# 1. Let X be a random variable with probability density function

1. Let X be a random variable with probability density function
f(x)=c(1-x^2): -1<x<1
0 elsewhere
a) What is the value of c?
b) What is the cumulative distribution function of X?

2. What can be said about the estimated slope coefficient for a regression of Y on X,
versus the estimated slope coefficient for a regression of X on Y?
a) The slopes are reciprocals
b) The slopes are not reciprocals
c) The slopes are the negative of each other
d) The slopes are identical

3. Consider the height/weight data provided. The first column contains the height in
centimeters and the second column the weight in kilograms.
a) Make a scatterplot of the height/weight data using Stata.
b) What is the sample size? Explain briefly.
c) Determine the sample mean height and weight using Stata.
d) What would the sample mean height and weight have been if they were measured in
feet and pounds rather than centimeters and kilograms?
e) Determine the sample variances of the height and weight data using Stata.
f) Determine the sample covariance of height and weight using Stata.
g) Without further use of the computer, determine the sample correlation of height and
weight. What do you conclude?

4. The body mass index (BMI) is the weight in kilograms divided by the squared height in
meters, i.e. w/h2.
a) Determine the body mass indices B1,…,Bn for all n observations in your sample (using
Stata).
b) Determine the sample mean BMI.
c) Determine the sample variance BMI.
d) Test the null hypothesis that ?B = 25 against the alternative that ?B ¹ 25 using the CLT.

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