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Problem 53 Hard Difficulty

According to the Federal Highway Administration's 2006 highway statistics, the distribution of ages for licensed drivers has a mean of 47.5 years and a standard deviation of 16.6 years [www.fhwa.dot.gov]. Assuming the distribution of ages is normally distributed, what percentage of the drivers are:
a. between the ages of 17 and $22 ?$
b. younger than 25 years of age?
c. older than 21 years of age?
d. between the ages of 48 and $68 ?$
e. older than 75 years of age?

Answer

a .0289; b .0869; c .9452; d .3787; e .0485

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Video Transcript

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