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By hand, compute the z-score for a sample of 25individuals with a mean HR of 71 BPM and assuming that the SD forHR found in the NHANES 2001 dataset is the true popu...

Question

By hand, compute the z-score for a sample of 25individuals with a mean HR of 71 BPM and assuming that the SD forHR found in the NHANES 2001 dataset is the true population value.Then, determine the probability of occurrence of this mean in anormal distribution and comment on the relative frequency of suchan observation. NO DATA IS MISSING.

By hand, compute the z-score for a sample of 25 individuals with a mean HR of 71 BPM and assuming that the SD for HR found in the NHANES 2001 dataset is the true population value. Then, determine the probability of occurrence of this mean in a normal distribution and comment on the relative frequency of such an observation. NO DATA IS MISSING.



Answers

Construct a $95 \%$ Z-interval or a $95 \%$ $t$-interval about the population mean. If neither can be constructed, state the reason why. For convenience, a normal probability plot and boxplot are given. The heights of 20 - to 29 -year-old males are known to have population standard deviation $\sigma=2.9$ inches. A simple random sample of $n=15$ males 20 to 29 years old results in the following data: (FIGURE CANNOT COPY)

The following is a solution for # seven. And this gives us a generic example where the sample Is size 15, but it comes from a normal distribution, so the population is normal. Therefore that sample size of 15 is okay. Now, had it not been normal? That sample size needs to be at least 30, but since it's normal, that sample size can be, you know, whatever, so 15 is fine, and the sample mean of this particular sample is 23.8. The sample standard deviation was 6, 3, And we're supposed to test at the significance level .01 If the mean is different than 25. Now, first off, we're going to use the T test, and the reason why we have to use a T test here is because we're given the sample standard deviation, were not given the population standard deviation sigma were only given the sample standard deviation. So, if we were given sigma, we could use the Z test, but since we're not we're given s we have to use the T test. So we're gonna do a five step hypothesis test, and the first step is to state our hypotheses and the null hypothesis h not Always assumes some sort of equality in this case, since we're testing a population mean μ equals 25. And the alternative, It says that we're just testing to see if it's different, so if it doesn't say up or down or greater than or less than, then we just assume that it's not equal to so different than 25, so not equal to 25, the 2nd and 3rd step, kind of, they kind of fall in line with each other. So I'm gonna do these together because um you can do it with the formula, but it's gonna take a little bit longer, so I'm gonna use technology. So if you have technology available, I really suggest you use that. If not, then you can start plugging in, you know, this stuff up here into your formula, but the T. Is stands for the test statistic, and then the p value is a probability. So if you go to the T. I 84 and go to stat and then air over to test, we're gonna use this T test right here, so go down, go to the T test and then make sure summary stats is highlighted. The mu not that's the hypothesized value of 25 in this case, that's just your h not The x bar, is your sample means that's 23.8 S stands for the sample standard deviation, and that was 63 And then the sample size remember was 15. And then we get down to this alternative and it's already set up as not equal to so we can just leave that the same. And then when wherever we calculate it gives us a lot of good stuff but this uh the T value and the P value are really the only things that I care about. So the T value is negative 0.74 and then the p value is 0.47 So let's go and write those down and then we'll talk about them. So the t value, the test statistic is negative zero point 738, which is pretty close to zero. So this is the so the p value is quite large, so 473 And what we do with that PVP PV I really is the most important thing um in this process. And step forward we explicitly compare the P value with our significance level alpha. And in this case it's significantly larger than alpha. P value is very much larger than our alpha value. So whenever it's larger than or whenever the PV is greater than alpha, then we fail to reject H nine. So we failed to reject the null hypothesis that μ equals 25 or more or less. We're accepting that the population mean is in fact 25 or at least we we say that it's we can't reject that. It's not or we can't say that it's not. Now. Had this been less than had the P value been less than alpha than we would reject H dot, but it's not. So we failed to reject. And then step five, we just kind of package that together into a nice conclusion and we'll say there is not sufficient evidence to suggest that the population mean, I'll just calm you for short. The population mean is different from 25. So whenever we fail to reject, that means there's not sufficient evidence to say that the whatever the alternative hypothesis is. Okay, so that's your five step T test using the P value method.

Pro Clinton like the degrees of freedom is ableto end minus one, which is seven minus one. So it's equal to seize. Yeah, and the critical values X squared of one Minnesota. It is open 676 and X squared off Alfa, which is 18.548 So the boundaries or the confidence interval all the standard deviation is the squared off and minus one over X squared off over two times s, which is seven minus one over 18.548 times square. Root off 0.64 approximately equal to open 455 the square root off n minus one over x one minus alpha times s She is equal to seven wireless one over. Also in 676 times square load off 6760.64 which approximately equal toe 53 mhm three. So the boundaries off the variance will be the square off these values, which is off 455 square and 2.383 square, which is opening toe awesome and 5.6 sevens, right

That's a hypothesis test given the following information. X bar or mean for the sample is 21 sample size and equals 32 Population standard deviation sigma equals four H. Not or no hypothesis new equals 22 H. An alternative hypothesis and use less than 22 significance level alpha equals 220.5 Since we have the population center deviation, we're going to be conducting a one mean Z test for this hypothesis test. So there are three steps will go through them one x 1. First. We have to compute disease statistic associated with this sample. So dizzy stat is given by the formula expo minus mu, divided by a single room and plugging in our values we get Z equals negative 1.41 next part B. We have identified the critical value in critical region For a confidence of 5%. The critical value for a one tailed test Is -1.6.5. The critical region therefore is everyone in less than 1.645, or rather negative 1.645. Next part C, we can make a conclusion for this test. That conclusion is since she is not in the critical region, we cannot reject it, not. We fail to reject the null hypothesis. That means that we lack evidence that new is less than 22.

All right, and it's moved to question 1 84 Um, so in this question, our sample size can is equal to 80. The now hypothesis ISS, the population mean is equal Teoh 200. And if we use me to represent the population mean it ISS equipment and to say our no hypothesis is is mu equaled Teoh 200. All right, part be the alternative hypothesis ISS population mean is different from 200. Or we can just employees say our mute is now equal to 200. All right, um parte see, what's the value for Alfa? You hear our alphas? He co two Thera playing there a one because we're interested in on Thea Aeroplane there a one level of significance. All right, parte de, what's the is that value for Alfa divided by two? So here it is, the critical value off Seth for the to tout situation as we desired. So we can go Teoh table for four b and find the corresponding Call him for the, um, Alfa Value. Which is there a point there? What? And we will have the critical value equaled Teoh 2.58 Okay, in part B What is the value from you May sound the null hypothesis is so, as we said in part a, our mu is 200. All right, on, then. What is a value for ex parte here? Export is how a simple mean And according to the question, our simple mean is equal to 205. All right, and, uh uh, this standard deviation, which is sick a lot. Yes, he called to 15. It is given in a question and, uh, cart age ciccolo off export. Or so it is a standard arrow off the mean and it is equal to speak a lot of item by square root off school which is 15 divided by a squared off 80 and IHS, he could chew one point 68 All right. And we need to calculate the past a statistic that star. So according to the formula that star is, you go to your ex four minus the hypothesized value for you over sick. A lot divided by square enough. And and here the denominator is also he quote Teoh seeking law export, which is the center arrow for the mean all right way won't just plug in the numbers, Um, using the original formula. So the simple me is to go five minus 200. Go far 15 divided by a squared. Rode off Katie. And what we have, IHS two point 98 So this is ours. Test a statistic. Oh, here, we need Teoh. Make the decision. Can We've already have the value for the Zet off Gulf over to it as you go to 2.58 And we also have our test, a statistic that star is equal to 2.98 So to make the comparison between though and we found power is that star is greater than the that alfa over to you. Which means Oh, where is that star falls into the critical region. So we're going Teoh reject. Well, no hypothesis. All right. And if we draw a curve, Um imagine the critical value. Sad elf over two is here. And then as he go to 2.58 here we have the negative one negative 2.58 And the area for this How in this towel they ended up together is equal Teoh health, which is their applying? There are one, but our test of statistics that star and somewhere here and as 2.98 So obviously, um, can't falls into the critical ranging. So we're going Teoh reject the null hypothesis.


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