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.