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Let X equAC Lhe widest dinmeler (in nilliteters) ol the letal head measunxl betwCe the I6th andl 25th weeks of pregIAnCy. Assule Lhat the distribution of X is N(46....

Question

Let X equAC Lhe widest dinmeler (in nilliteters) ol the letal head measunxl betwCe the I6th andl 25th weeks of pregIAnCy. Assule Lhat the distribution of X is N(46.58. 40.96). Lut X bthe satple IeAn ol FAIot saiple 16 obenations o X_Find ELX]Fiud Var(F).Fiud P(44.42 < X < 48.98).

Let X equAC Lhe widest dinmeler (in nilliteters) ol the letal head measunxl betwCe the I6th andl 25th weeks of pregIAnCy. Assule Lhat the distribution of X is N(46.58. 40.96). Lut X bthe satple IeAn ol FAIot saiple 16 obenations o X_ Find ELX] Fiud Var(F). Fiud P(44.42 < X < 48.98).



Answers

Suppose that $X$ has an exponential distribution with $\lambda=2 .$ Determine the following: (a) $P(X \leq 0)$ (b) $P(X \geq 2)$ (c) $P(X \leq 1)$ (d) $P(1<X<2)$ (e) Find the value of $x$ such that $P(X<x)=0.05$.

12 question A. The mean off the export distribution is equal to the mean off the population, so its able to find which shouldn't be on the interval of X War it's bigger than six does not contain the population means Mu is equal toe find. Then we know that the distribution with the smallest standard deviation will have the smallest for ability poverty X bar, because than six. Since more values will be centered about the population means the Sun Division is smaller since the Center division on the X Bar distribution, it's Sigma over square Ruto of end with Sigma the Population some division We know that the civil division is smaller in the sample sizes larger that the probability is the smallest for the sample size and is different to 81 question. See, um, for the interval off war and six which contains the musical toe five. Then we know that the distribution with the small standard deviation will have the largest probability Oh four x bar six. Since more values will be centered about operation mean instant division is smaller. Since the start, the division off export distribution is Sigma was worried about with Sigma populations from the deviation. We know that the Central Division is smaller if the sample size is nausea, so the probability is smallest foreign sample size of 81.

From number 74. We're talking about bursts throughout the year, so we're told that it's a uniform distribution. So a is a uniformed distribution from week one 53 for 52 weeks B s for the graph of the situation. So from one 2 53 in the height would be won over 52. So one over B minus a so 53 minus 51 and then we just make a rectangle c f of X. So these all look that basically the same. So the probability that is one over 52 or it is zero. So it's one over 52 between one and 53. Otherwise, in a zero, the D is looking for the mean a plus B over too. So one plus 53 over too, which is 54 over, too. And that's 27. He's looking for the standard deviation and these problems. The standard deviation is the square root of B minus A. The quantity squared over 12. So 53 minus one squared over 12 which is 52 squared over 12. Swear to that and that gives you approximately 15.1 huh? Find the probability that a person is born at the exact moment 19 weeks, so X equals 19. So in the uniform distribution, any single point has a probability of zero G. The probability that its excess between two and 31 so 31 minus two times one over 52. The base times I 0.55 77 a. Find the probability that a person is born after 40 weeks, so the probability that X is greater than 40. So that would be between 40 and 52 or 53 so 53 minus 40 times one over 52. That's 0.25 The Probability X is less than probably 12 is less than X given X is less than 28. So the numerator of that is when both things have to be true. So both things are true between the numbers 12 and 28. So the probability 12 is less than X is less than 28 and the denominator is the given part. X is less than 28. The Probability X is less than 28 to the numerator is 28 minus 12 times one over 52 over 28 minus one times one over 52. So those cancel and we're left with 16 over 27 which is 0.59 26 j find the 75th percentile. So the 75th percentile it's 0.7. So that's basically the area. So in the percentile means the bottle. So that is for the base some number K minus one times the height, which is one over 52. So multiply both sides by 52. That gives us 36 point for equals K minus one 52 1 over 50 to cancel out had won the both sides Que is 37 point for and for letter K find the minimum of the upper quarter, so the upper corner is 25.25 So since it's the upper corner, it's from 53 to some number. So we have 53 minus K. That's what we're trying to find for the base. Once again, the height is one over 52 so multiply both sides by 52 that gets us 13 equals 53 minus K. Subtract 53 from both sides. Negative 40 equals negative k times, both sides by negative one k equals 40

Right and problem in the 48 there are six parts and all six parts do with the fact that the average is 28 and the standard deviation will be Southern. So in part A, you're asked to determine what the what is the probability that X is less than 28. We also knew that the variable was normally distributed 28. Since it was the average would be right in the center of the bell and the probability that we have picking X value less than 28 which would represent half the bell, which would be 0.5 for Part B. We're looking to do the probability that X takes on a value between 28 38. So again, I would highly recommend drawing that bell shaped curve again. 28 is in the center and 38 would be over here. So we're going to utilize Z scores in order to solve this problem. And the formula for Z scores is that Z equals X minus mu divided by sigma. So the Z score associated with 28 would be 28 minus 28 all over seven or zero, which we should have known that just because the center of the bell does represent a Z score of zero and the Z score associated with 38 would be found by doing 38 minus 28 divided by seven, and you do get an answer of 1.43 So when we posed the question, what's the probability of being between 28 38? It's no different than the question that says, What's the probability that Z is between zero and 1.43? You can then calculate the probability that Z is less than 1.43 and from that subtract the probability that Z is less than zero. You would then go to your standard normal table in the back of your textbook. You would get a value of 0.9236 for the probability that Z is to the left of negative one of positive 1.43 and the probability that Z is less than zero would be 00.5000 for an answer for Part B two B 20.4 to 36 in part C, making a little bit more difficult as we go along in part C. We're asking, What's the probability that ex falls between 24 40? So again, I highly recommend drawing that bell shaped curve, putting 28 in the center because that is our population mean and we want to be between 24 and 40. So we're going to find the Z score associated with each and the Z score for 24. We'll do Z equals 24 minus 28. Divide by the standard deviation of seven, and we get negative 0.57 and then the Z score associated with 40 would be 40 minus 28. Divide by seven, which yields a value of 1.71 So when we say, what's the probability that X is between 24 40? It's no different than saying What's the probability that Z is between negative 400.57 and positive 1.71? We would then find the probability that Z is less than 1.71 and subtract from that the probability that Z is less than negative. 0.57 If you used the table in the back of your book, the standard normal table you will find the probability that Z is less than 1.71 to be 0.9 five 64 and the probability that Z is less than negative 0.57 you will find to be point to eight four three. Thus, the answer to Part C would be 0.67 to one in Part D. We're looking to calculate the probability The X Files between 30 and 45. So here's our picture. We have 28 here, so 30 and 45. So because both of those values or above 28 both of those thescore should be positive values. So we'll find the Z score associated with 30 by doing 30 minus 28. Divide by seven. And when we do that, we will get a value of approximately 0.29 and the Z score associated with 45 would be found by doing 45. Divide by 20 of minus 28 divide by seven and you get approximately a 2.43 We can put both those on the bell at the 30 mark. We're gonna have the 300.29 and at the 45 mark, the 2.43 so when the problem is asking you was the probability that the X values between 30 and 45. We can also say the problem would be What's the probability that Z is between 0.29 and 2.43? We're going to rewrite this then as the probability that Z is less than 2.43 minus the probability that Z is less than 0.29 and using what's in the back of your book. The probability that Z is less than 2.43 would be 0.99 to 5. From that, you're going to subtract 0.6141 for an answer to Part D of 0.3784 in party. Very similar style problem. Part E is asking us what's the probability that your ex value is between 19 and 35. So again, we're gonna have our bell shaped curve. We've got 28 in the center, so 19 would be to its left and 35 would be to its right. We're going to find the corresponding Z scores, so Z equals 19 minus 28/7. And when you do that, you get a negative 1.29 and Z equals 35 minus 28/7, which gets you a Z score of nice clean one. So we're gonna place those on our bell. So this is a Z score of one and this is Aziz Score of negative 1.29 Someone were asking, What's the probability that X is between 19 and 35? It's no different than asking What's the probability that Z is between negative 1.29 and one? We can then solve this problem by calculating or looking up the probability that Z is less than one and subtract from it the probability that Z is less than negative 1.29 So from your standard normal table, the probability that Z is less than one is 0.8413 And the probability that Z is less than negative 1.29 is 0.985 for an answer to Part E of 0.74 to 8. And then finally, for part F, the problem is asking you to determine what's the probability that X is less than 48. So again, one more time. Here's the image. Here's 28 so 48 would be to the right. We're going to calculate the Z score, the easy score. Oops, sorry about that. The Z score would be found by doing 48 minus 28. Divide by seven, and you get a Z score value of 2.86 So when you're asking, what's the probability that X is below 48? It's no different than asking. What's the probability that Z is less than 2.86? And looking up in your chart, you would get an answer of 0.9979

So in this question we have X being a normally distributed random variable mean is minus 25. Standard deviation is four. And we're basically asked to find the probabilities of X. So we're basically going to convert X to our standard normal variable Z, which is x minus mu over sigma. We're going to do those conversions for each of those values of X. So for a we have probably X less than minus 27.2. That's the same as probably to see less than minus 27.2 minus negative 25 over mom. So that's probability. See less than minus 0.55 just equals 2.29124 part games. We're going to apply the same transformation for part B through deep. So part B quality X less than minus 14.8 is the same as the probability of seeing less than 2.55 from our table. That value is 0.9946 Part C we have probability effects more than minus 33.1 which is the same as probabilities. Zing more than minus 2.0 three. Rounding after two decimal places. So that's one minus probability easy. Less than minus 2.3 from our table. This value one minus this 0.212 So that gives us a probability of 0.97 88 Part C. For pardon? Teen we have probability of X more than 2.13 That's the same as probability not X. So we basically converted our X here to 2.13 That's probably one minus probabilities. E less than 2.1 thing. That is one minus 10.9834 which is 0.166


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