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Problem 9. (10 pts) Following is a randomly selected list of body mass indexes (BMI) from recent Miss America winners.19.5,20.3,19.6,20.2,17.8,17.9,19.1,19.8,17.6,1...

Question

Problem 9. (10 pts) Following is a randomly selected list of body mass indexes (BMI) from recent Miss America winners.19.5,20.3,19.6,20.2,17.8,17.9,19.1,19.8,17.6,16.8 Use a 0.01 significance level to do a one sided test of the claim that recent Miss America winners are from a population, which has a standard deviation equal to 1.34 compared to the alternative that the standard deviation is less than 1.34_

Problem 9. (10 pts) Following is a randomly selected list of body mass indexes (BMI) from recent Miss America winners. 19.5,20.3,19.6,20.2,17.8,17.9,19.1,19.8,17.6,16.8 Use a 0.01 significance level to do a one sided test of the claim that recent Miss America winners are from a population, which has a standard deviation equal to 1.34 compared to the alternative that the standard deviation is less than 1.34_



Answers

Conduct the appropriate test. A simple random sample of size $n=19$ is drawn from a population that is normally distributed. The sample mean is found to be $0.8,$ and the sample standard deviation is found to be 0.4. Test whether the population mean is less than 1.0 at the $\alpha=0.01$ level of significance.

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.

The following is a solution to number nine and were given that there was a random sample of 40 noticed they didn't tell you what the population distribution was, because it doesn't matter because that that uh sample size of 40 is big enough, where the samples of the population doesn't matter, The distribution of the population doesn't matter. So, um if If it had been like 25, for example, though you would need to have a population that's normally distributed, but in this case 40 is big enough, so we don't care about that. And uh were also given the The sample mean of 108.5 and a sample standard deviation of 17.9. And we're supposed to test at the 5% level of significance If the population mean is greater than 100. All right, So, first off, let's say what kind of tests for using? We're going to use the T test because we are given the sample standard deviation, not the sigma. So, if you're given the sigma, if you're given the population standard deviation, then you can use the Z test. But since you're not given the population standard deviation, you're only given the sample standard deviation, you have to use that T test. So we're gonna do a five step process ah five step hypothesis test. Where the first step, you state your hypothesis and the null hypothesis always has something to do with uh equality. So we're saying that the mean is equal to 100, so it's always the uh the parameter equaling whatever they say it is, And then the alternative is you kind of have to do a little bit of reading here, but it says, is there enough evidence at the 5% level of significance that the mean is in fact greater than 100. The population mean is greater than one. That's why we said great of him. Steps two and three are basically the same thing. Well, not the same thing, but you find them the same way whenever you have technology. So I'm going to do these kind of together the T stands for the test statistic. And there is a formula in your book. You can certainly use, I'm gonna use technology because it goes by much more quickly and it's easier to see. And then uh p value is it stands for probability. And that's also found with technology. Usually you can use a chart as well, but I think technology is a little bit easier. So I'm gonna use the t. because I think it works quite nicely. And if you go to stat an air over two tests, we're going to go to the second option here. The T test. So it's a T. Test. And summary stats needs to be highlighted. The stats is there and then you just start filling in your stuff. So um you not, that's the hypothesized value, that's your null hypothesis, and that was 100. The X. Bar. Is your sample mean? And remember that was one of 8.5 S sub x stands for the sample standard deviation set 17.9 in this case, And then your sample sizes the ends, that's 40. And then you get to this alternative hypothesis here and we need to switch this over to greater than mu not so it's greater than the null hypothesis. And then whenever we calculate that gives us basically everything we need. So there's a lot of good stuff here, but the T. And the P. Or what I really care about. So the t. value is about 3.003, and we have a really small p value of .002. So let's write those down and then we'll talk about them. So the T. Value Is 3.003 And then the P value is .002. So what we do in step for because the p value really is the most important thing. We explicitly compare the p value with the alpha value. So I have a P value of 0.2 and I'm comparing that 2.5 and that is less than five. So I have a P value that's less than alpha. And any time the p values less than alpha, you reject the null hypothesis. So we're rejecting H. Not now had the p value been greater than alpha than we would fail to reject H. Not so there's not enough evidence to say that me was greater than 100. But since the P values less than alpha, we are rejecting this null hypothesis, meaning we're accepting this alternative hypothesis as true. So a step five we basically just summarize that. So we're gonna say there is because we're rejecting there is sufficient evidence to suggest That the population mean I'll just say μ is in fact greater than 100. So since that we rejecting the null hypothesis, there is sufficient evidence to suggest that μ is greater than 100. That's the five step hypothesis test for a T. Test whenever sigma is unknown.

Mhm. MS question given that end it's 40. What X is 1 25? Yeah. Mhm. And because Standard deviation as 12 and nine. Okay. Uh D T up over to value for 99 confidence interval. Thanks 39 degrees. Mhm. Freedom Okay. is 2.70 and uh value table T Alice. So now to calculate 99% confidence interval, the opposition, you can use this general, you know, it's funny that So plugging in the values that we know 20.5 plus or minus 2.7 earlier Times, Over the square root of 40. Well I should give us two distinct values for the upper and lower bounds. Um Expectedly She will be 1 14.98. Lower bound and the upper bound will be 126.02. So this is the region for the 99% Clinton's level.

Okay. This problem is for number 55. Um, we're estimating the B m I with the standard deviation being 7 45 And, um, we're estimating the mean of the and I, uh, with a plus minus one. Okay, with 99 percent confidence level. So in this case, if you're looking for the minimal sample size gonna be an minimal We have formula is the ALF of the two times sigma and over e and then square over here. Standard deviation 7.5. Where is gonna be the eat? So there's there's gonna be the signal within plus minus one. So we didn't call to being one, which is the maximum era for 99% confidence level, You know, disease, or what goes case will be to going 57 So if we know that we can just go ahead and plug everything we have, we're into the Cingular to coin fired. Stellar five times. Sigma 75 desire by, uh, Mac Margin of the era can large in the era which your one square I was in cockpit or you have 19.31 25 inside of the open Percy's. We're square. That getting 372 point mind. Seven. Because this is the minimum sample size, we need to round this up too. 373 so the minimum sample size would be 373.


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