Question

Consider the observations taken on the continuous random variable Y given below.

1. Graph the data using an appropriate plot. Comment on the key features of the graph.
2. Assess the normality of the data set. What do you conclude?
3. Compute the 5-number summary.
4. Treat the sample mean and sample standard deviation as if they are the true population mean and standard deviation. Find P (Y > 90).
5. Treat the sample mean and sample standard deviation as if they are the true population mean and standard deviation. With a sample size of n = 50, what is the sampling distribution of the sample average?
6. For a sample of size n = 50, find P(Y  > 88.73).

x/y	0	1	P(X=x)
0	0.72	0.08	0.80
1	0.08	0.12	0.20
P(Y=y)	0.80	0.20	1

marginal probability

• P(X=x) = 0.80, x=0

             = 0.20, x=1

• P(Y=y) = 0.80, y=0

             = 0.20, y=1

P (Y=1 | X = 1) = P(Y=1, X=1)/P(X=1)

                                     = 0.12/0/20

= 0.6

P (Y=1| X = 0)= P(Y-1, X=0)/P(X=0)

= 0.08/0.80

= 0.1

P (X = 1 ∪ Y = 1) = P(X=1) + P(Y=1) – P(X=1, Y=1)

= 0.20 + 0.2 – 0.12

= 0.28

E(X) = , P(X=x) = 0.20

• E(Y) = , P(y=y) = 0.20
• E() = , P(X=x) = 0.20, E() = 0.20

Var(x) = E() - (x) = 0.16, var(y)=var(x) = 0.16

• Cov(x,y) = E(xy) – Ex)(Ey)
• E(xy) =  P(X=x) P(Y=y) = 0.12
• Cov(x,y) = 0.12 – = 0.08

Answer 1

Sol: