Scores on a certain nationwide college entrance examination...

Scores on a certain nationwide college entrance examination follow a normal distribution with a mean of 500 and a standard deviation of 100. Find the probability that a student will score... a) over 680 b) less than 580 c) between 550 and 650 d) If a school only admits students who score over 650, what proportion of the student's pool would be eligible for admission? e) what limit (score) would you set that makes the top 30% of the students eligible? f) if 25 exams are randomly chosen and their mean is calculated what is the standard deviation of the average score ? g) of the 25 exams what is the probability the average score is over 630

Transcript

00:01 For this question, there is given a normal distribution and the mean, which is denoted by mu, that is given as 500.

00:10 And standard deviation, which is denoted by sigma, that is given as, let's say this one, which is 100.

00:22 So let's say x be the random variable for this normal distribution.

00:26 It has the mean value and the standard deviation is 100.

00:29 So in part a, what we have to do, we have to find the score, the probability that the score is over 680.

00:39 So the probability of random variable x, which is greater than 680.

00:44 So what that means, if i just graph the situation here, this is the mean score.

00:50 So the 680 is here.

00:53 So we need to get the area of this region.

00:56 So practically, we can just find this area by using the normal c.

01:00 F function of the ti calculator.

01:03 When we look at the shaded region, the minimum value is 680.

01:07 The upper boundary, that goes to positive infinity.

01:10 So i can put 1a99, which is very big number.

01:13 And the mean value is 500, and the standard deviation is 100.

01:17 Let's get this value here.

01:20 Just press the button second and distribution.

01:22 There is normal cdf here.

01:24 Lower boundary, 680.

01:27 And the upper boundary is, this is 1a ,000.

01:30 99 and 500 and the standard division 100.

01:36 So we got the results, so we're gonna give the answer with how many decimal places? is that given in the question, no, i'm gonna give the answer with four decimal places.

01:45 0, 3, i mean 0, 0, 0, and 59.

01:52 Let's take okay, part b.

01:55 That is less than 580, so the probability of random variable x, which is less than 580.

02:04 Again, i'm going to use normal cdf.

02:09 Lower boundary, so in this case, so this is 580, so we need to get the area of this region here.

02:16 So the minimum value just goes to negative infinity.

02:20 I'm going to put negative 1 in 99, and the upper boundary is 580, mean value 500 and 100.

02:28 Let's put all these values on the calculator.

02:31 This is negative 1 99 and the upper boundary is 580 the mean value is 500 in the standard division 100 so just put a comma here okay and enter so the answer is 0 .788 and 1.

02:54 What about for part c that is between 550 and 650 so the probability of x which is 650 and 550.

03:05 Again, let's use the normal cdf function.

03:10 In this case, the lower boundary is 550, 650 and the mean value is 500, and the standard deviation, 100...

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