Question

Investigators studied the relationship between screen time (measured in hours/week) and body- mass-index, BMI= wt/ht2 (wt = weight in kg and ht = height in meters), in men age 25 - 45 . They surveyed 6234 U.S. men in this age group, the scatter diagram for the data was homoscedastic and the BMI data was approximately normally distributed both unconditionally and in each vertical strip.

The investigators generated the following summary statistics:

X=35 SDX=12

Y = 27 SDY = 6 r = 0.8
where X = hours of screen-time per week, and Y = BMI.

a) Approximately what percentage of U.S. men, age 25 - 45, have BMI of 30 or higher?

b) Approximately what percentage of U.S. men, age 25 - 45, who consume 50 hours of

screen time per week have BMI of 30 or higher?

Answer 1

a)

for normal distribution z score =(X-μ)/σ
here mean=       μ=	27
std deviation   =σ=	6.00

P( BMI of 30 or higher ):

probability =P(X>30)=P(Z>(30-27)/6)=P(Z>0.5)=1-P(Z<0.5)=1-0.6915=0.3085~ 30.85 %

b)

μx =		35
μy =		27
σx =		12
σy =		6
ρ=		0.8
x=		50
E(y|x) μy|x=μy + ρ*σy/σx(x − μ1) =		33.000
E(Var(Y|x))=σ2y|x=σ2y(1 − ρ2) =		12.96
σy|x=σy√(1 − ρ2) =		3.60

probability =P(X>30)=P(Z>(30-33)/3.6)=P(Z>-0.83)=1-P(Z<-0.83)=1-0.2033=0.7967~ 79.67 %