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Problem 48 Hard Difficulty

Given that $x$ is a normally distributed random variable with a mean of 28 and a standard deviation of $7,$ find the following probabilities.
a. $\quad P(x<28)$
b. $\quad P(28<x<38)$
c. $\quad P(24<x<40)$
d. $\quad P(30<x<45)$
e. $\quad P(19<x<35)$
f. $\quad P(x<48)$

Answer

a. 0.5000
b. 0.4236
c. 0.6720
d. 0.4765
e. 0.7428
f. 0.9979

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Video Transcript

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