5

Point) retrospective study was conducted to investigate relationship between the colors of helmets worn by motorcycle drivers and whether they are injured or killed...

Question

Point) retrospective study was conducted to investigate relationship between the colors of helmets worn by motorcycle drivers and whether they are injured or killed in crash: Results are given in the table below (based on data from Motorcycle Rider Conspicuity and Crash Related Injury: Case- Control Study;" by Wells et al., BMJ USA Vol 4). All of the following questions should be rounded to the nearest percent:Colour of Helmet for Motorcycle Riders involved in CrashNot Injured Injured/Kille

point) retrospective study was conducted to investigate relationship between the colors of helmets worn by motorcycle drivers and whether they are injured or killed in crash: Results are given in the table below (based on data from Motorcycle Rider Conspicuity and Crash Related Injury: Case- Control Study;" by Wells et al., BMJ USA Vol 4). All of the following questions should be rounded to the nearest percent: Colour of Helmet for Motorcycle Riders involved in Crash Not Injured Injured/Killed 491 213 377 112 Black White Yellow/Orange (iv) Red 170 (v) Blue Total 1124 429 What percent of the Motorcycle riders recorded for this study wore black helmets? 0.4533 What percent of the Motorcycle riders recorded for this study were injured or killed? 0.2762 What percent of the white helmet-wearing Motorcycle riders were injured or killed? 0.261 What percent of the red helmet-wearing Motorcycle riders were injured or killed? 632 What percent of the yellowlorange helmet-wearing Motorcycle riders were injured or killed? 0.0186 What percent of the blue helmet-wearing Motorcycle riders were injured or killed? 0.0606 Do the colour of motorcycle helmet worn and injury/death chance appear to be independent?



Answers

A case-control (or retrospective) study was conducted to investigate a relationship between the colors of helmets worn by motorcycle drivers and whether they are injured or killed in a crash. Results are given in the table below (based on data from "Motorcycle Rider Conspicuity and Crash Related Injury: Case-Control Study", by Wells et al., BMJ USA, Vol. 4). Test the claim that injuries are independent of helmet color. Should motorcycle drivers choose helmets with a particular color? If so, which color appears best?

All right, So, we have a few proportions that are given to us by the National Highway Traffic Safety Administration. They published reports about motorcycle fatalities and helmet use. So, they have a distribution that shows the proportion of fatalities by the location of injury. For motorcycle accidents. The first step always is to draw the observation table very important. So this is location of injury. Right? So the first one is multiple locations. Multiple locations. Then we have head, neck, abdominal, tor, axe, head, neck, abdominal, abdominal oratory accesses right, abdominal to Iraq. We have lumber spine. All right, number. Mhm. Fine. Ah Right now we have the distribution that is given to us. Right. The proportion. So this is the proportion the proportion that is given to us by the National Highway Traffic Safety Administration. Okay, So for multiple locations at this .57.57 for head, it is .31 For Nick, it is .03 For abdominal authorities at this zero six, Then it is .03 again for lumbar or spine 0.3 All right. Now, what else do we have now? What we have is the data that shows the location of injuries and fatalities for 2068. Riders who were not wearing helmet. Which means our sample size is given to us as 2068. All right. So, let's just Right. This is the observed frequencies. Right? So, this is the column for the observed. Okay, let's just draw this little neatly. Alright, So from multiple locations, it is 1036. Then it is 864. Then it is 38, then it is 80 three, Then it is 47. All right. Now, We have to use 0.05. Level of significance. All right. And we have to It leads to the conclusion Does the distribution of fatal injuries for riders not wearing a helmet? For the distribution that is given to us for all writers? Okay, So let us form the nail in the alternative hypothesis. What is another hypothesis? And I like What is that? The distribution of fatal injuries? The distribution of fatal injuries for riders not wearing a helmet not wearing a helmet is similar is similar to the distribution mm Given Bye, the Safety Administration, Traffic Safety Administration by the Traffic Safety Administration. Right, Okay, What will be the alternative hypothesis? The alternative hypothesis will be that the two distributions that we have, the two distributions that we have differ. That is the distribution of fatal and juries for riders who are not wearing the helmet is not similar to the distribution given by the Traffic Safety Administration. All right, now, that we have another in the alternative hypothesis. The first step in a chi square analysis is to find the expected values. So the expected values for all the categories. So let's say that the expected value for category I Right. This value will be given by the sample size. The sample size multiplied by the probability for each category or the probability for category I in this case the probability for category I. Right. So let's look at this in action. So I think we had 2068 vehicles. Yes. 2068 writers. So this edition is 20 succeed. So this is our sample size. Okay, now, let us look at the first category. That is the category where the location of injury is given us multiple locations. So the proportion or the probability according to them is 0.57 So this is going to be 2068 multiplied by 0.57 which is 1178.76 This is 117 8.76. This is the column for the expected values, right? The expected values Then we have 20 succeed multiplied by .31, which is 641.08 6 41.08. Then we have 2068 in 2.03, which is 62.04, 62.04. Okay then we have 2068, multiplied by .06, which is 124.08 1 24.08. Then again, we have 2068 in 2.03, which is 62.04, which is similar to the neck category. Right now that we have the expected mandy. The next step is to find the chi square statistic. How do you find the chi square statistic for all the categories that you have? What are you going to do is you're going to find the difference between the observer and the expected values. You'll square them, you will divide this by the expected value. And in the end you will air all of these up. Let's look at this formula in action. Are you for the first category? That is where the injuries in multiple locations? The difference that we observed and the expected is 1178.76 -1036. Now, we square this which is 142.76 square. And they want this by (117 8.761178.76. Which is 17.2897 or 17.29 17.29. Right Then we have 864 -6 41.08. All right, we square this 2 22.92 square. Okay? And divide this by 6 41 6 41 Which is 77.52, 77.5-77.52. Now, just looking at this one value, I can say that. My God, this is way higher. So this is definitely we are going to reject the null hypothesis. Well, let's carry on. So this is 62.04 -38. We square this And divide this by 62.04, which is 9.31. Then we have 1 24.08 -83 square. This Divided by 1 24 points you'll eat which is 13.6, 13.6. And then we have 62.04 minus 47. You square this. And they went there were 62.04. This is 3.646, 3.646. All right. Now, what we have to do is we have to add all of these up. Okay, So this is what we are going to do Some one of these up. So this is 17.29. That's 77.52 plus 9.31 plus 13.6 plus 3.6463 point 646 1 21.36. This is ridiculous. 1 21.366. Absolutely ridiculous. Mhm. So we can say definitely 100 that we are going to reject and hypothesis the distribution of fatal injuries for writers not wearing helmet is not similar to the distribution that is given to us by the traffic safety administration. I have not completed my analysis for just looking at this value. I can say that yes, this is going to be our answer. But again, that does follow the steps. The next step is to follow the degrees of freedom is to find the degrees of freedom. This is given by the formula number of categories that we have Number of Categories -1. All right. How many categories do we have? 12345 So we have five different categories. So this is going to be 5 -1, which happens to be four. Right? Somebody who's a freedom is for now in order to further continue my analysis, I have two methods. The first one is the P value method, the peak value method, and the second one happens to be the critical value method. The critical value method. Okay, so in order to find the P value, what I need is Mic eyes question to stick and my degrees of freedom. Now I can use either apply square table to calculate the P value, or I can use a statistical software or a calculator, which is what I want to do. We're here, this is 1 21.366 And degrees of freedom is for. So this is 121.366 and the degrees of freedom is four. My significance level is 0.05 and my p value is 0.000000001. That is very close to zero is what I mean to say. So my p value is less than 0.00001. My alpha For this question is 0.05 0.05. And since my p value is less than alpha, I will reject minor hypothesis, reject my final hypothesis each, not How about the critical value method for critical value method? I just need to find that critical value for the chi square statistic beyond which I will reject minor hypothesis. Right? So my alpha is 0.05 and my degrees of freedom is for I put in these two values in this calculator, I had calculate. This is a critical value calculated by the way, and I get my critical value is 9.488. So my critical value critical Value is 9.488. Which means if this is my chi squared graph, write something like this, 9.40 let's say is over here 9.488 Beyond this is the rejection region that 5% Right, the area of 0.05. Now, my value is 1 21 point something which is if I extend this all the way where it will lie on the way. So this is definitely in the rejection region. So I will reject my age. Not. We see the similar answer from both the methods. So what is going to be my conclusion? I would say that At 0.05 level of significance level of significance, we have enough statistical evidence. Do you suggest that the writers without helmet without helmet have a different distribution, have a different distribution? Mm Then that proposed by the proposed by the what was the name? The Traffic Safety Administration proposed by this? Traffic Safety Administration. Traffic Safety Administration Administration are great. Okay. Now we have the part to says parties has compared the observer and the expected counts for each category. Okay, what does this information tell us? All right. We noticed that the observed count for head injuries is much higher than the expected ones. If I'm not wrong, that is just go here. This is what we notice, right. The observed value For head counts is much greater than the expected was. You see the 77, we see such a big jump. We reached one radio and just solely because of this one valley. Right. So the observed count for head injury is much higher than expected, while the observed count for all the other categories are lower. So this can lead us to conclude that the motorcycle fatalities from head injuries are the more frequently for riders not wearing a helmet. And this is how we go about doing this question.

For this problem were given the following table shown right here hurt a states to use the data to compute estimate of probability than a motorcyclist wears a dot compliant helmet. To do this, we take the number of favorable outcomes over the total number of possible outcomes. So the total number of people that wear dot compliant helmets are 350 divided by the total number of possible outcomes, which would be 350 last, 170. This gives a probability of 0.6731 Herbie States The probability that a motorcyclist war dot compliant helmet five years ago was point for eight, and last year this probability was 80.63 Would the National Highway Traffic Safety Administration be pleased with the most recent survey results shown above in the most recent survey? As calculated in part a. The probability of wearing a dark, compliant hum? It is 0.6731 This is greater than both 0.48 and 0.63 So, yes, the Highway Traffic Safety Administration would be pleased with the results because the probability of wearing a dot compliant helmet is increased. Burt see asks what is the probability of dot compliant helmet use by region of the country in which region has the highest probability of wearing a dot compliant helmet? To do this, you take the number of favorable outcomes for each region and divided by the total number of possible outcomes in the region. So for the Northeast, number of favorable outcomes would be 96. Because 96 people in the region war dot comply at home, it's divided by the total number of people in the region, which is 96 plus 62. This gives a probability of 0.6076 For the Midwest. It would be the number of favorable outcomes which is 86 divided by the number of possible outcomes, which is 86 plus 43 to give a probability of 0.6667 For the South, it will be the number of favorable, which is 92 divided by the number of possible, which is 90 to close 49 which would give a probability of 0.65 to 5. For the West, it would be 76 divided by 76 plus 16 to give a probability of 160.8261 Of all the probabilities, the West has the highest and so therefore, the region with the highest probability of dot compliant helmet use is the West.

So part is truly just asking us to make two matrices for each of these, uh, piece, uh, maps of information, these diagrams, these tables of information. So if you do one for 2000 and eight, we're making one for 2000 and eight. Then I'm simply just going to write the numbers for the compliant column in order. So 45 67 61 71 Right Then I'm just going to create another column for the non compliant, which would be 8, 16 14 and five. Fall under the same thing for 2000 and nine. Simply again. Just going to create another matrix using the same data. The data from 2009 diagram table. So 61 67 65 83 and then for non compliant 15 86 and four. Not really much to that. You're just kind of combining me. You're just writing them as you see them. Part B ask you to use the two matrices from far day to write a matrix showing the change in helmet usage change. Meaning they want to know the difference between these two. Meaning Essentially, they're asking us to subtract thes two matrices and write the answer matrix that we get for part B Well 45 minus 16. Because remember, if we add or subtract matrices, they just need to be the same size. And so, since both of these matrices are the same size, therefore by twos, we just subtract each corresponding entry. So 45 minus 61 give me negative 16 eight minus 15 would give me negative seven 67 minus 67 0 16 minus eight is eight 61 minus 65 is negative. Four 14 minus six is eight 71 minus 83 is negative. 12 and five minus four is one That is your part. B matrix truly is a simple is that is just doing subtraction. Part C then wants us to analyze the results from part B and discuss the extent to which changes from away tau nine differ from one region to another. This is a pretty open ended question. As far as things were considered, there really isn't a specific answer for part C that you need to give. Like what I can say what I want to say. But that doesn't have to be the exact same thing that you have as I'm looking at this stuff you could talk about how in the Northeast region, the number of non compliant helmets actually increased from O A 209 whereas everywhere else it looks like they decreased Midwest, South and west. Right? Um, you can see that by the fact that we had a negative differential here, but all the other three were positive. You could talk about that. You could talk about comparing the amount of compliance to non compliance and how that changed from away 209 There's multiple things you could talk about here, and there's not a specific answer that you should have for this. So whatever you see is probably worth writing down as far as an answer for Part C.

A survey of bicycle riders in the U. S found that 60% of 1000 and 20 bike writers war, helmet or own a helmet Let's calculate a 95% confidence interval. Remember that the formula for the 95% confidence interval is a do P prime plus or minus the maximum air estimate. So for this example, R P Prime would be six 12 out of 1020. I got that by doing 1020 times 10200.60 and I get 600 and 12. You're also could just leave it at 0.60 So the 95% confidence interval first thing I have to do is find Z for this example doozie of an Alfa of 950.5 because one minus 10.95 point 05 divided by two. And I'm gonna look up on my table 0.25 and I see that my Z value is 1.96 You might start memorizing some of these Z values because you get so used to using them and there's not very many of them. So this 1 95% is 1.96 times 0.6 times one minus 10.6, which is 0.4 divided by 1020 and this value would be 0.30 So that's my maximum error. Estimate the value of E. Then I'm gonna take that I want to do 0.6 minus 0.30 point six lust 0.30 Well, I get 0.57 to 0.6 three to my 95%. Confidence interval is somewhere between 57% to 63% if you like to express it as a percent. So we can say that there is compliance for a little bit over half of US bicycle writers. Some could argue that this percentage to be much higher because if you're riding a bike you should be were wearing a helmet. And so I might be happy that at least you know, 57 to 63% of American writers are writing their using their helmets. So let's look at different sample sizes so we can use a formula to determine this. If we there's a specific margin of error that we would like and for a specific confidence interval, we can determine what sort of sent sample size we need, and to do that, we use the formula of we do the Z value squared times p Star Time's Hugh Star and you divide that whole thing by the value of E squared. It looks complicated, but let's check it out. So our first example we have a say. We want a margin of error of 0.2 with 95% confidence, so we would do The Z value that corresponds with 95% is 1.96 He is 0.6 times Q Star, which would be one minus 10.6, which is 0.4. And it tells us that we want maximum error estimate of 0.2 squared, which is equal to 2304.96 which we're going to round 2 2305 No matter what the decimal is, you always want around up. Alright, what about a maximum air estimate of 0.4 within the 95% confidence in not much will change for this one Still do 1.96 squared times 0.6 times 0.4, divided by 0.0 for Spared, which is equal to 5 76.24 which we're going to round 2 577 No matter what the decimal is for sample size, we always round because of always want to make sure we have enough. Um, the people, whoever that end is for the experiment. And if we have 5 76 we might not get that 760.4 margin of error that we're looking for. 90% confidence interval with a maximum error estimate of 0.2 You do 1.645 I did that by looking up in my chart. One minus 10.9. They get Alfa, which was 0.10 divided by two. On my chart, I looked Z of 20.0 eyes and I got 1.64 fat last 0.6 times 0.4 divided by 0.2 squared, and I get 1623.615 which is 1624. That's the sample size I would be looking for there. So what happens to the maximum error and the sample size? What can we compare there? So let's look at when the confidence intervals are the same. So right here with 95% we have 950.4 point 02 Let's see what happened when my maximum here estimate. So I went from 0.0 for the 0.2 that went decreased. The sample size has the increase. There's the larger the sample size, the close, the less that margin of error. It's So what happens to as the confidence level, this 90% and what happens to the sample size? So if I look at 90% to 95% my maximum error stayed the same. So, uh, my confidence level from 90 Teoh 95 went down, So 90 to 95 went well, actually, let's say 95 to 90. Your confidence level is going down. What happens to the sample size with a sample size also decreases. That's because you're now less confident than you were before that you're gonna get that maximum that you know that your actual proportion is going to be in there so your sample size can can decrease. Um


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