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  • You are hired by the governor to study whether a tax on liquor has decreased average liquor consumption in New York. You obtain from a random sample of n individuals in New York, each person’s liquor consumption both for the year before and for the year after the introduction of the tax. From this data, you compute Yi​​ ="change in liquor consumption" for individual i = 1,…. n. Yi​​ is measured in ounces so if, for example, Yi​​ = 10, then individual i increased his liquor consumption by 10 ounces. Let the parameters​​ μy​​ and​​  σy2​​ of Y denote the​​ population mean and variance of Y.

    • You are interested in testing the hypothesis H0​​ that there was no change in liquor consumption due to the tax. State this formally in terms of the population parameters.

    • The alternative, H1, is that there was a decline in liquor consumption; state the alternative in terms of the population​​ parameters.

    • Suppose that your sample size is n = 900 and you obtain estimates μy​​ =​​ -32.8 and σy= 466.4. Report the t-statistic for testing H0​​ against H1. Obtain the p-value for​​ the test [use Table 1 in Stock and Watson, p. 749-750]. Do you reject at a 5% level? At 1%​​ level?

    • Would​​ you​​ say that the estimated fall in consumption is large in magnitude? Comment on the practical versus statistical significance of this​​ estimate.

    • In your analysis, what has been implicitly assumed about other determinants of liquor consumption over the two-year period in order to infer causality from​​ the tax change to liquor​​ consumption?

 

  • Let Y be a Bernoulli random variable with success probability Pr(Y=1) = p, and let​​ Y1,...,Yn

be​​ i.i.d. draws​​ from this​​ distribution. ​​ Let​​ ​​ pˆ​​ ​​ be​​ the fraction of​​ successes​​ (1s)​​ in this​​ sample.

  • Show that​​ ​​ pˆ​​ ​​ =​​ Y

  • Show that​​ ​​ pˆ​​ ​​ is​​ an unbiased​​ estimator​​ of p.

  • Show that var(​​ pˆ​​ ) =​​ p(1-p)/n

  • Let​​ Y1,​​ Y2,​​ Y3,​​ Y4, be independently, identically distributed random variables from a population with mean​​ ​​ and variance​​ . Let​​ Y​​ = (1/4) (Y1+Y2+Y3+Y4) denote the average of these four random​​ variables.

 

  • What are the expected value and variance of​​ Y¯in terms of​​ ​​ and​​ 2?

  • Now, consider a different estimator of​​ :​​ Ỹ​​ =(1/8)Y1+(1/8)Y2,+(1/4)Y3+(1/2)Y4. This is an example of a​​ weighted​​ average of the​​ Yi.’s. Show that​​ Ỹ​​ is also an unbiased estimator​​ of

. Find the variance of​​ .

  • Based on your answer to parts (a) and (b), which estimator of​​ ​​ do you prefer,​​ Y​​ or​​ ?

  • Suppose​​ Y1,​​ Y2,​​ Y3,​​ Y4​​ follow​​ a​​ Normal​​ distribution​​ with​​ mean​​ 5​​ and​​ variance​​ 2=3. What is the distribution of​​ Y​​ and​​ ?

 

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  • Suppose at Columbia University, grade point average (GPA) and SAT scores are related​​ by the conditional expectation E(GPA|SAT) = .90 + .001​​ SAT.

 

  • Find the expected GPA when SAT =​​ 1600.

  • Find​​ E(GPA|SAT=2200)

  • If the average SAT in the university is 2000, what is the average​​ GPA?

 

  • Let​​ u​​ and​​ X​​ be two random variables that satisfy E [u|X] = 0 and​​ E[u2|X]=2.

  • Find the​​ unconditional mean and variance of​​ u.

  • What is the covariance between​​ u​​ and​​ X?

 

 

 

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Following questions will not be graded, they are for you to practice and will be discussed at​​ the recitation:

 

  • [Practice question, not graded] SW​​ 2.3

 

 

Rain (X=0)

No Rain (X=1)

Total

Long Commute (Y=0)

0.15

0.07

0.22

Short Commute (Y=1)

0.15

0.63

0.78

Total

0.30

.70

1.00

 

Using the random variables X and Y from Table 2.2 (given above), consider two new random variables W = 3 + 6X and V = 20 – 7Y.​​ Compute:

    • E(W) and​​ E(V).

    • σ²W​​ and​​ σ²V.

    • σW,V​​ and​​ Corr(W,V).

 

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  • [Practice question, not graded] SW 2.6

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.

 

 

 

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

0.050

0.950

1.000

 

  • Compute​​ E(Y).

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

  • Calculate the E(Y|X=1) and​​ E(Y|X=0).

  • Calculate the unemployment rate for (i) college graduates and (ii) non-college​​ graduates.

  • 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?

  • Are educational achievement and employment status independent?​​ Explain.

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  • [Practice question, not graded] SW 2.14 [Hint: Use SW Appendix Table 1.]

In a population E[Y] = 100 and Var(Y) = 43. Use the central limit theorem to answer the following questions:

  • In a random sample of size n = 100, find Pr(​​ Y¯≤101)

 

  • In a random sample of size n = 165, find Pr(​​ Y¯>98)

 

  • In a random sample of size n = 64, find Pr(101 ≤​​ Y¯​​ ≤103)

 

  • [Practice question, not graded] SW​​ 3.12

To investigate possible gender discrimination in a firm, a sample of 100 men and 64 women with similar job descriptions are selected at random. A summary of the resulting monthly salaries are:

 

Avg. Salary (Y¯)

Stand Dev (of Y)

n

Men

$3100

$200

100

Women

$2900

$320

64

 

  • What do these data suggest about wage differences in the firm? Do they represent statistically significant evidence that wages of men and women are different? (To answer this question, first state the null and alternative hypothesis; second, compute the relevant t-statistic; and finally, use the p-value to answer the​​ equation.)

  • Do these data suggest that the firm is guilty of gender discrimination in its compensation politics?​​ Explain.

 

  • [Practice question, not graded] SW​​ 3.2

Let Y be a Bernoulli random variable with success probability Pr(Y=1) = p, and let​​ Y1,...,Yn

be​​ i.i.d. draws​​ from this​​ distribution. ​​ Let​​ ​​ pˆ​​ ​​ be​​ the fraction of​​ successes​​ (1s)​​ in this​​ sample.

 

  • Show that​​ ​​ pˆ​​ ​​ =​​ Y¯

  • Show that​​ ​​ pˆ​​ ​​ is​​ an unbiased​​ estimator​​ of p.

  • Show that var(​​ pˆ​​ ) =​​ p(1-p)/n

 

 

  • [Practice question, not graded] SW 2.10 [Hint: Use SW Appendix Table​​ 1.]

Compute the following probabilities:

  • If Y is distributed N(1,4), find​​ Pr(Y≤3).

  • If Y is distributed N(3,9), find​​ Pr(Y>0).

  • If Y is distributed N(50,25), find Pr(40≤Y≤52).

  • If Y is distributed N(5,2), find Pr(6≤Y≤8)

 

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  • [Practice question, not graded] SW​​ 3.3

In a survey of 400 likely voters, 215 responded that they would vote for the incumbent and 185 responded that they would vote for the challenger. Let p denote the fraction of all likely voters​​ that preferred the​​ incumbent​​ at the time of​​ the survey,​​ and​​ let ​​ pˆ ​​​​ be the fraction of

survey respondents that preferred the incumbent.

  • Use the survey results to estimate​​ p.

  • Use​​ the​​ estimator​​ of the variance​​ of​​ ​​ pˆ​​ ,​​ ​​ pˆ​​ ​​ (1​​ -​​ ​​ pˆ​​ )/n to​​ calculate the​​ standard error of your estimator.

  • What is the p-value for the test H0: p=0.5 vs.​​ H1:p≠0.5?

  • What is the p-value for the test H0: p=0.5 vs.​​ H1:p>0.5?

  • Why do the results from (c) and (d)​​ differ?

  • Did the survey contain statistically significant evidence that the incumbent was ahead​​ of the challenger at the time of the survey?​​ Explain.

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