Question

Assume that the population of all sister-brother heights has a bivariate normal distribution and that the data below were sampled from this distribution.

Sister height (x) 69 64 65 63 65 62 65 64 66 59 62

Brother height (y) 71 68 66 67 70 71 70 73 72 65 66

Heights of n=11 pairs of siblings

(a) Consider the population of all sister-brother heights. Estimate the proportion of all brothers who are at least 5′ 10′′.

(b) Suppose that Carol is 5′ 1′′. Predict her brother’s height.

(c) Consider the population of all sister-brother heights for which the sister is 5′ 1′′. Estimate the proportion of these brothers who are at least 5′ 10′′.

Answer 1

a) Proportion of brothers who are are more than 70 inches ie 5 feet 10 inches = 6/11 =0.545

b)

R Code

> x<-c(69,64,65,63,65,62,65,64,66,59,62)
> y<-c(71,68,66,67,70,71,70,73,72,65,66)
> summary(lm(y~x))

Call:
lm(formula = y ~ x)

Residuals:
Min 1Q Median 3Q Max
-3.5909 -1.2273 -0.9545 1.1136 4.0000

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 31.1818 18.7584 1.662 0.1308
x 0.5909 0.2929 2.018 0.0744 .
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 2.379 on 9 degrees of freedom
Multiple R-squared: 0.3114,   Adjusted R-squared: 0.2349
F-statistic: 4.07 on 1 and 9 DF, p-value: 0.07442

Then the predicated value for for 5' 1" ie. 61 is =

y= 31.1818 +0.5909 x =31.1818 + 0.5909*61 =67.2267 ie. 5' 7"

c) The sister in 5 ' 1" in none of the cases .Then the proportion of brothers being atleast 5' 10" is 0.

But if we consider the sisters to be atleast 5' 1",then the proportion of brothers being atleast 5' 10" = 6/10 =0.6