A sample of 121 provided sample mean of 77.3. The population standard deviation is known to be 16.5 , Determine the p-value: and using 063 lest the above hypotheses_ (Hint: uSe the 2 distribution table t0 findOur p-value il you do nol have the TI-84]



Answers

Find the P-value for the hypothesis test with the standardized test statistic $z .$ Decide whether to reject $H_{0}$ for the level of significance $\alpha .$ Two-tailed test $z=-1.68$ $\alpha=0.05$

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

Even the following information were asked to find a P value, associate it with the given information here and compare our P value to an Alfa 0.5 to test these hypotheses. So the first thing we need to find is a test statistic. And because we're working with proportions as given by this P bar here, we are going to compute a pool variance. So the formula for our P bar is the first sample size times the first sample proportion plus the second sample size times a second sample proportion over some of the first and second sample sizes. So in this situation that is equal to 200 times 0.22 plus 300 times 0.16 divided by 300 plus 200 and we get a test statistic of 0.184 So this is our P bar and now we can use this this value to compute a Z score. So our Z score is equal to the difference of our proportions. Divided by the square root of are pulled variants times one minus our pool variance times one over our first sample size plus one over our second sample size and all of that under a square root. So this is equal to 0.22 minus 0.16 over the square root of our P bar. We just found out to be 0.184 times one minus 10.184 one minus 10.184 times. Um, one over 200 plus one over 300 and all of that square rooted. And we get a Z test statistic of approximately 1.70 so we'll draw our normal curve. Z equals zero lies in the middle. Z equals 1.7 lives here and now we can check our oh, so part, eh? Heart lead. Um, so important. We were asked to find a p value, but we need to find the direction of, um of the hypothesis in order to discover our p value. So since this difference of means is greater than zero sorry, the difference of proportions is greater than zero. We're going to be doing a right tail t test Z test. Sorry. So we're looking at this value here, and this value represents the probability that Z is greater than or equal to 1.7, and that's hard to compute. So what we can do is find this area over here to the left of P of Z equals 1.7 and subtract that from one. So after that, we get a P value of 0.0 for four. You're a 0.44 So for a part, ay, our answer is P equals 0.44 And now we're asked to test this Against an Alfa of 0.5 Our P value is 0.440 point 044 Sorry, because 0.44 is less than our alphabet 0.5 We can reject no hypothesis. So does that mean there's sufficient evidence to support the claim that sample one has a larger proportion in the sample?

Okay, um, in this question, I'm going to use the software are to help us, Um, calculate the Peabody owes. Okay, that's snowcapped heart to pay. The alternative hypothesis is Pete want greater than p two, and our that start is equal to 2.47 So, um, our p value as you go to the probability off they were in them their bows that greater than our, um God, it was that star. Well, I know that this that IHS representing a random bearable which is following Southern normal distribution. And this that star isjust a value board. A number we have, and it is our test statistic. Okay, um, that start with, um are well use function copy, Norm. And how include poor words that star brother. Um, because, um, here, this operator is greater than so We're looking for the right held. So we need Teoh indicate in the function that lower towers, because to fall us here we crack our, um, key value for part A, which is equal to Is there a point they're owed there? Oh, 68 All right, um, and if we draw grabs, who have, like this point to be ours that star 2.47 and our P value here is just the area for this rightto. Okay, let's move to part B part B. The H A as p a does not go to pp can't That star is equal to negative 1.33 So because of this, operators were looking for, um the to tout part. So we're interested in the area both right and left And the summation off. Thesis Blue of Blue Part is our people do. And this Ah, going to be positive. 1.33 And here his hours at Star, which is negative 1.33 Okay, so let's couple live for just 11 part the norms records of 1.33 and we multiply this value by to cause Ping because we just calculated for, um for one site. So our p brother for parts three Azaz there a 0.18 35 Oh, right. We can move to part C part saying we have bought one people on Linus Pete too smaller than their own. And that star is now. Good tip. There a point k five. So we're looking for the probability off that smaller than negative zero point pay. Five. Show pink Norm MCE 2.8 fried and we have our people loot. Which is he? Quote Teoh Arrowpoint 1977 All right, and heart D he l minus p f greater than their own. That star Aziz 3.4 So the P value should be equal to the probability off. Sad, greater than 3.4 Again, that is a random variable which is following the standard normal the distribution. Okay, so it is going to be the right to tell. You're in 3.4 And our P bateau is just the area from 3.4 to the positive infinity. All right, let's calculate this. Number 3.4 lower toe is and we cat P bateau is equal to zero point there. Oh, there are 12

So in this question, we are given a board off a T I 84 close. So if we look at the board, were given the value off RZ statistic were given our P value which yes, 0.443 were given the mean off the first sample population X one, the second sample population X two and the sample size and so on. And what we're asked is to use the display to make a decision to either. So this is the now hypothesis. So we're asked to make the decision to either reject or fail to reject the now hypothesis at the level off significance given here. We're asked to make the decision using the standardized test statistic as well as the P value. So if you look here are people us 0.3 which is less than 0.5. So using our p value, we can reject the now hypothesis. Now, rZ statistic is 2.956 Approximately so. So for our Z statistic, we know the critical value for Z for a two tailed hypothesis test at Alfa Aiko's point or five is 1.96 And so since our value of C is more in our critical value of C. Again, we can reject no hypothesis. This is the answer to the question.


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