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Week 4 Tutorial Exercises: Review of Basic Statistics

Questions:

1. The following table gives the joint probability distribution between employment status and college graduation among those either employed or looking for work (unemployed) in the working age US population based on the 1990 US Census.

Joint Distribution of employment status and college graduation in the US population aged 25-64, 1990.

| Unemployed (Y=0) | Employed (Y=1) | Total |

Non college grads (X=0) | 0.045 | 0.709 | 0.754 |

College grads (X=1) | 0.005 | 0.241 | 0.246 |

Total | 0.05 | 0.95 | 1 |

a) Compute E(Y).

b) The unemployment rate is the fraction of the labor force that is unemployed. Show that the unemployment rate is given by 1- E(Y).

c) Calculate and

d) Calculate the unemployment rate for (i) college graduates and ii) non-college graduates,

e) A randomly selected member of this population reports being unemployed. What is the probability that this worker is a college graduate? A non-college graduate?

2. Suppose that X and Y are random variables with expected values and variances . Let Z = (X+Y)/2.

a) Find E(Z)

b) Find Var (Z) assuming that cov (X,Y) = 0.

c) Find Var (Z) assuming that cov (X,Y) = 0.5.

3. The following table contains the ACT (American College Test, score from 11-36) scores and the GPA (grade point average, score from 0 to 4) for 8 college students. Grade point average is based on a four-point scale and has been rounded to one digit after the decimal.

(i) Estimate the relationship between GPA and ACT using OLS; that is, obtain the intercept and slope estimates in the equation

Comment on the direction of the relationship. Does the intercept have a useful interpretation here? Explain. How much higher is the GPA predicted to be, if the ACT score is increased by 5 points?

(ii) Compute the fitted values and residuals for each observation and verify that the residuals (approximately) sum to zero.

(iii) What is the predicted value of GPA...