Introductory Statistics
STAT 230
Assignment 3
Instructions:
You must show significant steps to get full marks!
This assignment is out of 25 points.
1. Formally prove the following:
(a) (2 points) Let X be a continuous random variable that is normally distributed
with mean µ and standard deviation σ. Show that approximately 95% of the
area under the normal density curve is within 2 standard deviations of the mean.
(b) (2 points) Let Y1 and Y2 be uncorrelated (independend) random variables and
let U1 = Y1 + Y2 and U2 = Y1 − Y2. Find Cov(Y1, Y2) in terms of the variance of
Y1 and Y2.
2. (4 points) Given the following probability density function,
f(x, y) = (
2x, 0 ≤ x ≤ 1, 0 ≤ y ≤ 1
0, elsewhere
Find the covariance of X and Y .
3. (4 points) Let X and Y be discrete random variables with joint probability distribution shown below.
x
y −1 0 1
−1
1
16
3
16
1
16
0
3
16 0
3
16
1
1
16
3
16
1
16
Show that X and Y are dependent, but have zero covariance.
4. A chemical process has produced, on the average, 600 tons of chemical per day. The
daily yields for the past week are 585, 604, 590, 593, 602, 598 and 593 tons.
(a) (6 points) Do these data indicate that the average yield is less than 600 tons
and hence that something is wrong with the process? Test at the 5% level of
significance.
(b) (1 point) What assumptions are required for the valid use of the procedure you
used to analyze these data.
(c) (1 point) Would you decision change if the level of significance was α = 0.01?
Explain.
(d) (1 point) Would you decision change if the alternative hypothesis was two-tailed
with α = 0.05? Explain.
(e) (3 points) Construct a 95% confidence interval for the true mean weekly yield
and interpret the interval.
(f) (1 point) Is the result consistent with your conclusion in part (a)? Explain.
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