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A simple random sample of size n = 40 is drawn from a population: The sample mean is found to be 108.5,and the sample standard deviation is found to be 17.9. Is the...

Question

A simple random sample of size n = 40 is drawn from a population: The sample mean is found to be 108.5,and the sample standard deviation is found to be 17.9. Is the population mean greater than 100 at the & = 0.05 level of 'significance? (You should be able to test using critical values by hand and p-values by using calculator)

A simple random sample of size n = 40 is drawn from a population: The sample mean is found to be 108.5,and the sample standard deviation is found to be 17.9. Is the population mean greater than 100 at the & = 0.05 level of 'significance? (You should be able to test using critical values by hand and p-values by using calculator)



Answers

Conduct the appropriate test. A simple random sample of size $n=40$ is drawn from a population. The sample mean is found to be $108.5,$ and the sample standard deviation is found to be $17.9 .$ Test the claim that the population mean is greater than 100 at the $\alpha=0.05$ level of significance.

For the given parameters, soviet, given years, uh, example being the samples and attribution Publishing Media. We have already businesses that Russian is better than a 100 and european european level of significance and supported the tragic here he's given us expand negative meal or s by. Okay. Oh just so all this These are things coming up with three point obviously three points. No they don't. Now the p value at this probability value at this technology here 0.0 to tweet is right and the value 0.00 country that is the right edge P value zero point with a little boat. Now he's compared me with alpha and has two piece coming out to be less than open. Right. Therefore really Jack now the bulletin so we reject the politics. That means we can say that there is sufficient evidence that coalition meat is better than 100. There is sufficient evidence that population means is greater than a 100 from you.

I let the statement here. The question to verify the given parameters In the working medium and equal 90 Example meaning your pointed little consideration is kind of the .4. That's what the question means. Less than one point a to point to a woman. Never seeing it again. So population and for example mediums given simple celebration if you want. I wanna be a given temperature division. We just go for artigas. So There we go. 40 desk 40 days. Depending on the value. He's coming up to be expo negative mu or we have a sore throat so we know the better meters here. But the value we get the T Values 92.179. I'm going to use the technology basically They use technology. I will get the P value. That is 0.0214. As AP value is alpha is 0.01. Right? So the values read an alka liberal. We been two reject molly for this with them to reject Malibu this, this and this Porsche. So I was able to reject the multiple. This is so we don't have any. We don't have sufficient evidence to say that population means less than 1.0. We don't have sufficient evidence to say that obviously means less than one point of view and

The following is a solution to number four. And this says that a random sample of 19 people, I guess uh from a normal distribution. So that's important. The population distribution is normal. That makes that sample size okay. If it weren't normal, then we would need a sample size of at least 30. So the 19 is okay. And the sample mean of that particular uh Uh there is no context of this problem. So the sample mean of this sample is .8 and then the sample standard deviation is .4, and we're supposed to test that alpha equals .01 if the mean is less than one. Um so first off, because we don't know the population standard deviation, we were never given sigma, were only given the ass, we need to use the T test. Alright, We cannot use the Z test because we don't know sigma's we have to use the T test and it's a five step process, like I said. So first off, the uh the first step is to state our Nolan are alternative hypotheses. So the null hypothesis h not always has some sort of equality and we're testing a population means. So you're always gonna use greek symbols here. So mu is 1.0. And then we're testing if it's less than 1.0. So the alternative is that mu is less than 1.0. The 2nd and 3rd step, I'm actually going to write together because we're going to use technology to get it. So the second step is to find the test statistic and you can use a formula for that. But if you have a calculator, if you have, you know, Excel or some sort of software, um it does a lot of the work for you. So that's what I'm going to use just to save some time. So t is the test statistic and then the p value is also going to be given to me. So let's go to the calculator. Now I'm on ATI 84. And if you go to stat on the T- 84 and go down to tests, it's the second option here. The T test. And we need the summary stats here, so make sure summary stats is highlighted And the mute not, that's the hypothesized value. So in this case it's 1.0 or just one. So it's the null hypothesis. The X. bar is your sample mean, and that was given as .8 SX is the sample standard deviation, which was given us four, And then in is the sample size and that was given as 19. And then you get down here, and this is the alternative hypothesis. So it's either not equal to less than or greater than and if you look back that alternative is less than one. So that's all good. And now we can calculate and that gives you some stuff here. But really this is the T. And the P. Is what I need. So the T. Is negative 2.179. So let's go and write that down -2.179. and then the P value Is .02 0.2 And the p value really is probably the most important thing on that what you do. And step forward as you explicitly compare the P value with the alpha value. And remember the alpha value here is 1.2 is greater than one. And whenever the p value is greater than alpha than we fail to reject the null hypothesis. So failed to reject H. Not had that P value been less than alpha than you would reject H not, but since it's greater than alpha than you failed to reject, so any time you failed to reject your conclusion, which is the last step step is going to be something along this line, there is not sufficient evidence to suggest that The population means I'll just use mu is less than 1.0, so there is not sufficient evidence to say that the alternative hypothesis is true, so therefore we're for lack of a better phrase, were accepting that the null hypothesis is true.

The following is a solution to number five and this is a proportion test where were asked the question or 200 people are asked the question. Um these people have a valid driver's license. Do you drive an American made um vehicle? And of the 200 people, 115 of them said Yes. And we're supposed to test at the alpha are at the significance level of .05. If more than half the people drive an american car. So first off this is called a one I'm going to abbreviate here but it's a one prop Z test. So one proportion Z test will be um the name of this and it's a five step process. The first step is to state our hypotheses and our hypotheses, the knoll always assumes some sort of equality. So mean, I'm sorry not the mean it's the population proportion so P equals 50.5 and that's that's where they get the half. So half is 0.5. And then the alternative it's asking for us to test if it's greater than half or more than half. Okay so that's those are first two steps are first step and it's the two hypotheses. The 2nd and 3rd step are kind of the same thing as the calculator is going to give it to us. So we find the test statistic which all this called Z. And then we also find the P. Value. So you can use the formula here to find Z. And then use you know tables and whatnot for the p value. But I think a calculator is a much easier way to go about this. So if you go on a T I. T. For at least so you can use any sort of calculator you wish. But I think the T I A for is the easiest to use. So if you had a stat and then tests And you're going to go down to this fifth option here where it says one prop Z. Test. So go to option five and then the Peanut, that's the hypothesized value. So that's .5 X. was the number of favorable. So that's 115 in this case. And as the sample size and remember there were 200. And we're checking to see if um If it's greater than 50% so greater than half. So that's greater than Peanut. Alright so everything should be good to go and then we can calculate and it gives us you know the P. Hat and it gives us a lot of information but really it's just the Z. And P. That really make a difference. So the Z value is 2.12 and the p value is 0.17 So don't get that mixed up with P happy hat is the sample proportion? The statistic P stands for the p value. Okay so let's write these down and then we'll talk about a little bit. So the The Z. is like I said 2.1-1 and then the p value Is 0.017. So the P value, like I said is probably the most important thing on here. What you do in the fourth step is you explicitly compare the P value with your significant level alpha. And this time we have a P value of point to 1.2 is going around and an alpha value of .5. So in this case the p value is less than alpha anytime. The p values less than alpha than you reject the null hypothesis. Now had this been greater than alpha? The p value you would fail to reject but it's less than the alpha, so you reject h not. So we're rejecting this null hypothesis that it's equal 2.5, and we're for lack of a better phrase, we're accepting that The actual population proportion is greater than 50%. So our conclusion here would be something to this effect. There is convincing statistical evidence to suggest that more than and the scroll down. Bear with me, more than half of individuals with valid driver's license, our licenses dr american cars or vehicles. Okay, so we are uh we do have sufficient statistical evidence to say that more than half the people drive american vehicles.


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