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0 / 1 ptsQuestion 4Speeds of cars in an expressway are monitored by high-speed camera units. The minimum speed of cars in the expressway is 60 kph and the maximum s...

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0 / 1 ptsQuestion 4Speeds of cars in an expressway are monitored by high-speed camera units. The minimum speed of cars in the expressway is 60 kph and the maximum speed is 100 kph: Cars going below the minimum; or going above the maximum; are apprehended: Suppose that speed of the cars on the expressway are normally distributed with a mean of 75kph and a standard deviation of 10 kph: What percentage of the cars travelling on the expressway will not be apprehended?Note: Write your answer as perce

0 / 1 pts Question 4 Speeds of cars in an expressway are monitored by high-speed camera units. The minimum speed of cars in the expressway is 60 kph and the maximum speed is 100 kph: Cars going below the minimum; or going above the maximum; are apprehended: Suppose that speed of the cars on the expressway are normally distributed with a mean of 75kph and a standard deviation of 10 kph: What percentage of the cars travelling on the expressway will not be apprehended? Note: Write your answer as percentage with two decimal places (i.e: 87.26%).



Answers

The speed of vehicles on a highway with speed limit 100 km/h are normally distributed with mean 112 km/h and standard deviation 8 km/h.
(a) What is the probability that a randomly chosen vehicle is traveling at a legal speed?
(b) If police are instructed to ticket motorists driving 125 km/h or more, what percentage of motorists are targeted?

Mhm. In this question, we were given that the speed of vehicles on some highway with a speed limit of 100 km/h. We have that they follow a normal distribution In which the mean speed limit is 112 km/h And the standard deviation is eight km/h. And we're asked to find the probability that we drive legally that some random vehicle is driving an illegal speed. And also what percentage of motorists will be targeted if they're driving past 125 km/h. Let's deal with the first case. 1st we have we're I get I wrote down the normal distribution for you, given a parameter mu and a parameter sigma, which are the mean and the standard deviation, respectively. Yeah. And so what we're going, what we're going to do is we're just going to be plugging in our values from you and sigma. So in doing so we're going to have that this is going to be the exponential function of X -1, 12 Squared over two times 8 squared It will be two times 8 squared Which is 1 28. Uh huh. And we're going to have all of the speed divided by eight times the square root of two pi and for the probability That X is less than 100. Actually, it doesn't matter whether I put the equality or not on this inequality because it won't change the answer. All we need to do is we need to integrate this function from 0 to 100. And obviously because this is some integral of the form E to the minus X squared, we can't find an elementary anti derivative for this. So what we're going to have to do is we're going to go into Wolfram Alpha and you're going to Just type the sense you're going to type integral from 0 to 100 of the above function in green. And what you're going to do and what you're going to get is you're going to get that this is approximately 0.068. 0668. Yeah, so that's how you do that one. Okay. And the second one is we want the probability that X our random variable X is greater than or equal to 1 25. This will let us know how many of us, how many will get targeted. Mhm. So since we want this greater than 125, we don't care where it ends. We're going to apply the same concept and integrate this function From 1 25 to Infinity. And using the same logic as before because this is an integral of the form E to the minus X squared. We're going to have to put this into Wolfram Alpha and when we do so we're going to get that this is about 0.05-1. Alternatively you could use Simpson's rule for this but we won't do that because that takes too long. And plus using Wolfram Alpha is way easier. But that's how you do this question or at least the set up of it.

Speeds of vehicles on a section of highway have a bell shaped distribution. So, because of that statement, you can apply the empirical rule, and the empirical rule says that the mean falls at the center of that distribution, and if you Go one Standard Deviation in each direction, it will account for 68% of the data. Now, because the bill Shape is symmetric, that means there would be 34 In this section and 34% in this section. The empirical rule also says that if you go two standard deviations away from the mean in each direction, it will account for 95% of the data. So if Two of these four sections is 68%, that means there's 27%,, that has to be split evenly between the remaining two sections. So therefore we will find 13 4.5 in each of these sections. The empirical rule also says that if you go three standard deviations in each direction, then that will account for 99 7% of the data, Which means that each of these little sections would be 2.35 and the entire bell Would represent 100 of the data. So in this particular problem We have a mean of 60 mph, So 60 will go in the center of our curve And we have a standard deviation of 25. So that means we're going to add to five And we'll be at 62.5, We'll add to five again Will be at 65. If we subtract 2, 5, We'd be at 57,5, And if we subtract 2,5 again We'd be at 55 oh, so I'm going to create the four sections here, So this would be 34%,, 34% 13 5% 13.5%. And keep in mind that half the bell To the left would be 50 and half the bell to the right Would be 50%. So the entire bell would be 100%. So in part a The question is asking if the speed limit is 55 mph, what proportion of vehicles are speeding? So technically we're trying to find The proportion when X is greater than 55, So greater than 55. We would have 50% of the bell to the right, We have 34 And we have 13.5%. So if we add all of those up, then we could say 97 5 of the cars or the vehicles would be speeding. Yeah, for part B, you're asked to determine the median speed of vehicles on this highway. Well, whenever it's bell shaped, then you are mean your median and your mode are all equal. So therefore, since the mean was 60, The median would also be 60 mph in part C. You are trying to determine the percentile rank Of the speed at 65 mph. So for part C I'm going to redraw the picture Again. We had 60 in the middle And then the first line to the right represented 62 5 And the next one represented 65. And over here we had 57.5 and 55. And we want to know the percentile rank for 65. Well anytime you're talking percentile, If you are at the 50th%ile, that meant 50% scored lower than you. So if you were at the 80th%ile, that meant 80% scored lower than you. So we need to know what percent scores lower. So again, keep in mind that the bottom half of the bell is 50% This was 34 And this section was 135%. So therefore the percentile rank for 65 mph would be 50 Plus 34 Plus 13 5%. So it would then be 65 mph Is at the 97 5 percentile. And finally part D What speed corresponds to the 16th%ile. So again, I'm going to draw the bell shaped curve and in the center we have 60 And we know that 50% is over here and We have 57.5 and 55. So we're trying to figure out where 16 percent would be below a certain score. So that means if 16% is in the bottom, then 84% has to be in the top or above it. So we know that this is 34%,, And we know that this is 135. So at this location right here, we would have 84% above it, Which means that we'd have 16% below it. So therefore the speed At the 16th%ile would be 57 5 MPH.

In this problem. We want to construct a frequency table. So, uh, we've actually, um, general the data for a frequency table already, So it's just gonna rewriting it. Um, so we'll say Ah, let X equal to this speed and, uh, second call in frequency. Eso the intervals are going to be 60 to 75 75 to 90 90 to 105 105 to 1 20 And then 1 20 2135 and then x greater than or equal to 1 35 The frequency is given to us already. That's 20 70 1 10 1 50 40. And so in part B, we would like to ah, draw hissed a gram that illustrates this data. So Part B set up a chart for us to do the history ground. I can draw a straight line. So the intervals, except we're starting at 60. So I'll give it some space. 60 ah, 75 90 1051 20. And then one 35 beyond. Okay. So Ah, the frequencies. It looks like the highest it goes is 1 50 So let's go by twenties. 2040 60 80 120 40 and 1 60 So we can draw our bars now. Ah, First interval. 60 to 75. 20 second interval. There are 70 third interval. We have 110 and then the next 11 50 1 21 forties on one fifties around there and then back down to 40 and then 10. Okay, so we've got our bar graph. There are not bargain for history. I'm sorry. Part C is to draw the I believe it to draw the cumulative frequency curve. Good. So we have to set up another chart. Except this time it's got to be based on ah, cumulative frequency. So the X isn't gonna change. I'll just say 60 90 uh, 1 20 And then we'll leave it like that on the why the cumulative frequency is 400. So I want to make sure spaces out properly. So let's do 50 each. That 400 top. Okay, so the cumulative frequency curve, remember, we used the upper limit. So at 75 will have 20 twenties. We're halfway from the first bucks, and it's cumulative frequency. So at 90 we're gonna put the addition of the two frequency so that will be 90 and then adding another 1 10 That'll be 200 for the next intervals 200 and then the next year, or heading one fifties would be 3 50 3 50 and then we add another 40. So be 3 90 and then 400 and ah, we can draw our curve by connecting the dots. Perfect. And then, ah, lastly part d ah, It is to use this cumulative frequency curve to estimate the percentage of drivers that drive faster than the speed limit. On the speed limit is 100 and 30 kilometers. So 130 is where we're going to draw a line up. So it was 1 20 then the next interval was 1 35 1 30 Somewhere around here, if you draw it up and then across, it is not a not a great curve, not very exact, but it's about somewhere above 3 50 below 400. So, uh, for the sake of your sanity here and rounding will just say 3 75 Um, and that means 25 people drive above that sleep speed limit. So that means it's 25 out of 400 people and we want to convert it to a percentage so that will be equal to roughly equal to 6.25%.

We're told that the speed of cars on a freeway near Toronto is normally distributed with the mean of 119 kilometers per hour and a standard deviation of 13.1 kilometers per hour. Now, for part A were asked for the probability that the speed of a randomly selected car is between 101 120 kilometers per hour. So this is the probability that X is between 101 120. And now if we transform to standardize, we take 100 minus the mean divided by the standard deviation. This is equal to the probability that minus 1.46 his less than report his head, which is also less than or equal to zero point 0763 And now you can look up these values either in a zed table or using software. So I'm using our in which case I used the command p norm. So for the for example, the first term I would use P norm of 0.76 And so we get 0.530 minus 0.72 which gives us a probability of 0.458 So that is the probability that a randomly selected car on that freeway is traveling between 101 20 for Part B were asked to determine the speed that characterizes the fastest 10% of all speeds. So if they're distributed normally like this, we're looking for this speed that cuts off the fastest 10% from the other 90%. So that means we're looking for the 90th percentile, which is given by this formula. Now again, you can look up this value either in azzet table. I'm doing it in our So I used the command que norm 0.9 that Q stands for Quanta, which is basically the same thing as percentile. This comes out to 135.8 kilometers per hour. So 10% of the vehicles on the freeway are traveling faster than 135.8 kilometers per hour now for Part C were asked to find the percentage of vehicles that are exceeding 100 kilometers per hour. So the probability that a randomly select cars exceeding 100 is equal to one minus probability that it is going at most 100. Now we standardize that probability comes out to 0.96 which is equal to 92.6 percent. So we can say that 92.6% of the vehicles on the freeway exceed 100 kilometers per hour or part d. We want to know the probability that at least one car among five randomly selected cars is not expecting exceeding the speed limit. So let's say not exceeding the speed limit is a success with the probability. So we found the probability that a car is exceeding the speed limit, so the probability of not exceeding the speed limit is zero point 074 And we want to find the probability that at access at least one at least one car is not exceeding the speed limit. The number of cars exceeding the speed limit is a random variable that is a binomial random variable based on a sample of five and a probability of success of 0.74 This probability is equal to one minus probability that X is equal to zero. That's one minus 0.681 which gives a probability of 0.319 So if he randomly select five cars from the freeway, the probability that at least one is not exceeding 100 kilometers per hour is 0.319 And now for party. We're trying to find the probability that the speed of a randomly selected car exceeds 70 MPH. Since our distribution that is defined in units of kilometers per hour, it's easiest to convert the 70 MPH, two kilometers per hour and 70 miles per hours approximately 112.65 kilometers per hour. So now we're looking for the probability that X is greater than 112 0.65 So here we are, standardizing it to the normal distribution, and that comes out to 0.686 The probability that a randomly selected vehicle is exceeding 70 MPH on that freeway is 0.686


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