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Refer to the data in question 3 on the growth of COVID-19 confirmed cases in South West Ontario from March 8 t0 June 23 . (See Project Data File) Make sure that the...

Question

Refer to the data in question 3 on the growth of COVID-19 confirmed cases in South West Ontario from March 8 t0 June 23 . (See Project Data File) Make sure that the distribution is applicable (Normal or approximately normal) Build a histogram with the bin values provided (bin width 75) and check whether it is approximately symmetric and bell-shaped Use Descriptive Statistics function from Data Analysis. State the mean, median and mode What is the degree of skewness? Construct 90%, 95%, and 99% c

Refer to the data in question 3 on the growth of COVID-19 confirmed cases in South West Ontario from March 8 t0 June 23 . (See Project Data File) Make sure that the distribution is applicable (Normal or approximately normal) Build a histogram with the bin values provided (bin width 75) and check whether it is approximately symmetric and bell-shaped Use Descriptive Statistics function from Data Analysis. State the mean, median and mode What is the degree of skewness? Construct 90%, 95%, and 99% confidence intervals for the average number of new cases in South West Ontario. Interpret each confidence interval. #Show the output of confidence interval (the margin of error by Excel), and then calculate LCL (lower confidence level) and UCL(upper confidence level) by labeling "LCL" and "UCL" for the answers clearly. (Round to whole number) Which interval is narrower? Why?



Answers

Please provide the following information for Problems $11-22$. (a) What is the level of significance? State the null and alternate hypotheses. (b) Check Requirements What sampling distribution will you use? Explain the rationale for your choice of sampling distribution. Compute the appropriate sampling distribution value of the sample test statistic. (c) Find (or estimate) the $P$ -value. Sketch the sampling distribution and show the area corresponding to the $P$ -value. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level $\alpha ?$ (e) Interpret your conclusion in the context of the application. Note: For degrees of freedom $d . f .$ not given in the Student's $t$ table, use the closest $d . f .$ that is smaller. In some situations, this choice of $d . f .$ may increase the $P$ -value by a small amount and therefore produce a slightly more "conservative" answer. Ski Patrol: Avalanches Snow avalanches can be a real problem for travelers in the western United States and Canada. A very common type of avalanche is called the slab avalanche. These have been studied extensively by David McClung, a professor of civil engineering at the University of British Columbia. Slab avalanches studied in Canada have an average thickness of $\mu=67 \mathrm{~cm}$ (Source: Avalanche Handbook by D. McClung and P. Schaerer). The ski patrol at Vail, Colorado, is studying slab avalanches in its region. A random sample of avalanches in spring gave the following thicknesses (in $\mathrm{cm}$ ): $\begin{array}{llllllll}59 & 51 & 76 & 38 & 65 & 54 & 49 & 62 \\ 68 & 55 & 64 & 67 & 63 & 74 & 65 & 79\end{array}$ i. Use a calculator with mean and standard deviation keys to verify that $\bar{x} \approx 61.8$ and $s \approx 10.6 \mathrm{~cm} .$ ii. Assume the slab thickness has an approximately normal distribution. Use a $1 \%$ level of significance to test the claim that the mean slab thickness in the Vail region is different from that in Canada.

The following is a solution to number nine and this were given the sample mean X bar is wanna wait and the sample standard deviation not the population standard deviation. The sample standard deviation is 10 and we're asked to find the 96% confidence interval whenever n equals 25. Now, since I only know s I only know the sample standard deviation, I need to use the tea interval instead of the Z interval. So you can do manually, I'm going to use the T I 84 because it doesn't really nice. So if you go to stat and then tests, it's going to be this eighth option down here, the T interval. So usually if you know sigma you're gonna use the Z interval, but this time I'm going to go to the eighth option, the tea interval, oops, Do it again, so stat tests and then eight. Okay, And then make sure you're covered over stats here, Um that's highlighted, and then you can start typing in your summary stats. So X. bar is 108. S. is 10 and then the sample size was 25 and then the sea levels .96. And whenever we calculate that this first line here that's going to be the uh confidence intervals. So one of 3.66 all the way up to 1 12 point 34. Okay so one of three 0.66 And 1 1234. Okay now we're gonna do the 96% confidence interval whenever we decrease the sample size to 10. So let's see what happens here. If we go back to stat And you can probably guess what's going to happen and then test and then we go down to this eighth option again and then everything stays the same except this time the n equals 10 And everything else stays the same. So we calculate and that gives us this confidence interval 100.42 all the way up. 215.58. Okay so let's write that down and compare 142 All the way up to 115.58. Okay. So what's the relationship between N. And the margin of air or the interval size? Well as you can see as in decreases the margin of air is going to increase. Making making the confidence interval wider. Okay. And this is super typical. So we've seen this before with the Z. Interval um The less you know or the smaller your sample size, the more variable that's gonna be especially the T. Distribution. You got to think of that. Um ah T distribution as in decreases that makes that t distribution more variable or more spread. Um So that's why that confidence intervals wider so that the lower bound is smaller and then the upper bound is a little bit higher. So what happens that's with the sample size. Now? What happens with the confidence interval this time we're back to the 25 is there in? But we need to find the 90% confidence interval. So we're gonna go back to stat Tests and then it's the 8th option again. And this time. So let's change this back to 25 And then the sea level we're going to change 2.9 for 90%. So we calculate what do we get? 1? 11.42. So let's write that down 104 0.58 All the way up to 111 42. So what's the relationship between the confidence level and the interval width? So if you compare this to part A it's a little bit narrower. Internet, so we go from one of 3 to 1 to force the lower bound is increased and then the upper bound 1 12 to 1 11 is decreased. So the interval the confidence interval is narrower than in part A. Okay, so this shows you that as confidence level decreases, the margin of error also decreases, making that that interval more narrow. Okay. And then the last part just asks um let's it says in the very very beginning that this population comes from approximately normal distribution. So could we conduct this test if the population were not normal? And the short answer would be no. If the population is not normal. Mhm. Then in needs to be at least 30 for Z or T. But in this case the T. Distribution. Okay, so since it's not normal, it's okay. If it is 25 or 10 or whatever, but if it's not normal, then that end needs to be at least 30 in order for us to use that central limit theorem and conduct this test.

The following is a solution to number 19 and this is where X. Bar the sample manner. The point estimate is one. Oh wait. And then the standard deviation for the population. So we do actually know what sigma is, is 13 and we're asked to find the 96% confidence interval whenever and equals 25 96% confidence interval whenever and equals 10. So a smaller sample size and then an 88% confidence interval whenever and equals 25. So a lower confidence level. And I should note that they do actually say that the population from which these samples were drawn is approximately normal. That allows us to use these smaller sample size. If we didn't know what the population distribution was or if there was skin is to the population distribution then we would need to have a larger sample size greater than equal to 30. And we kind of talk about that a little bit down the road. All right, so let's go to our calculator. So I'm gonna go to my calculator and um you know you can use the form if you want but calculators much quicker and it's always accurate. So I'm gonna go to stat and then there are over two tests and I'm gonna go to the seventh option here the Z interval. So since I know that I I know what the population standard deviation is, that I was given that sigma that allows me to use that Z interval. And here you're given either data or stats. Now we're given the summary stats. I'm gonna go ahead and just keep it on stats And the Sigma is 13. The x bar is 108. And then the end at least for this first time is 25. And then that confidence levels .9696% confidence level. And then whenever I calculate I get 102.66 So 1 13.34. So I'm gonna write that down Before I forget it. So 102 0.66 All the way up to 113.34. That's my 96% confidence interval whenever and equals 25. So now I'm going to do a 96% confidence that or whatever and equals 10. So almost the same thing. I go to stat Tests and it's that 7th option, Everything else stays the same. But this time I want to change that end to 10 And then I calculate and I get 99.557 and 116.44 99.557 In 116.444. So if you compare these two real quick um the confidence interval for part B is much wider than Part A. And that's because it's a smaller sample size and that kind of tells you what that second part of the answer is as in decreases. The margin of error is going to increase. So the smaller number that you divide by means that it's going to give you a bigger number. So the margin of error increases and therefore the confidence interval is wider. So the smaller the sample size, the wider the confidence interval. Now let's look at part. See this time if we go to stat tests Z interval this time the ends back to 25 but we're going to do an 88% confidence interval And that's gonna give us one of 3.9 612.04 103 0.96 And then 1 12.04. So between one and 3.96 and 112.04. And it says compare part C to a well, this is narrower. They're over than part A. Okay. So the reason is as the confidence level decreases, which is what happened went from 96% confident to 88% confident, then the margin of air Emmy also decreases. Therefore the confidence interval is narrower. Okay, so if you decrease the confidence of all, that will make the confidence interval narrower because the margin of error is going to be narrower too. All right. And then partied says, could we compute the confidence interval if the data is not normal? So if the population from which the data was taken, if it's not normal, can we compute a confidence interval? And the short answer is no um reason is because and is not at least 30. So if n is Great and equal to 30, then it doesn't matter what the population is. If it's normal, if it's not normal, if it's skewed, whatever Central limit theorem applies and in equals 30 and it's created equal 30, then we're fine. But um With these sample sizes, this is 25, 10 and 25, these aren't big enough. So the the population has to be normal in order to conduct these confidence intervals, which such with such small sample sizes and then the last part part e If there are three outliers above the main, how would that, you know, affect everything? So if there our three outliers above the mean? Well, first off, we probably couldn't do a confidence interval because that would skew the data and our sample sizes aren't big enough, but let's just assume that, you know, we may be truncated or something. Um In theory, the confidence in the rule would b shifted up, or it would kind of increase. Everything would just shift up towards the outliers. Thank you. So that's what's going to happen if you apply a few outliers, at least on the high side. If they the outliers on the low side, the opposite would happen. Everything would just kind of shift down a little bit.

Following the dissolution to number 10. And were given the summary stats that X. Bar, the sample mean is 50. And then the sample standard deviation that have a sample standard deviation, not the population standard deviation. So we're given S is eight. Now, since we're given S and not sigma, we're going to use the tea interval here, not the Z interval. So any time you don't know the population standard deviation, you have to use the tea interval. So we're asked to find the 98% confidence interval whenever in equals, 20, They were asked to find the 98% confidence interval whenever it is 15. So we decrease that sample size, Then we're asked to find the 95% confidence in a role. So we decrease that confidence level whenever in his back to 20. And we're just gonna basically just compare these two are these three. Um So I'm going to use technology here now, you can use the formula or you know, any technology one, I'm gonna use the T. I. T. Four because it does this quite nicely. So if you go to stat and then tests And it's the tea interval here says this 8th options to go to eight and the summary stats should be highlighted here. So we don't have any data. We just have summary stats and the ex partner I've already done this. So X bar is 50 S. Is eight In was 20 for this first time. And then the sea level the confidence level was 98% of 98. And then we calculate that in this top Row here, that's our confidence interval. So let's go and write that down now. All we have to do is just we don't have to interpret it at all because there's no word problem associated with this. So 45 457 All the way up to 54 point 541 543 star. So that's our confidence in everybody. That's about 10 apart. 10.1 or so. Um now we're gonna do a 98% confidence interval but we're gonna decrease that in. So if we go back to stat and then test it's the eighth option and everything stays the same. Except I'm sorry I think I've had 10, it should be 15. So this is going to go to 15 and We calculate and it gives us this so it goes from 44 to 55. Let's write that down and we'll compare. So 44 point five was it to nine or 79 seven 79 To 55.4- one. Okay. So what does this say about the relationship between the sample size or the degrees of freedom and the margin of error? Or the interval length or the inter with? So as in decreases which is what happened here notice that the interval got wider. Now if the interval is wider that means that margin of error needs to be larger. So the margin of error increases making the confidence in the rule wider. Okay and you can see that here. So here we have a confidence interval of about 9 9.1. And then here we got a confidence interval with of about 11 or 10.9 or so. So this is um the relationship between in and the margin of error or the the width of the confidence interval. Now what happens whenever we're back to 20? But we decrease that confidence level of 95%. So we go Tests and then eight And we're back to 20 But this time it's 95 instead of 98 And we get this here 46.3, let's say. And then 53.7. So let's write that down and then compare it. So 46 .256 All the way up to 53 point 744 So what happens with see compared to a well the lower bound got higher and the upper bound got lower. So this with appear was about nine Um units long 9.1. This with here looks like it's about seven or 7.5. So the interval is actually a narrower. So the interval, the confidence interval is narrower than in part. Hey, So that means as the confidence level decreases, which is what happened, it went from 98 to 95. The margin of error also decreases because it's a narrower interval. And then this last part, uh it just says that if this population were not normal now, in the very beginning, it said, assume that this population is approximately normal. Let's say that it's not normal. Can we still use this procedure? And the answer is no, because these sample sizes aren't big enough, so no. If the population is not normal Then in needs to be at least 30 for the T. Distribution in order to use the T. Distribution. Okay, so since it's normal in the original problem, that's that means that we can use these smaller sample sizes. But if it's not normal, that means those sample sizes are going to work, they need to be at least 30.

The following is a solution to number 20. Where were given the summary statistics X. Bar the sample mean is 100 and 23 then the population standard deviation. So we actually do know sigma this time, the population standard deviation is 17. And this allows us to use the Z interval instead of the T interval. And we're asked to find the 94% confidence interval if N is greater than area, if any is equal to 20, 94% confidence interval of in his 12 and then 85% confidence interval whenever it is back to 20. And then we're gonna compare those those three confidence intervals. Now the reason why we can have sample sizes so small is because the population of the original data was normally distributed. So the fact that it's a normal distribution that allows us to have these smaller sample sizes. So let's go and get started. I'm gonna use technology that you can use the formula if you so wish. But I'm gonna use the T. I. T. For I think it works quite well. So if you go to stat and then air over two tests and since we have the population standard deviation we can go ahead and find the Z. Interval. So there's the intervals that seventh option and here you have data or stats. Now we're not giving any data, we're just giving summary stats. So keep that stats highlighted And then the signal was 17 and the the mean 123 The sample size is 20 and the confidence level was .94 for 94%. And whenever we calculate this top line that gives us the confidence interval. So 115.85 to 130.15. I'm gonna write that down. 115 0.85 to 130 point 15 So we're 94% confident that it's between those two values. Now, let's go back to the calculator and go back to stat tests and it's the seventh option. This time we're going to decrease the sample size to 12, we're going to keep the confidence level the same, but we're going to decrease the sample size to 12 and let's see what happens. Well, that's 113.77. All the way up to 132.23 113 0.77 To 132 0.23 And let's just kind of look at this and see how these compare. So this second one, part B is quite a bit wider than Part A. The lower bound is smaller and the upper bound is larger. And so that just goes to show that as in decreases as you decrease your sample size, the margin of error increases, which means that confidence interval will be wider. Okay, so the less you know, or the smaller your sample size, that margin there will be bigger. Um And that is going to make the confidence interval wider. All right. So then last one. This is where we go to 85%, but then back to the original sample size. So if you go to stat tests and it's the seventh option again. So everything else here is the same. So we're gonna change this to 20. So n equals 20 and then the confidence level um this time is 85%. So we'll save .85. And then whenever we calculate this we get a (117.53 and 128.47 1 17.53. All the way up to 128 4 7. So if you compare this with part a. This is actually narrower so the lower bound is higher and the upper bound is lower. This means that part see is narrower. Mhm than part A. Okay, so as the confidence level decreases, the margin of error also decreases. Yeah, meaning the confidence interval is narrower or skinnier. Okay. And then part D. It says can we conduct a confidence interval if the original population is not normal and the quick answer is no. Um If population is not normal then And must be at least 30 And if you look back to the sample sizes, these sample sizes are not big enough. 2012 and 20, those are not big enough. It needs to be at least 30 if the original population is not normal or if it's skewed or have outliers of some kind, and then the last part part e if there is an outlier greater than the mean, how does that shake things up? So if there is one outlier greater than the mean, the confidence interval would be shifted up towards that outlier. Okay, so outliers tend to skew the data and excuse it towards the direction of that outlier, especially the mean. The mean is pretty non robust against outliers, so the mean is gonna get shifted up towards that outlier. And so will the confidence confidence that will follow that that print that statistic.


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