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Don"t drink and drive: highway safety council particular vear: Followlng frequency reported that there were 4012 fatalities among drivers In auto accidents in ...

Question

Don"t drink and drive: highway safety council particular vear: Followlng frequency reported that there were 4012 fatalities among drivers In auto accidents in place distribution of their ages. Approximate the mean age; Round your answer to one decimalAgeNumber of Fatalities 354 1534 85011-20 21-303 -4041-5051-6044961-70128Send duta to ExcelTho mean approximately

Don"t drink and drive: highway safety council particular vear: Followlng frequency reported that there were 4012 fatalities among drivers In auto accidents in place distribution of their ages. Approximate the mean age; Round your answer to one decimal Age Number of Fatalities 354 1534 850 11-20 21-30 3 -40 41-50 51-60 449 61-70 128 Send duta to Excel Tho mean approximately



Answers

Drunk Drivers In the last section, we saw that the age of a randomly selected, alcohol-impaired driver in a fatal car crash is a random variable with probability density function given by
$$ f(t)=\frac{4.045}{t^{1.532}} \quad \text { for } t \text { in }[16,80].$$
Source: Traffic Safety Facts.
(a) APPLY IT Find the expected age of a drunk driver in
a fatal car crash.
(b) Find the standard deviation of the distribution.
(c) Find the probability that such a driver will be younger than 1 i standard deviation below the mean.
(d) Find the median age of a drunk driver in a fatal car crash.

In this exercise were given that the probability is 0.4 that a traffic fatality will involve an intoxicated or alcohol impaired driver or non occupant in the first part of the problem. A We're supposed to find the probability that the number y of fatalities that involved into intoxicated or alcohol impaired drivers or non occupant is exactly three, at least three. And at most three. Now, before we can do that, uh we we would need to create the probability distribution for that uh random variable Y. And first, why can take on the value zero All the way to 8? Because we are looking at eight traffic fatalities. This make the numbers all the way 28 And then the next column will be for the probabilities. So six. See Yeah, you need to fill up the table with a different probabilities for different numbers. So we use a binomial distribution. Yeah. Number of fatalities is zero out of eight trouts. And the successful ability is 0.4. Mhm. Since and and it's a false and I yeah we copy the formula all the way through for the eight different values of y. Now we have our probabilities, the probability distribution. And in part a of the question We're looking at three different probabilities. The first one. The probability but why is exactly three? That means that we're looking at Y equals three six. Yeah. Oh And come up to the table. The value is zero points. So you just need to put that they're 0.278. Which you can round off to make it 0.279 67 Sure. Next looking at the probability That the number is at least three. So at least three means It could be three four or or four or 567 and eight. So the probability that y is at least three. So sad sleep. You obtained as follows. You have to get uh the probabilities for zero one and to then we add them up and from then and then we subtract from one. The some of these probabilities. So it will be equal to one minus Some of your abilities of 01 and two. See That gives us 0.684605 Michigan round off to 0.685. Next we're looking at the probability that the number is at most three. I believe that by he is at most three. Yeah this is Now for white we at most three it means why could be 012 or three but not any number Greater than three. So we need to get the some of these four probabilities. So we put the formula equals the sum first. For That will be 0.594-086. Which you can round up to zero five and four. So x. In part B. We're supposed to find the probability that the number is between two and four inclusive. That means they're going to be looking probability that it's two or 3 or four. So we focus on these three probabilities and get there some. So the some of those three possibilities is given by the formula equals some. It's three powerful. That's going to be 0.719954. Which can round off 0.7 20 But see we're supposed to find and interpret the mean of the random variable. Y. Now the mean of the random variable by meal is given by N. P. In this case n equals eight. So you put the formula equals eight Times here, which is 0.4 success probability. So the mean is 3.2 fatalities. So we can interpret it as follows that on average 3.2 of every eight traffic fatalities involved an intoxicated or alcohol impaired driver or non occupant. Yeah. And lastly, but d sorry, supposed to obtain the standard deviation of why? Now the standard deviation is given by the square root of n times p times one minus P. Which we can compute using the formulas with the equal sign and then square uh where it off? Yeah. Uh n which is eight times p which is 0.4 Times one may not be, which is 0.6 plus, And you have not at 1.3 uh which we can approximate it to the 1.4385641, which is 1.4 traffic fatalities. Mhm. Yeah.

For exercise or problem number 53. There are five different parts, and all the parts utilize the fact that the average is 47 0.5 and the standard deviation is 16.6. So now this problem is about, ah, the Federal Highway Administration's 2006 highway statistics, and it's ages for licensed drivers. So the 47.5 is the average age of a licensed driver, and the standard deviation is 16.6. So in this particular set of problems for part A, you want to find the percentage of drivers who are between the ages of 17 and 22. So we're gonna do is we're gonna first find the probability, and it does tell us that ages are normally distributed. So therefore, we're going to have to utilize our bell shaped curve. And in the center of our bell shaped curve, we're going to put that average of 47.5. And in this case, we are trying to find the percentage of drivers that are between 17 which would be way left and 22. So we will need to use our Z score. So to refresh your memory on the formula for Z score. Z equals X minus mu over Sigma. So we're going to find the Z score associated with 17 by doing 17 minus 47.5, divided by the standard deviation of 16.6, and you will end up with a Z score of negative 1.84 And then we're gonna do the Z score associated with 22 serving 20 to minus 47.5, divided by the 16.6. And this time you'll get a Z score of negative 1.54 So I like to always go back and put those on the bell. So negative 1.84 is associated with the 17 and negative 1.54 is associated with the 22. So when they're asking us, what's the probability that we're dealing with a driver between the ages of 17 and 22? It's also saying, What's the probability that are? Z score is between negative 1.84 and negative 1.54 so we can rewrite this problem and we could say it's the probability that Z is less than negative 1.54 minus the probability that Z is less than negative 1.84 and then you would rely on your standard normal table in the back of your textbook. The probability associated with Z being less than negative 1.54 is 0.618 and the probability associated with negative 1.84 is 0.3 to 9. And therefore, when you subtract, you get 0.289 But the question said, What is the percentage of drivers? So we would have to transition this into a percent, so the percentage of drivers would be 2.89% between the ages of 17 and 22. Now let's move on to Part B and in part B, similar set up in Part B. We are trying to find the ages of drivers that are younger than 25. Well younger than 25 translates into X is less than 25. So again, we're going to use our bell shaped curve. We know that the average is 47.5, so 20 five is going to be to the left, and we want the Z score associated with 25 So we're gonna do Z equals 25 minus 47.5, divided by the standard deviation, which was 16.6. And for part B, the Z score turns out to be negative 1.36 So we're gonna put a negative 1.36 on our bell. And when we talk about being less than 25 we're also talking about the Z being less than negative 1.36 So we look that value up in your standard normal table. You're going to find an area of 0.869 which transitions into 8.69% of drivers are younger than 25. And let's take a look at part C in part C, we are asked to find what's the percentage of drivers that are older than 21? So we're going to do the probability that excess be greater than 21. So again, we're going to draw our bell curve. I'm gonna put her 47.5 in the center, and this time we want 21 we want greater. So we're gonna find the Z score associated with 21 7 21 minus 47.5, divided by 16.6. And our Z score is negative 1.60 so we can put that on our Bell Native 1.60 So when we're talking about drivers older than 21 it's no different than saying, What's the probability that are Z score is greater than negative 1.60? Well, because we are going greater than we would have to do one minus the probability that Z is less than negative 1.60 We would look in the chart and Z being less than negative 1.60 with the 0.5 for eight. So we're talking 0.9452 which translates into 94.52% of drivers are older than 21. Let's move on to Part D. In Part D. You were asked to determine what percentage of drivers are between the ages of 48 68. So again, I'm a big fan of that curve to give you a picture. Representation average was 47.5 and we're going between 48 and 68 so we're going to calculate the Z score for each Z equals 48 minus 47.5 over 16.6 and we get a Z score of 0.3 and then the Z score for 68 would be 68 minus 47.5, divided by 16.6, and we get a Z score of 1.23 So when you're talking about being between 48 68 you're also talking about Z being between 680.3 and 1.23 So we can tackle this by finding the probability that Z is less than 1.23 And from its subtract the probability that Z is less than 0.3 We would go to our standard normal table in the back of the textbook, and the area to the left of 1.23 would be 0.8907 and the area to the left of point No. Three would be 30.5120 resulting in 0.3787 or 37.87% of drivers are between the ages of 48 and 68. And then finally, for part E you are asked to find the probability or the percentage of drivers older than 75. So we're going to transition that into the probability that X is greater than 75. We're drawing that bell with 47.5 in the center. That's not quite center. So let's back that up. A second centers more right there, and we want to place 75 on here and find it Z score. So the Z score would be 75 minus 47.5, divided by 16.6, which yields a Z score of 1.66 So when we're talking about being older than 75 it's no different than saying the probability that Z is greater than 1.66 So we're going to do one, minus the probability that Z's less than 1.66 and that would be one minus 10.9515 or 0.0 for 85 which transitions in 24.85% of drivers are older and 75

Over the function of the number of fatalities in car crashes is given as an X equals to 0.33 X. X cube minus 0.118 x square Plus 0.215 x plus 0.7. Where X is from 0 to 7. Okay. And X equals to zero corresponding to the age group. That is 50-54. Okay. And same x equals to one for the age group. That is 55-59 and same. Okay. And we have to find out part a what is the and is the fatality rate? Okay. That is per 100 million vehicle miles driven? Okay. So we have to find out in part a what will be the fatality rate for the age group? 50 to 54 per 100 million vehicle miles. Okay? So we're able to find out. And okay then age group 50 to 54. Okay. 50 to 54. Age group that is we can say X equals to zero at X equals to zero. So we will find and zero that is we will put X equals to zero in this equation. And it will be 0.3360 cube -0.1181 sq plus 0.215 Multiplied by zero plus 0.7. Okay, So when we solve this will become zero this field zero. This will zero and the rest will be 0.7. So 0.7 per 100 million medical mile driven will be the fertility rate. And for the age group 50 to 54. Okay. And now part B of discussion that is we have to find out. And for the age group 85 to 89. Okay, so 85 to 89 age group, so we can say 50 to 54 it is X equals to zero and 55 to 59. That is X equals to one. Then we will compare then 85 to 89. It will be at X equals to seven. Okay. And now we have to find out. And seven that will be we will put X equals to seven here. And it will be 0.03367 Cube -0.1187 sq plus 0.215 Multiplied by seven plus 0.7. Okay so when we saw this this will be This will be and seven it will be the fertility rate per 100 million. We call driven. It will be 7.95 and this will be the final answer. Apart. View of discussion. Thank you.

For the problem, given here 36. We see that with aging drivers, the number of car accidents, we can see the increase. So this is gonna be and of X equals 0.336 X. Where minus zero point 118 X squared plus zero point 215 X. Last 0.7. This is the graph that we end up getting X. Is the age group of the driver. So we want to know um the fatality rate of 100 per 100 million vehicles for the average driver in the 50 to 54 category. So that's going to be an of zero. Let me see, that's only 0.7. Um It's the Fed tolerate for 100 million. Then we have the 85 to 89 group that's going to be N. F seven. So I end up seeing that the fatality rate is going to be. Um Since this is N. F. Seven is for people through 89 we end up getting this is our fatality rate with an zero. This just needs to be cute though, so we get 7.9 as our answer.


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