2. The SAT college admissions test consists of three...

2. The SAT college admissions test consists of three sections: writing, mathematics and critical reading. Each is scored between 200 and 800 points, and the scores are normally distributed. For the writing section the mean is ? = 484 and standard deviation ? = 115; for mathematics, ? = 511 and ? = 120; and for reading, ? = 495 and ? = 116. An overall score between 600 and 2400, which is the sum of these scores, is also reported. For this problem we will assume that the scores on the three sections are independent. (a) Find the distribution of the overall score, and the overall score giving the 25th percentile of all scores. (b) Find the distribution for the average of the writing and reading sections of the test. What is the probability that this will be over 600? (c) Find the probability that the score on the mathematics section of the test will exceed the average of the writing and reading sections.

Transcript

00:01 Hello students, the sat college admission test consists of the three sections.

00:07 The first section is writing, second section is mathematics, third section is reading.

00:20 Then let writing is x1, this is x2 and this is x3.

00:27 And for this mean and the way standard deviations are given, mu1 equal to 484, sigma1 equal to 115, mu2 equal to 511, sigma2 is equal to 120, mu3 is equal to 495 and sigma3 is equal to 116.

00:47 Now we have to find the first distribution of the overall score.

00:53 Then overall score is equal to mean.

00:58 Let overall score is equal to x and we have to find mean related to the overall score, that is a mu1 plus mu2 plus mu3.

01:10 Put the values here 484 plus 511 plus 495.

01:15 And after simplifying we get the mean of the overall score is 1490.

01:25 Then overall standard deviation also we have to calculate.

01:28 But how we can calculate it directly because the observations are given independent, that's why.

01:34 Then sigma1 square plus sigma2 square plus sigma3 square.

01:38 They are independent and the value of covariance is zero here.

01:42 Then sigma1 square is 13225, sigma2 square is 14400, sigma3 square is 13456 here.

01:57 And after simplifying we get this value as 41081.

02:02 And after taking the square root we get value of sigma equal to 202 .6845.

02:07 Then overall score is x follows normal distribution with mean mu equal to 1490 and variance is equal to 202 .6845 square.

02:20 Then we have to find probability of not priority overall score giving the 25th percentile of the score.

02:27 Then x is less than x which is equal to 0 .25.

02:32 Then simplify it x minus mu upon sigma is less than x minus mu upon sigma.

02:39 Put the values which is equal to 0 .25.

02:44 And we know that this follows standard normal distribution.

02:48 Then 1 minus probability of z is greater than x minus 1490 upon 202 .6845 which is equal to 0 .25.

02:59 After simplifying we get z is greater than x minus 1490 upon 202 .6845 which is equal to 0 .75.

03:09 Now find the inverse probability x minus 1490 upon 202 .6845 which is equal to minus 0 .67.

03:21 Because this value is not exist we have to find probability for this.

03:24 Then after simplifying we get this value as x equal to 1354 .20145 and approximately equal to 1354...

Question

Please give Ace some feedback