Example 4.42 introduced a bivariate normal model for X= SAT...

Example 4.42 introduced a bivariate normal model for $X=$ SAT Critical Reading score and $Y=$ SAT Mathematics score. Let $W=$ SAT Writing score (the third component of the SAT) which has mean 488 and standard deviation $114 .$ Suppose $X$ and $W$ have a bivariate normal distribution with $\rho_{X, W}=\operatorname{Corr}(X, W)=.5 .$ (a) An English department plans to use $X+W,$ a student's total score on the non-math sections of the SAT, to help determine admission. Determine the distribution of $X+W .$ (b) Calculate $P(X+W>1200)$ (c) Suppose the English department wishes to admit only those students who score in the top 10$\%$ on this Critical Reading + Writing criterion. What combined score separates the top 10$\%$ of students from the rest?

Example 4.42 introduced a bivariate normal model for $X=$ SAT Critical Reading score and
$Y=$ SAT Mathematics score. Let $W=$ SAT Writing score (the third component of the SAT)
which has mean 488 and standard deviation $114 .$ Suppose $X$ and $W$ have a bivariate normal
distribution with $\rho_{X, W}=\operatorname{Corr}(X, W)=.5 .$
(a) An English department plans to use $X+W,$ a student's total score on the non-math
sections of the SAT, to help determine admission. Determine the distribution of $X+W .$
(b) Calculate $P(X+W>1200)$
(c) Suppose the English department wishes to admit only those students who score in the top
10$\%$ on this Critical Reading + Writing criterion. What combined score separates the top
10$\%$ of students from the rest?

Key Concepts

Bivariate Normal Distribution

This concept refers to a joint probability distribution for two normally distributed random variables that may be correlated. In the bivariate normal system, any linear combination of the two variables is also normally distributed. This property simplifies the calculations involving sums or differences of the variables and allows for the use of standard techniques for handling dependence between the variables.

Linear Combination of Normal Variables

A linear combination of normal random variables involves summing the variables each multiplied by a constant. When the variables are normally distributed, their linear combinations remain normally distributed. The mean of the combination is the corresponding weighted sum of the means, and the variance is computed as the sum of the squared constants times the variances plus twice the product of the constants times the covariance for each pair of variables involved.

Standardization and Z-Score

Standardization is a process used to convert a normally distributed random variable into a standard normal variable (Z-score) by subtracting the mean and dividing by the standard deviation. This transformation is essential for utilizing standard normal tables or computational tools to find probabilities associated with particular values of the variable.

Percentiles and Quantile Calculation

Percentiles in a normal distribution are values below which a given percentage of the data falls. For instance, to determine which score separates the top 10% of students from the rest, one would compute the quantile corresponding to the 90th percentile. This involves finding the Z-score that matches the desired cumulative probability and transforming it back to the original scale using the known mean and standard deviation of the linear combination.

Transcript

00:02 Referring to an example in this section, example 4 .42, for the bivariate normal model for the random variable x, which was the sat critical reading score, and the random variable y, which was the sat mathematics score.

00:20 Define the random variable w to be the sat writing score, the third component of the sat.

00:27 We're told this is a mean of 488 and standard deviation of 114.

00:34 We're asked to assume that x and w, so the critical reading and the writing portions have a bivariate normal distribution, where the correlation coefficient of x and w is .5.

00:51 In part a, we're asked to find the distribution of x plus w.

01:11 Well, we know that x and w are bivariate normal.

01:22 This is what we're given as an assumption.

01:28 It follows that the sum, x plus w, has a univariate normal distribution.

01:38 In particular with a mean of the expected value of x plus w which we call it the expected value of a sum is the same as the sum of expected values so this is the expected value of x plus the expected value of w we're given that the expected value of x previously excuse me was 496 and we're given that the expected value of w is 488 so we have 496 plus 496 plus 496 plus 496 plus 496 plus 488 and this sums up to 984.

02:38 So this is the mean and it has a standard deviation well determined by the variance, so we'll actually find the variance first.

03:06 And the variance of a sum of variables, this is going to be the sum of the variances, so variance of x plus the variance, variance of w plus 2 times the coefficients in x and w, which is 1 and 1, times the covariance of x into w.

03:34 This is the same as the variance of x plus the variance of w, plus 2 times the correlation coefficient row of x and w, times the standard deviation of x and w, times the standard deviation of w, times the standard deviation of w, by the definition of correlation coefficient.

04:01 In substituting, we have that the variance of x, well, the standard deviation of x was 114, so this wouldn't be 114 squared.

04:11 Standard deviation was of w was also 114, and so this is 114 squared, plus 2 times the correlation coefficient row, which we're told, was 0 .5 times the standard deviation of x, which again is 14, or 114 i mean, times the standard deviation of w, which is also 114...

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