(a) The following results were obtained for about 1,000 families: average height

Question

(a) The following results were obtained for about 1,000 families: average height of husband 68 inches, SD 2.5 inches; average height of wife 63 inches, SD 2.5 inches, correlation coefficient r = 0.6. Of the men who were married to women of height 60 inches, what percentage were under 64 inches? Assume normality wherever necessary. (b) For the first-year students at a certain university, the correlation between SAT scores and first-year GPA was 0.60. Assume the distribution of the scores is jointly normal. Predict the percentile rank on the first-year GPA for a student whose percentile rank on the SAT was (a) 90% (b) 30% (c) 50% (d) unknown

Explanation / Answer

Hi, we are allowed 1 question at a time. I'm doing (a). Kindly repost (b)

(a) Since in this question, we dont need to make a predciction we find the mean and standard deviation and do the question as below.

For Wifes (X): x = 63 and x = 2.5

For Husbands (Y): y = 68 and y = 2.5

Covariance () = 0.6

To Find P(Y < 64/X = 60) (Probability that husbands height is < 64 in, given that wifes height = 60 in)

Y/X = X ~ N (y + [* y* (X - x)/ x], (1 – 2)2y)

Y/X = (68 + [0.6*2.5*(60-63)/2.5], 1 – (0.6)2* 2.52)

Y/X = X~ (66.2, 1)

Therefore [Y/X] = 66.2 and [Y/x] = 1= 1

To Find To Find P(Y < 64/X = 60, Z = (Y – [Y/X])/[Y/X] = (64 – 66.2)/1 = -2.2

The Required probability at Z = -2.2 = 0.0139

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