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QuestionLet represent Ihe nurnber of Iimies cuSiomAI enlering superrarket Assume (his Is Ihe probability distribullon of X;1-week-period.X=L23 otherwiser6x)Find mon...

Question

QuestionLet represent Ihe nurnber of Iimies cuSiomAI enlering superrarket Assume (his Is Ihe probability distribullon of X;1-week-period.X=L23 otherwiser6x)Find monn; E(X) and variance, V(X)(5 marks)E(3X) - V(5 2X)MATkS)Basud on pnal experlonce, Iha managor 0f an Engineering company osllmated Ihat of Ihe tolal numbor o projocts secured by his company; 91% succoss lo bu comploted on tlmo_What Is Ihe probabillty Ihat atmost of Ihe noxt 6 projecis (all lo be completed on Ilme? (3 marks)Whal is tho

Question Let represent Ihe nurnber of Iimies cuSiomAI enlering superrarket Assume (his Is Ihe probability distribullon of X; 1-week-period. X=L23 otherwise r6x) Find monn; E(X) and variance, V(X) (5 marks) E(3X) - V(5 2X) MATkS) Basud on pnal experlonce, Iha managor 0f an Engineering company osllmated Ihat of Ihe tolal numbor o projocts secured by his company; 91% succoss lo bu comploted on tlmo_ What Is Ihe probabillty Ihat atmost of Ihe noxt 6 projecis (all lo be completed on Ilme? (3 marks) Whal is tho average number of projocts Ihal failod t0 complole on lime Ihe company can gel= 50 projecis Ihis year? marks)



Answers

Stop the car! A car company has found that the lifetime of its disc brake pads varies from car to car
according to a Normal distribution with mean $\mu=55,000$ miles and standard deviation $\sigma=4500$ miles. The company installs a new brand of brake pads on an SRS of 8 cars.
(a) If the new brand has the same lifetime distribution as the previous type of brake pad, what is the
sampling distribution of the mean lifetime $\overline{x} ?$
(b) The average life of the pads on these 8 cars turns out to be $\overline{x}=51,800$ miles. Find the probability that the sample mean lifetime is $51,800$ miles or less if the lifetime distribution is unchanged. What conclusion would you draw?

Here in this problem we are told to refer back To Exercise 4.36. Now it's important to remember that on that exercise we found that the expected value of acts Was there No 1? Right. And the variance of X was equal to two. Mhm. Now we're going to use this in order to find the given values here. Now we're told that Z is five X plus three. Uh huh. And so they expected values the is the expected value of five X plus three Which is five times the expected value. That's plus three. Yeah, just five times one plus three. Which is agents are expected value. The state the variance was the There's a variance of five, they're supposed three which is five squared times the variance of Acts, You know the variance of access to and so this is 25 times two which is 15. So the variance of the that's 50.

So we have some scenarios here, and we're gonna be finding the mean and the standard deviation. The easiest way to do this is with these two equations that I think you should have written down. So I mean, is equal to N. P. I'll say what that means in a minute, and standard deviation is equal to the square root of NP, one minus p. And just as a disclaimer, these equations only work with a binomial random variable. That means there are only two possible outcomes, and you'll see that all of these problems there are only two outcomes, so we can use these equations. The end is the number of trials, and the P is the probability. So let's start with the 1st 1 So number of tales seen in 50 coin tosses so 50 is R N, and our probability is 1/2. And that's just implied because it is a coin ties so for mean, is equal to 50 year times. One happen that gives us 25 and for standard deviation will take the square root of 50 times 1/2 times one minus 1/2. I can make that look meter on this gives us a 3.536 So the number of left handed students in a class of 40 students and we know that 11% are left handed it so 40 is R N and 11% is RP. However, we need to convert this to a decimal. So we're not gonna put a well, then we're gonna put 110.11. So I mean, is equal to 40 times point. Well, then, which is? 4.4 and standard deviation. Anna, this equal, truly square root of 40 times 400.11 times one minus point. Well, then and this is equal to 1.979 Okay, the number of cars that have unsafe tires from 400 cars and we know that 6% of, um, having unsafe tire. So we're gonna use 400 not six, but 60.6 who won it as a decimal. So mean is equal. Chill 400 times 4000.6 which is 24. And standard deviation is equal to the square root of 400 times 4000.6 times one minus 10.6 And this is 4.750 The number of melon seeds that germinate when a package of 50 year planted and 88% of them. Germany, too, so are mean is equal to 50 times 500.88 which is 44. And our standard deviation is equal to the square root of 50 times 500.88 times one minus 10.88 which is equal to 2.298 and we are done.

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.

Problem 16. We want to and enjoy this probability density function over the center. Then find the means the parents and the sender division. Without integration. We can write F of X to be in the full five divided by three. Not to blow it by E. There's a part of minus five divided by three. Want to blow it by X. Then it's in the fall A But to blow it by E, it's about minus A. X. And from this in terrible X Is greater than or equals zero. This is the fall of the exponential probability density function, then F of X is big soup potential. And for the exponential distribution we can get you of X or mule As one divided by He here is five divided by three. Then it's one divided by five, divided by three equals to be divided by five Or 4.6. We can get the variants equals square root Sorry, equals one divided by a square Equals one, divided by five, divided by three. Old, squared Equals nine, divided by 25. Or it equals 4.36. To get the standard division, it equals square root of the variance, it equals square root of 4.36 equals or 0.6. And this is the final answer our.


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