Question
Assume Ihat the random variable X is normally distribuled Wilh mean 50 and standard deviation 0 = 7 Compute the probabilily Be sure t0 draw normal curve with the area corresponding to the probability shadedP(X <47)Which of the following shaded regions corrosponds to P(X < 47)?P(X s4T) = (Round (0 four decimal places as needed )
Assume Ihat the random variable X is normally distribuled Wilh mean 50 and standard deviation 0 = 7 Compute the probabilily Be sure t0 draw normal curve with the area corresponding to the probability shaded P(X <47) Which of the following shaded regions corrosponds to P(X < 47)? P(X s4T) = (Round (0 four decimal places as needed )
Answers
Assume the random variable $X$ is normally distributed with mean $\mu=50$ and standard deviation $\sigma=7 .$ Compute the following probabilities. Be sure to draw a normal curve with the area corresponding to the probability shaded. $$P(40<X<65)$$
Again. This is a continuation of the previous two questions. We have a standard normal curve. Uh, sorry. A normal cove. Where mean is 50 and sanity Aviation is seven. Okay, now what do I want to find? I want to find probability that access less than equal to 45. Okay, So access less than equal to 45 Probability that access less than equal to 45. We are going to use a Z statistic. What is the formula for that? The formula for resisted the stickies X minus mu by sigma it ain't. My ex is 45 minus mu is 50. Sigma is given to me at seven. So this is equal to minus five by seven. Or this is nothing but minus zero point 714 Right. This is my Z statistic. So minus 0.714 This is my Z Valley. Okay, Now, if this is my standard normal curve, this is minus 0.714 and I want to find the area in this tale that is to the left of this. Right. This is mute. This is zero. So what I'm going to do is I'm going to find the P value of this. And this is going to give me the probability that I want. So the P value of minus 0.714 So this is minus point 714 This turns out to play 0.2376 So this is the area 0.2376 So the probability that I'm looking for is 0.2376 This is my answer.
Okay, so over here. Mhm. Our standard cove is this where commune is 50. This is the mean and Sigma is the standard deviation is seventh. I want to find the probability that my ex is greater than 38 less than equal to 55 Probability that my X is greater than 38 less than equal to 55 where I studied somewhere around here. Where is 55? Somewhere around here. So I want this probability. Okay, so how do I find this? What I can do is I can find the p value for the statistic corresponding to 38. Right? Which will give me that is just draw standard Normal co, this is zero. Okay, let me just find those that statistics for these 2, 38 55. So we know the value. Sorry. The formula to find the statistic is X minus meal by sigma. So 38 minus 15 is minus 12 by seven. So this is nothing but minus one point 71 four. Okay, so what is this? Z statistic? Minus 1.714 and the P values minus one points minus one point 714 I had calculate and I get my just a moment. Yeah, my P value is 0.43 So, uh, so this is minus 1.714 And this p value that is this region is 0.43 0.43 Similarly, I can find the Z statistic for 55 its corresponding p value. So the 55 minus 50 is 55 by seven. Happens to be 0.7 14 Okay, so what is the P value for 0.714? This is 0.714 I had to calculate, and I get as 0.2376 point 2376 So this is 0.714 and its corresponding P value that is area of this shaded region is 0.23 76.2376 So I want the area between these two right in this area. So this is going to be one minus point 0.2376 That is this shaded region minus this shaded region minus zero point 0433 Okay, so let us use the calculator This is one minus 0.2376 minus 0.433 this 0.7191 So this probability, the area over here in between is 0.7191 So I can say that this is 0.7191 and this is my answer.
All right. So I have a standard normal curve with immune as 50. And standard deviation is seven. So this is 50 and standard deviation is seven. What is the formula for the Z statistic? It is X minus mu by sigma. Okay, what do I want to find? This is question number 23. So I modify the probability that X is greater than 35 probability that X is greater than 35. Okay, so X is 35 minus 50 upon Sigma seven. So this is minus 15 by seven. Right? This is two point, So this will be minus two point 14 Okay, so I get my Z statistic as minus 2.14 Okay, minus 2.1 force. Okay, so this is the standard normal curve. This is zero and minus 2.14 will be somewhere around here. Right. So I want the area to the left of this, right, Because I want to find probably that exists. Where is this? Great. Okay, I want this to be greater than okay. So what I want is test probability, right? That is to the right. This has to be right. Tailed I want the area in the right tail. So how do I find this? What I can do is I can find the p value for this and subtracted from one, because the overall area has to be one so one minus the p value for minus 2.14 Right. P values the area to the left of this. So let's do this. Uh, this is minus 2.14 2.14 And this is one day I had to calculate, and I get my p values 10.16 point 016 Okay, so this is one minus 0.16 which happens to be 0.9. This happens to be 0.98 four if I'm not wrong. So this is zero point 984 So the probability that X is greater than 35 is 0.98 food. This is my answer.
Okay, So this time I want to find the probability that my ex is greater than 65. Right? This is similar to the last question where I had this normal curve where my mind was 50 and one. Standard deviation was seven. Okay. And this is the formula that we are going to use again. Z is equal to X minus mu by sigma. Okay, so what is x x 65? So they're sadistic. Will be 65 minus 50 upon Sigma seven. Okay, Now what I can do is I can take the help of the last example. In the last example, this was 35 minus 50. Right This time I have 65 minus 50. So, just instead of this minus 15, I'll get plus 15. So my Z statistic will be Instead of minus 2.14 it will be 2.14 Great. So my that statistic will be 2.14 And this time I need to find the probability that it is greater than this thing greater than 2.14 Because I want X greater than 65. So I just want the p value of 2.14 And if I calculate that, I get that a zero point 016 So this is my answer. This turns out to be 0.16 Right? This is my answer.