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Problem 70 Hard Difficulty

Use a computer or calculator to find the probability that one randomly selected value of $x$ from a normal distribution, with mean 584.2 and standard deviation 37.3 will have a value
a. less than 525.
b. between 525 and 590.
c. of at least 590.
d. Verify the result using Table 3.
e. Explain any differences you may find.

Answer

a. 0.0562
b. 0.505544
c. 0.438215
d. 0.0559,0.5077,0.4364
e. Probabilities found by both methods are same except round off error.

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Video Transcript

in problem number 70 were going to use a computer or calculator. I'm going to show you how to use the T I graphing calculator, the Texas Instruments, and we need to find the probability that one randomly selected value of X from a normal distribution with a mean of 584.2 and a standard deviation of 37.3 will have a value that is less than 525 for part A. This is part A. So when it comes time to use your calculator, um, on the graphing calculator and I'll pull my graphing calculator in, you're going to hit the second key. We're gonna hit the second key, and then you're going to hit the There's Key. Something hit the Bears key against second key than the variables cake, and then we're going to use number two, and that's your normal C D. F. So it's a cumulative density function. So when we use normal CD F notice below it, it talks about lower bound upper bound, so we're going to bring it to her home screen. So we're going to have our lower boundary. We're going to have our upper boundary. We're then going to have our average and our standard deviation. So in this particular problem are lower bound is infinitely into the left tail. So if I think about what the bell would look like, here's 584.2. We want to be less than 525. So we're going all the way into this tale. So in order to do that, our left bound is going to have to be a really big negative number. So what we're going to type in is we're gonna type in negative one times 10 to the 99 which is a super, super super negative number. So we'll have negative one times 10 raised to the 99th Power. Then we're going to do a comma. We're gonna do our upper boundary. So what we have just put in is a number all the way back here, so our upper boundary is going to be the 525. So we're going to type in 5 to 5, followed up with a comma. Then we have to go with our average, and our average was the 5 84.2. So we'll type in the 5 84 0.2 and we'll follow it up with our standard deviation. And our standard deviation was 37.3. So we're getting an answer for our problem. The probability that X is less than 525 is 5250.5 six to they took that away. I'm going to finish up what was hidden there. Remember that last argument for the A cumulative density function is going to be that standard deviation. Okay, so for part A, what's the probability that a randomly selected X value is less than 525 would be 5250.562? And we used our graphing calculator. And again I used the T I graphing calculator, the 84 or the 83 wood. We're all right. So let's go to part B in part B. You are asked to find the probability that X lies between 5 25 and 5 90. So the probability that X is between 525 and 590 again we're going to use our normal probability normal CDF, which means cumulative density function. And remember, it goes lower bound, upper bound average standard deviation, so we'll use normal CDF in this case are lower bound is the 5 25. Our upper bound is the 590 the average was the 5 84.2 and the standard deviation was 37.3. Some have slide my calculator in, and I'm going to do the second. There's select number two. This time lower is 5 25 separated by the comma, 5 90 separated by the comma, 5. 84.2, separate by the comma and 37.3. So for the answer here we get 0.5055 and I may have to move my calculator. All right, so let's go to port. See, Part C is a probability of at least 5 90. Probability will at least 5 90 means it's going to be greater than 5 90. So if you think about what the Bell would look like, then we're talking about going into this tale so our upper boundary is gonna have to be a super large number. So we're going to set it up by saying normal CDF again. It's lower boundary, upper boundary average and standard deviation, so we're gonna dio a lower boundary of 5. 90 because that's gonna be right here the beginning of our shaded region. Our upper boundary has to be a very large number. So we're going to say, one times 10 to the 99th power, our average was 584.2 and our standard deviation was 37.3. So I'm gonna slide my calculator in. We're going to set that up. Or did you? Second, there's number two. Our lower boundary is 5 90 separate by comma one times 10 to the 99th power separated by a comma put in our average, which was 5 84.2 and follow it up with our standard deviation of 37.3. So in this case or probability that X is greater than 5. 90 is going to be 0.4 three eight to So now that we have used the graphing calculator parte de wants us Teoh, verify these three results by going back to the old fashioned way and using the table three from the back of your book. So let's go back to we're going to party again the old fashioned way and part A Waas. What's the probability that X was less than 5 25? So the formula way is going to be to construct that bell curve, put the average in the seven or 5 84.2, we want less than 5. 25. So we will need a Z score. Remember our Z score formula this X minus mu divide by sigma. So we're going to find the Z score to be 5 25 minus 5 84.2 divided by our standard deviation of 37.3. So our Z score turns out to be negative 1.59 So if we're talking less than 5. 25 were also talking that we are less than negative 1.59 as a Z score. So we would then go to our standard normal table in the back of our book and we would get a 0.559 We're gonna also, um, do Port e as well at the same time. So are part a answer. If we sneak back to it, we had point 0562 0.562 So that's pretty close. All right. So part A, we got the 0.56 to, but when we redid it using the table of values, we got 0.559 So they're close. They both around 2.56 But the reason for the difference And this is the part e the explain the differences. The reason is, when we found this Z score negative 1.59 we rounded, um, the calculator. If you're doing it in the calculator, it's not rounding an easy scores. It's using more than two decimal places. So actually, the calculator is going to be a little bit more accurate because of the fact that we had to round this Z score. So let's do a comparison for Part B as well. So in part B, we had an answer. Let's go back and get it of 0.5055 But now we've got to do it with the long hand, using the old fashion way using tables. So Part B, the question was asking you what was the probability that we were between 525 and 590 so on our bell shaped curve, we want to be between 5 25 and 590 you're going to have to calculate the Z scores for each of those. So the Z score for 5 25 will be 5 25 minus the average, which was 5. 84.2. Divide by the standard deviation of 37.3 and you get a Z score of negative 1.59 and we're gonna find the Z score for 5 90. So we'll do 590 minus 5 84.2, divided by the 37.3 and you're going to get a Z score of 0.16 So when we're talking about being between 5 25 and 5 90 it's no different than saying between negative 1.59 is less than Z, which is less than 0.16 So we would find the probability that Z is less than 0.16 in our standard normal table. And then we would subtract from it the probability that Z was less than negative 1.59 Utilizing the table, we're going to get an area of 0.5636 minus 0.5 59 for an answer of 0.5077 And again, we're close, not quite exact. And again, the reason we're not exact is because over here we rounded to two decimal places. So we rounded within the problem, which skews our results a little bit. And then finally, for part C, let's do a verification on the part C. Move up a little bit, Part C. If we scooped back Part C was 0.4382 we had point for 382 and the problem we were solving was the probability that our value was greater than 5 90. So our bell curve 5. 90 would have been over here. So we need a Z score so it would have been 5 90 minus 584.2, divided by 37.3. And that's going to get us a Z score of 0.16 So when we were talking about being greater than 5 90 it's no different than Z being greater than 900.16 And since the standard normal table always goes into the left tail, we're going to have to rewrite this as one minus the probability that Z is less than 10.16 When we look in the table, we will get a value of 0.5636 resulting in a probability of 0.4364 And again, here we are. We're close. We had a calculator answer, and we have a manual table. Answer there close. And the reason there's a difference is because back here we rounded to two decimal places.