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Answers for Chapter 3 1a) (5+6+5)/36 = 44.4%. 1b) The chance of rolling greater than 9 is the chance of rolling 10, 11, or 12. This is equal to (3+2+1)/36 = 16.7%. The chance of rolling less than or equal to 5 is the chance of rolling 2, 3, 4, or 5. This is equal to (1+2+3+4)/36 = 27.8%. 1c) The mean value is 7, the variance is 5.833, and the standard deviation is 2.41. 1d) The CDF is:

CDF for sum of two dice

0

0.2

0.4

0.6

0.8

1

1 2 3 4 5 6 7 8 9 10 11 12

2a) Its mean is 4, its variance is 3, and its standard deviation = 1.732. The CDF is:

CDF for unfair die

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0.2

0.4

0.6

0.8

1

1 2 3 4 5 6

2b) The mean of the unfair die is higher than the mean of the fair die, because the chance of a high roll 6 is substantially increased, and the chance of a low roll 1 is reduced. The unfair die also has a higher standard deviation, because the chance of an error of plus or minus 2.5 has increased from 33% to 40%, and the chance of an error of plus or minus 0.5 or 1.5 has decreased. 3a) The largest value is 3, the smallest is 1. 3b) The probability of Z>2 is 0.5*0.5*1 = 0.25 = 25%. The probability of Z<2 is 100% minus this, or 75%. 3c) Zero, because the CDF has a height of 0 at and above 4 on the horizontal axis. 3d) The expected value is less than 2, because the odds of numbers below 2 are higher and the range is symmetric around 2. 4a) 1*1/3 = 33%. 4b) The chance that X will be less than 0.5 is 0.5*1/3 = 16.7%. The chance that X will be greater than 1.5 is 0.5*2/3 = 33%. The chance that X will be less than 1.5 is 100%-33% =

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67%. 4c) The CDF is:

CDF of random variable X

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2

5. Both have probabilities that don’t add up to 1. For the first, the chances of outcomes A through D add up to 0.3+0.2+0.3+0.4 = 1.2, greater than 1. For the second, the area under the PDF is 3*0.25 = 0.75, less than 1. 6a) 13 of the 52 cards are clubs, so the chance of a club is 13/52 = 0.25 = 25%. 6b) The expected value is 0.25*1 + 0.25*2 + 0.25*3 +0.25*4 = 2.5. The standard deviation is 1.118. 6c) Now the chance of a spade (S=1) is 12/51 and the chance of any other value is 13/51. The expected value is 0.2353*1 + 0.2549*2 + 0.2549*3 + 0.2549*4 = 2.53. It rose because the chance of S=1 is lower than before, and the chance of any other value of S has risen. 6d) Now the expected value is 2.5 again because the chance of each suit is back to 25% (12/48). 7a) The lowest possible rate is 1%, the highest possible rate is 7%. 7b) The chance of a rate less than 3% is 25%, read directly off the graph. The chance of a rate more than 4% is 50%. 7c) It is equally likely, with a probability of 0 in both cases, that the rate will be exactly 2% or 4%, but it is more likely that the rate will be close to 4%, because the slope of the CDF is steeper at that rate. 8a) The z-statistic is equal to -0.8, which has a chance of 21.2%. 8b) The chance of being above 22 (z=0.4) is equal to 34.5%, the chance of being above 24 (z=0.8) is 21.2%, so the chance of being between 22 and 24 is 13.3%. 8c) It is more likely that the price will be between $22 and $24, because that range is closer to the mean of $20 than the range $12 to $14, and the normal PDF is therefore higher there. 9a) (125-120)/10 = 0.5, so the chance of this is 30.9%. 9b) 60,000 yen interest is equal to $550 if the exchange rate is 109 or less. The z-statistic is -11/10 = -1.1; the chance of this is 13.6%. 9c) 60,000 yen interest is worth $440 or more if the exchange rate is 136.4 or less. The chance

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of this is 95%. The odds that the US investment is better are thus only 5%. 10a) The expected value is equal to (0*0.5) + (1*0.5) = 0.5. 10b) It is expected to land heads 50% of the time, so we should expect 50 heads in 100 throws. 10c) The expected value becomes 0.6, and we should expect 60 heads in 100 tosses. 11a) The first firm gets a bid in four of the six cases, so the expected value of X is 4/6 = 2/3. The same is true for the second firm, so the expected value of Y is also 2/3. 11b) The covariance is -0.11. They are negatively correlated, because given that one firm gets the first contract, the chance that the second firm will get a contract falls to only 50%. 12a) GE and IBM have positive correlations, CSX has a negative one, and Microsoft’s appears to be uncorrelated. 12b) GE and IBM have weak correlations, and CSX has a fairly strong one. 12c) Buy CSX - if the market falls, CSX is likely to rise. 13a) The average value is 0.6136 pounds per dollar; the standard deviation is 0.046. A typical fluctuation is plus or minus 0.046, which is about 7.50% of the average value. 13b) Average Std. Dev Percent Germany 1.704 0.195 11.44% Ireland 0.659 0.050 7.58% Italy 1455.9 192.5 13.22% Japan 127.2 20.55 16.15% 13c) In percentage terms, Japan’s fluctuates the most (16.15%) and Britain’s the least (7.50%). The Japanese currency is riskiest.

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Answers for Chapter 4 1a) The expected change over a ten-day period is 10 times the one-day expected change, which is 10 times 0 = 0. The average change is therefore also 0. 1b) 50 cents divided by the square root of 10 = 15.81 cents. 1c) This requires an average daily fall of 5 cents, which is 0.316 standard deviations. The odds of this are about 38% if the stock is normally distributed. 2a) The estimated means are 11.56, 1.87, 5.96, and 2.80 respectively. The estimated variances are 1.129, 0.549, 1.907, and 0.564 respectively. 2b) The confidence intervals are [10.71,12.41], [1.28,2.47], [5.13,6.80], and [2.20, 3.40] 2c) They do not overlap. The US has lower expected unemployment. 2d) They do overlap. It is not possible to tell which country has lower expected inflation from these data. 3a) The estimated expected value is $594,171. 3b) Predict $594,171, because it is the estimated expected value. 3c) The estimated variance is $7,445,000,000; the estimated standard deviation is $86,300. We might expect to be off by about $86,000 on average. 3d) Now we would predict $694,171. 4a) $1.8 million divided by four months = $450,000 per month. 4b) The estimate is the same as the estimate for a single month, $594,171. The standard deviation of the estimate is $86,300/2 = $43,150. 4c) The estimated amount of $594,171 is greater than the required amount of $450,000; it is more than three standard deviations greater, because (594,171-450,000)/43,150 = 3.34. 4d) You should recommend it. The odds that sales will be more than three standard deviations below their expected value are extremely small, so this project is quite safe. 5a) $86,300/3 = $28,800. 5b) For 90% confidence and 8 df, the critical value of the t distribution is 1.86. For 95% confidence, the critical value is 2.31. The 90% confidence interval is $594,171 +/- $28,000 times 1.86, which is approximately [$540,000, $648,000]. The 95% confidence interval is approximately [$527,000, $661,000] which is slightly wider. 5c) It depends on the degree of desired confidence. For 95% confidence, 550 and 650 are reasonable, 700 is not; 550 and 650 are inside the confidence interval but 700 is not. For 90% confidence, 650 becomes unreasonable, since it doesn't fall inside the 90% confidence interval. 6a) The t-critical values are 1.68, 2.02, and 2.70 respectively. The confidence intervals are [3.328,4.672] for 90%, [3.192,4.808] for 95%, and [2.92, 5.08] for 99%. 6b) 4.2 is inside all three confidence intervals. 2.8 is inside none of them. We can believe that the multiplier is equal to 4.2 but not that it is equal to 2.8.

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6c) We should predict 4*$20 billion = $80 billion rise in GDP. The 90% confidence interval is 3.328*$20 billion to 4.672*$20 billion, which is approximately $66.6 billion to $93.4 billion. 7a) The sample average is also 3.5. The variance is 4.05/100 = 0.0405. The distribution is approximately N(3.5,0.0405), because we know the true variance under the assumption that the die is fair. 7b) The standard deviation is 0.201. The chance of getting an average above 3.6 is 30.9%. The chance of an average below 3.2 is 6.68%. 7c) 3.894. 7d) In this case the distribution would still be approximately normal; however, we would not know the true mean or variance, so we could not say exactly what it was. 8a) The expected value is 0.5 if fair and 0.75 if weighted. 8b) A fair coin is expected to produce 25 heads, a weighted one 37.5. 8c) The estimated mean is 26/50 = 0.52. The estimated variance is 0.2535, and the estimated standard deviation is 0.5035. 8d) 0.5035 / 7 = 0.0719. 8e) The confidence interval is [0.3954, 0.6846]. 0.5 is inside this interval and 0.75 is outside it. We conclude that the coin is fair, not weighted. 9a) The estimated fraction that prefer the company's product is 28/100 = 0.28 or 28%. The variance of this estimate is 0.002016. 9b) This sample produces exactly the same estimate because it also has 28% customers preferring the product. It has a different variance estimate, which is 0.0004. This is a smaller estimated variance because of the larger sample. 9c) The standard deviations are 0.045 and 0.020 respectively. The confidence intervals are therefore [0.190, 0.370] and [0.240, 0.320]. The second one is narrower. 9d) The variance is 0.28*0.72/N. To get the confidence interval to plus or minus 0.01 means the standard deviation must be 0.0051, and the variance must be 0.000026. Solving for N produces 7750 customers. 10a) The estimated expected value is 3.81. The estimated variance is 3.02, which is rather lower than expected. 10b) If fair, then this is 1.54 standard deviations higher than expected, so the chance of observing this or higher is about 6.2%. The chance of being this far off is twice that, 12.4%. 10c) The confidence interval is 3.81 plus or minus 1.98*0.173, which is [3.466, 4.154]. 3.5 is inside this range, which suggests the assumption of a fair die is acceptable.

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Answers for Chapter 5 1a) The average salary is $35,272, which is also the expected salary of another major. The estimated standard error is $1245, and the t-value is 2.11, so the confidence interval is [$32,645, $37,898]. 1b) [$36,373, $41,626] 1c) (35,272 - 39,000) / 1245 = -2.99. We reject because this is (in absolute value) greater than the critical value of 2.11. 2. The null hypothesis is that the slope is equal to 0. We should use a one-tailed test; we know that if the slope isn’t zero, then it’s positive, because supply curves don't slope down. The alternative hypothesis is that the slope is greater than 0. 3a) The average change is 3.21%. The sample variance of the data is 11.72, so the estimated variance of the estimator based on 10 observations is 11.73/9 = 1.30. 3b) The t-statistic is (3.21-0) / 1.14 = 2.81. This is greater than the critical value of 1.83, so we reject the null of no change. 3c) The test statistic is (3.21-(-1.20) / 1.14 = 3.87. This is more than the critical value of 1.83, so we reject the null of equality to the market return of -1.20%. 3d) Apparently good news, as they rose on a day when the market as a whole did. 4a) The mean is 2.72 and the sample variance is 3.525. The variance of the estimator is therefore 0.391 and its standard deviation is 0.625. The t-statistic for this test is 2.72/0.625= 4.35. This is greater than the critical value of 1.83, so we reject this null hypothesis. Airline stocks rose that day. 4b) The test statistic is (2.72-0.66)/0.625= 3.29, greater than the critical value of 1.83, so we reject this null hypothesis also. 4c) It would seem that it did have a continued effect. This may have occurred due to further developments in the case, however. 4d) It would make no difference. 5a) The estimated variance is 11.73. 5b) The estimated variance is 3.525. 5c) The F-statistic for this test is equal to 11.73/3.525 which is 3.328, and it has 9 and 9 degrees of freedom. The 95% critical value is 3.18. We can therefore reject, just barely, the null hypothesis of equal variances. Many different conclusions can be drawn. 6a) 14/25 = 56%, so the estimate is 0.56. The standard error is 0.101. 6b) The t-statistic is 0.06/0.101 = 0.59, below the critical value of 2.06 for a two-sided test. A two-sided test is more appropriate because there’s no reason to think the true fraction would be higher or lower than 50% if it is not equal to 50%. 6c) This belief implies the true chance is 80%. The t-statistic for this null hypothesis is (0.56-0.8)/0.101 = 2.38, which is above the critical value. The belief is not reasonable.

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7a) With 19 failures in 1500 products, the estimated chance of failure is 19/1500 = 1.267%. The variance of one trial is 0.0125, so the variance of 1500 trials is 0.000008337 and the SD of that many trials is 0.00289, or 0.289%. 7b) The critical Z-statistic is 1.96. (We use Z, not t, because the distribution of the individual observations is known to be non-normal). The rejection region is therefore 1% (the hypothesized value) plus or minus 1.96 standard deviations, which is greater than 1.566% or less than 0.434%. 1.267% is inside this, so we fail to reject. We use a two-sided test, since the alternative hypothesis allows the true failure rate to be either above or below 1%. 7c) The critical Z-stat is now 1.645. The rejection region is now greater than 1.475%. We still fail to reject, even on this one-sided test. A one-sided test may be more appropriate, since there are problems only if the manufacturer’s product is more defective than claimed. 8a) The expected value is µ, and the variance is µ*(1- µ). 8b) The estimate of µ is 18/50=0.36. The estimated variance is equal to 0.2327. 8c) The standard deviation of the estimated µ is the estimated standard deviation of Q divided by the square root of 50. This is equal to 0.0682. The critical Z value is 1.645. The rejection region is below 0.1878 or above 0.4122. We fail to reject the null hypothesis. 8d) In this case, the standard deviation of the estimate is the square root of 0.21 divided by the square root of 50, or the critical value is 1.645, the variance is 0.21, and the rejection region is below 0.1934 or above 0.4066. We again fail to reject the null hypothesis. 8e) The second test is more powerful; its rejection region contains the rejection region of the first test, so it will reject more often than the first one will when the null hypothesis is false. 9a) It has a t-distribution, because its variance is estimated, not known. 9b) The test statistic is 0.14/0.11 = 1.27. The critical value is 2.02, so we fail to reject. 9c) The confidence interval is -0.86 plus or minus 0.11*2.02, which is equal to [-1.08, -0.64]. We cannot tell; there is at least a 5% chance of the true elasticity of demand being on either side of -1. 9d) In this case the test statistic would be -0.29/0.11 = -2.63, the confidence interval would be [-0.93, -0.49] and we could definitely conclude that Acme was not a monopolist. We would reject the hypothesis that the elasticity is -1 and the entire confidence interval would be in the inelastic range. 10a) The standard deviation is 0.6. The T-stat is 0.25/0.6 = 0.42. This is less than the critical value of 1.99, so we fail to reject the null hypothesis. The rejection region is anything more than 1.99 standard deviations away from 10, that is, below 9.17 or above 10.83. b) If the true expected value is 11, then the chance of being below 9.17 is the chance of being 3.05 standard deviations below or lower, which is about 1%. The chance of being above 10.83 is the chance of being 0.28 standard deviations below, or higher, which is 61%. The chance that we will fall in the rejection region is thus 62%. The chance of falsely failing to reject is 38%; the power of the test is 62%.

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11a) The expected value is 3.5. 11b) We could not test whether the die is fair this way, because both possibilities for the distribution of this die make the same predictions about the expected value. 11c) If fair, the expected value is 0.167. If unfair, it is 0.3. The predictions are different, so an estimate of the expected value of X could help us tell the two possibilities apart. 11d) The estimated value is 0.27 with an estimated variance of 0.00199. Its estimated standard deviation is 0.0446, so the confidence interval is 0.27 +/- 0.0446*1.96 = [0.1808, 0.3592]. 0.3 is in this range and 0.167 is not, so the die is likely unfair. 11e) Test the number of 1s, or the number of 2s through 5s, or test the variance. Other answers are possible also.

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Answers for Chapter 6

1a) 0β̂ =83.57, 1β̂ =16.43. 1b) The predicted values are 149.29, 182.14, and 198.57. The residuals are 0.71, -2.14, and 1.43 respectively. 1c) The error terms are given by Y - (80 + 17*X). They are 150-148 = 2, 180-182=-2, and 200-199=1. The error is larger than the residual for observation 1 and smaller for observations 2 and 3. 1d) 0.71-2.14+1.43 = 0. 0.71*4 - 2.14*6 + 1.43*7 = 0. 2-2+1 = 1, not 0. 2a) The residuals are -6, -16, -9, -4, and -1 respectively. The SSR is 390. 2b) The residuals are 4, -2, -3, 8, and -1 respectively. The SSR is 94. 2c) The second set is better, because it has the smaller SSR. 2d) Neither could be, because the residuals don’t add to zero; the sum is -36 for the first set of estimates and +6 for the second. 3a) The predicted value is $3692. 3b) The difference is $95, which is a little more than two times the SER; fairly far off, but not impossibly so. 4. The value of the multiplier would then be 1/(1-0.65) = 2.86, and government spending would have a much less stimulatory effect on the economy. 5a) Costs rise by $5.018. The marginal cost is $5.018. 5b) 0 RVM cost -$1,519,000. Estimated fixed costs are predicted to be that amount, or 0 if one imposes the requirement that they can’t be negative. 5c) The cost of this is -1519+5000*5.018 = $25,090,000. 5d) The standard error of the regression is 6437. The average difference is about $6,437,000. 5e). The R2 is 0.888. The data lie quite close to the line, and the pattern looks reasonably linear. 6a) One more year of education adds $2182 to your income. This seems reasonable - it’s about 7.5% of the average value, which is $28,600. 6b) A high school graduate gets about $26,200, and a college graduate about $34,900. A college degree is worth $8,700 per year. 6c) The R2 is 0.101 and the SER is $20,200. This regression doesn’t forecast very well at all. 7a) $182 million. The effective marginal tax rate is 18.2%. 7b) The predicted values are $1422 billion, $1515 billion, $1601 billion, and $1694 billion respectively. 7c) The differences are +$77 billion, -$110 billion, -$153 billion, and -$180 billion. One is positive, the rest are negative. The average forecast error is not zero, because tax collections were unusually high 1997-1999 due to the economic boom.

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Answers for Chapter 7 1a) The demand curve should shift up and to the right, causing both price and quantity to increase. 1b) Higher income should lead to higher demand and higher expenditure, so the expected sign of β1 is positive. 1c) The estimate is 0.0129. A $1 billion rise in income is predicted to cause a $12.9 rise in automobile spending. This seems like a reasonable value - a little over 1% of the additional income would be spent on automobiles. 1d) The t-statistic is 29.5, and the critical value for a one-tail test is 1.65. We therefore reject the null hypothesis. This supports the theory of supply and demand by confirming its prediction about the effects of income on expenditure. 1e) The outcome would not be changed. The critical value for a two-tailed test is1.98, and the t-statistic is much larger than that. 2a) Spending is generally rising, but not smoothly; the graph displays ups and downs over time. 2b) The CPI is rising smoothly. 2c) The estimate is 0.00138. Car spending will rise by $1.38 million if income rises by $1 billion. 2d) We fail to reject the null hypothesis that income does not affect car spending. The t-statistic is 1.40 and the critical value is 1.65. 2e) This is not very easy to explain. Economically, one possibility is that car demand has a very low income elasticity, so that increases in income move the demand curve only a very small amount, not enough to be statistically significantly different from zero. 3a) The estimate is 103.94. The sign is positive, which is expected if players who hit more home runs are paid more. 3b) One extra home run increases salary by $103,940. Five more increases salary by $519,700. 3c) You should predict the intercept, which is $1,253,224. 3d) The estimate is 0.00154. This implies that $1,000,000 in salary adds 1.54 more home runs. 3e) It is more likely that home runs is exogenous and salary is endogenous, because salary depends on performance (really, on expected performance), whereas the number of home runs a player hits is probably not affected by his salary. The first regression is preferable because it puts salary on the left-hand side. Better would be to regress salary on home runs hit in the previous season, since that is known when salaries are negotiated. 4a) For a 90% confidence interval, the t-value is 1.65, for a 95% confidence interval it is 1.96. The confidence intervals are [4.83, 5.21] and [4.79, 5.24]. $4.90 is in both intervals, and so it is a believable true value. 4b) Because the true marginal cost is not random, there is no “chance” of this; it is either equal to $4.90 or it is not, and we don’t know which. 4c) Using a one-sided test, the critical t-value is 1.65 and the rejection region is the region

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greater than 846.4. Because the estimated value is actually negative, we fail to reject the null hypothesis. What is disturbing is that we are getting a negative estimate (and one that is significantly different from zero with a two-tailed test). This occurs because we are assuming costs are linear, which they probably aren’t. 4d) The average cost curve implied by this regression is AC = 5.018 - 1520/RVM. 5a) They should rise, as people and firms pay more taxes based on their higher incomes. 5b) I would expect β1 to be positive. The null hypothesis is H0: β1=0 and the alternative hypothesis is HA: β1≠0, because the alternative hypothesis is that higher GDP affects receipts without specifying which way. If the alternative hypothesis was that higher GDP raised receipts, then HA: β1>0 would be the correct alternative hypothesis. 5c) The t-statistic is 152.4, so we reject the null hypothesis that β1=0. 5d) The interval is [0.1805, 0.1853]. You should predict a rise of $1.829 billion. You should not accept a prediction of $2 billion; it implies that β1=2, and that value is outside the confidence interval. 6a) Real GDP is generally rising, but with brief periods of decline during recessions. Money declines at roughly the same time, but usually for longer periods and less sharply. 6b) The estimated slope is 0.092 and its t-statistic is 15.75. We find a significant effect of money on the CPI, which is what we expect, but it tells us nothing about the slope of the AS curve because both versions of the theory predict an increase. (It does tell us the AS curve is not horizontal, but economic theory does not predict that it should be horizontal.) 6c) The estimated slope is 3.77 and its t-statistic is 20.69. We find a significant effect of money supply on real GDP, which is consistent only with the sloped AS curve and suggests that AS is not vertical (at least in the short run). 6d) $3.77. 7a) The estimated slope is 0.051 and its t-statistic is 18.02. We find that deficits significantly increase CPI, as expected. 7b) The estimated slope is 2.02 and its t-statistic is 17.25. We find that deficits significantly increase real GDP, which suggests that AD is sloped, not vertical. This matches our finding in problem 6c. 7c) The estimates are 3.77 and 2.02, which are 1.75 apart. This is quite far apart, compared to the standard errors of the estimates, which are 0.183 and 0.117 respectively. 8a) β1 should be positive if higher education leads to higher salaries. The graph seems to match this idea. 8b) The test statistic is 8.62, and the critical value is 1.96. This rejects the null hypothesis that education does not affect wages. Education increases wages, as expected. 8c) The estimated variance is $20,204. This is pretty large compared to the average value of $28,595. A two-standard-deviation error is $40,408. Wages seem to be quite hard to predict.

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8d) There are more large positive ones. They look non-normally distributed because of the asymmetry between positive and negative errors. This suggests a better, but non-linear, estimator is possible.

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Answers for Chapter 8 1a) We might expect high-income states to give more, because they can afford more. We might expect states with high elderly populations to give less, since those states have other needs for their funds and might not have as much to give to schools. 1b) Only income is statistically significant. 1c) The F-statistic is (2.46e7 - 2.30e7)/2 / 2.30e7/45 = 1.565. The 5% critical value for (2,45) is 3.21, so we fail to reject this null hypothesis. Demographics don’t seem to matter; only wealth does. 2b) They do affect per capita school aid, just barely; the F-statistic is 2.64 and the 95% critical value is 2.59. 2c) School aid would rise by $3.18, but the effect is not significantly different from zero. 2d) School aid would rise by $3.50, also not statistically significantly different from zero. 2e) The only thing that matters is the elderly population. This is probably because elderly populations also have need for state resources that would take money away from schools. 3a) It was particularly strong around 1985, and particularly weak around 1995. 3b) We would expect a negative sign if high inflation rates lower the value of the dollar. Actually we get a positive estimate, 0.586, but it’s not significant; the t-statistic is 0.209. 3c) We would expect a positive sign, and we get one. The estimate is 1.411 and its t-statistic is 4.29. 3d) Now both parameters take the expected sign, and both are statistically significant. Both theories are supported by the multiple regression. 3e) A 1 percent rise in the inflation rate will decrease the value of the dollar by 7.61 points. A 2 percent rise in interest rates will increase the value of the dollar by 2*1.91 = 3.82 points. The change in both interest rates and inflation rates will cause the dollar's value to fall by 11.4 points. 4a) The demand curve shifts to the right. As a result, both ticket sales and prices should rise. 4b) We would expect β1>0, β2<0, β3=0. 4c) There is perfect multicollinearity because Wins+Ties+Losses = 16 always. 4d) A record of 0-16-0 would produce attendance of the intercept, which is 378,352. Adding a win adds 12,880 sales. Adding a tie adds 74,804 sales. 4e) You get the same answers, and it doesn’t matter which variable you drop. 5a) This hypothesis implies that β1>β2. We fail to reject the null β1=β2 against the alternative that β1>β2. A one-tailed test is appropriate, because wins should shift demand further than ties do, and the estimated value of β2 is larger than the estimated value of β1. (If we did a two-tailed test we’d still fail to reject.) 5b) There are so few ties that if a team has one, and happens to have unusually high attendance, the estimated parameter could be large. Therefore the estimate has a large standard error, and the estimated value isn’t significantly different from zero. (Washington had a tie and extremely

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high attendance in 1997.) Another possibility is that close games are more exciting than blowouts, and a team with a tie probably played some close games. 6a) All signs should be positive; spending more money should result in more students passing, if the production model of education is correct. 6b) None of the parameters are significantly different from zero. 6c) The hypothesis that state and local funds have equal effects on test scores implies that β1=β2. Imposing this produces math = β0+ β1*(localexp+stateexp). The F-test is (311,823-309,890)/1 / 308,890/681 = 4.24. The 95% critical value is 3.84. We conclude that the effects of state and local spending on test scores are significantly different from one another. 6d) For math, the test statistic is 2.26 with a critical value of 3.00. We fail to reject the null hypothesis that spending doesn’t matter. Either it doesn’t, or the model is poor. 7a) To test this null hypothesis, we use a t-test to see if β2=0. We fail to reject this null hypothesis at the 5% level, but would reject at the 10% level - the t-statistic is 1.67 and the critical values are 1.96 and 1.64 respectively. 7b) This is a one-sided test with the alternative hypothesis as stated. The 5% critical value is thus 1.64 and the t-statistic is 0.395. We fail to reject the null that spending doesn’t matter. 7c) We get the same results. 7d) We get essentially the same conclusions. 8a) The t-statistic is -0.97 against a critical value of 1.96. Poverty rates do not appear to have a significant effect on test scores. 8b) The t-statistic is 0.31 against a critical value of 1.64. Dropout rates do not have a significant effect on test scores either. Either student quality doesn’t matter, or poverty and dropout rate don’t measure student quality well, or the model is wrong to start with. 8c) The F-statistic is 1.01 against a critical value of 2.37. We can drop all four variables; they don’t affect scores. 8d) Substantially similar results; the F-statistic is 1.42 for this regression. 9a) Now β2, the parameter on the state share variable, is negative and significantly different from zero. 9b) β2, the parameter on the state share variable, is negative in this equation too. 9c) The evidence would suggest that higher rates of state spending reduce scores. This may be because state money comes with strings attached that reduce the effectiveness of education. Or it may be because states are choosing to give aid to districts where few students are getting high scores.

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Answers for Chapter 9 1. a) The marginal cost is equal to β1, which is estimated to be $5.02. 1b) The estimated elasticity is 1.178 when RVM=2000 and 1.065 when RVM=5000. It’s not quite constant but it’s not changing very much. 1c) In this model, the estimated elasticity is 1.118, and it is constant because of the log-linear functional form. 1d) They’re not terribly different. One reason to prefer the log-linear specification is because it avoids the problem of the predicted negative fixed cost. 2a) Marginal cost = β1 + 2*β2*RVM, which is estimated to be 4.85 + 0.000012*RVM. The graph is a line, sloping upwards, with an intercept of 4.85 and slope of 0.000012. 2b) For RVM=5000, the estimated marginal cost is $4.91. 2c) The null hypothesis is β2=0. We fail to reject this null hypothesis (t-statistic of 0.58 against a critical value of 1.65), suggesting that marginal costs are constant. 2d) There is not, because marginal costs start with a positive value and rise as output rises. The point where 4.85 + 0.000012*RVM = 0 is for a negative value of RVM, which is economically impossible. 3a) The formula is ε = β1 + 2*β2*logRVM, which for the estimated values works out to 0.866 + 0.035*log(RVM). 3b) log(5000) is equal to 8.517, so the estimated elasticity is 1.164. 3c) If elasticity is constant, then it must be the case that β2=0. We reject this null hypothesis; the t-statistic is 2.034 against a critical value of 1.96. We find that elasticity is rising as output rises. 3d) The elasticity is equal to 1 when 1 = 0.866 + 0.035 log(RVM), which is when log(RVM) = 3.826, or RVM=46. It has increasing returns to scale at any lower value of RVM and decreasing returns at any higher value. 3e) The average firm is producing 2696 output, so they have decreasing returns and they do not appear to be a natural monopoly. 4a) The estimates are 0.59, -0.011, 0.118, and 0.14 respectively. We would expect them to be positive, because increasing prices should increase costs, and between zero and one. Only one (driver labor) is significantly different from zero, but it is positive. 4b) The estimated elasticity of output is 1.06, and it suggests slightly decreasing returns to scale. The t-statistic to test if it is equal to 1 is (1.061-1)/0.0147 = 4.15, which means it is significantly greater than 1. This confirms decreasing returns to scale. 4c) The elasticity is equal to β5 + β6 log(RVM). The estimated values produce elasticity = 0.812 + 0.035 log(RVM). β6 is statistically significantly different from zero, so the elasticity is not constant; it increases as output increases. 4d) They are constant because there are no interaction terms for the log price terms. 5a) We reject this null hypothesis; the test statistic is 6.16 and the critical value for an F test with 1 restriction and 235 degrees of freedom is approximately 3.88.

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5b) We reject it again; the test statistic is still 6.00. 6a) The estimated value of β1 is 2812. Thus a rise of one year in education should increase wages by $2182. For a person with a salary of $25,000, an increase in education should produce a salary of $27,182. For a person with a salary of $2,000,000, the increase should be to $2,002,182. b) In this case, the increase in salary from a one-year increase in education is equal to 9.95% of the current salary, whatever that is. For someone with a salary of $25,000, salary would rise by $2488, to $27,488. For someone with a salary of $25,000, salary would rise by $199,000, to $2,199,000. c) The semilog function form is probably more reasonable. When wages are very high, because a person’s work is very productive, an increase in their productivity should produce a relatively large increase in wages. 7a) The expression is -0.046% + 0.808% education. From 4 years of education to 5, the increase in wages is 3.19%. From 13 years to 14 it is 10.46%. The fourteenth year is the more valuable. b) The null hypothesis that the effect of education is the same for all years is β2=0, and we do not reject this hypothesis; the t-statistic is 1.55. c) The t-statistic is -0.007, so we do not reject this hypothesis either. The percentage increase in education is given by β1 + 2β2education, so β1 is the return to education for a person who has none to begin with (education=0). We find that this increase is not statistically significantly different from 0. d) We reject this hypothesis: the F-statistic is 31.07. Education matters but we can’t say whether it is a linear or quadratic relationship. 8a) Our expectations are only partly correct; income increases expenditure but a stronger dollar reduces it. Expenditure rises by $21.7 million. b) The quadratic term is significant, with a t-statistic of -6.17. The increase is 0.027 - 0.0000038 income; for income=5000, the value is 0.008, and the increase is only $8 million. c) Again, income increases expenditure, dollar value decreases it. The estimated elasticity is 0.91; car expenditures rise slightly slower than income does. d) The quadratic term is again significant. The elasticity is 6.24 - 0.757*log(income), and when income=7000, the estimated elasticity is -0.20, suggesting that demand decreases as income rises. e) None are very satisfactory, but the best one is probably the second one. We cannot drop a significant quadratic term, but we do not want to conclude that income increases reduce new car expenditures. f) The answers are all very similar when wage income is used as the income measure.

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Answers for Chapter 10 1. a) β1 should be between 0 and 1. A value below 0 suggests that school districts spend less when more money is available, which is implausible, and a value above 1 implies that an increase in state aid is more than spent by school districts, also implausible. b) Only the parameter on income is statistically significantly different from zero. c) We fail to reject; the null hypothesis has a t-statistic of 1.11. In the new equation, the estimated value of β2 has risen by 0.01, which is about 50% of its initial value of 0.022; the estimated value of β2 is slightly lower than before. The standard error of β2 is about half of what it was in the old equation, and the standard error of β1 is smaller but only very slightly. d) This time we reject the null hypothesis that population can be dropped; the t-statistic is a very large 6.51. e) The correlation between income and population is 0.992. These variables are highly multicollinear. That is why when either one is dropped from the model, the other becomes highly significant. f) In this regression, β1 is significant, with a t-stat of 17.13. Total state aid has a correlation of 0.970 with income and 0.966 with population. All three of these variables are highly multicollinear. Omitting income and population from the model gives the impression that total state aid matters a great deal, whereas the reality is that (from this equation) it matters not at all. It’s just that big states have more spending and more state aid. g) From this regression we would say no, increased state aid doesn’t have any effect on spending as long as the size of the state (population and income) is held constant. 2. a) The variables are not very multicollinear; the correlation coefficients are 0.27, 0.23, and 0.13. b) Income and stateaid are significant; population is not. c) Population has no effect on state spending per pupil, but it does affect total spending; all else equal, a state with a larger population will have more pupils, and thus higher spending if spending per pupil is constant. d) Dropping population from the model is acceptable by t-test and causes virtually no change to the parameters of the model. e) The correlation of these variables is only 0.27. Multicollinearity is not a serious problem here. f) Now it appears that an increase in state aid does increase spending on schooling. This regression is more reliable because it is less affected by multicollinearity, and as a result its standard error of its estimate of the parameter on state aid is about half the size of the corresponding standard error in problem 1. 3. a) If the percentage of school age children rises by 1%, spending drops by $13 million. This seems unreasonable; it should rise, not fall. The wrong sign occurs because the parameter is not statistically significantly different from zero - the estimated value could take either sign. b) Rise with more school age children, fall with more elderly due to competition among needs for state funding. The estimated signs are the reverse of the expected one; schoolage is negative and elderly is positive.

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c) None are significant. The F-statistic for dropping all three is 0.12; we fail to reject the null that they can be dropped. 4. a) In this regression minority is significant, the other two are not. b) Dropping all three demographic variables has an F-statistic of 3.34 against a critical value of 2.84. We reject this null hypothesis. Dropping only elderly and schoolage has an F-statistic of 0.16 against a critical value of 3.23. We do not reject this null hypothesis. Demographics of race appear to matter; demographics of age appear not to. c) The preferred specification is the one that drops elderly and schoolage. Thus, changes in schoolage have no effect on spending. d) A 1 percent increase in minority population decreases school spending by $30.5 per pupil. This seems like a relatively small amount, given that average spending per pupil is $5114. However, the difference between a state with a 5% minority population and a 30% minority population would be $760 per student, and that is a substantial difference. 5. a) The regression cannot be estimated because of perfect multicollinearity among the variables. Hits, singles, doubles, triples, and home runs are linearly dependent because hits is equal to the sum of the other four variables. b) Five parameters (the ones on atbats, singles doubles, triples, and strikeouts) are significant at the 5% level, and one more (battingavg) is significant at the 5% level but not the 10% level. c) Dropping these variables is acceptable; we fail to reject the null hypothesis. d) Different specifications are possible. e) and f) Answers vary depending on the final specification selected. One acceptable specification produces answers of $2.27 million for e) and $2.06 million for f). 6. a) Answers can vary, but personal income and interest rates should be good measures, and CPI and priceoil should not be. b) The Republican and Democrat dummy variables add to 1, so the regression cannot be estimated due to perfect multicollinearity on the right-hand side. c) Only Republican (or Democrat if you keep that one) is significant at the 5% level, although investment is significant at the 10% level. d) The correlation between personal income and consumption is over 0.998, and the correlations between investment and the other two variables are over 0.95. All three of these variables grow proportionately as the economy grows. Obvious choices for the second pair are tbillrate and corpbondrate (which are strongly correlated - the coefficient is 0.87) and inflation and unemployment (which are not - the correlation coefficient is positive, not negative as expected, and only 0.29.) e) Answers can vary depending on which variables you choose to drop - there are several valid final specifications. f) Revenues are lower with Republican presidents in any acceptable final specification; the estimate is normally about $40 or $50 billion lower.

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Answers for Chapter 11 1a) For men, the average is $28,695, the median is $26,020. For women, the average is $19,437, the median is $17,475. The gap is $9,258 for average values and $8,545 for median values, both higher for men, both substantial differences. b) For education, the average value is 14.42 for men and 12.21 for women, a fairly substantial difference. For age, then average value is 37.2 for men and 38.9 for women, not so large a difference. c) All four parameters are statistically significant. d) One year of education adds $2494 to your wage income. 2.21 years of education adds $5512 to your wage income, which is more than half the gap. e) One year of age adds $520 to your income. In this sample, women are a little older than men, so this suggests women should have salaries $884 higher than men. f) The remaining difference between men and women is $4669, men having the higher value. This is what we might expect, but it’s still a very large gap, about 16% of the average male income, about 24% of the average female income. 2a) For women it equals 0. For men it is equal to the number of years of education they have. 2b) A woman’s wage rises by β1; a man’s rises by β1+β3. Education is more valuable for men if β3>0. 2c) The difference is β2+β3*education. If β3>0, then the gap is larger for more educated people. 2d) The interaction parameter is not statistically significant, nor is the parameter on the female variable significant any longer. 2e) The null hypothesis is β2=β3=0. We reject this hypothesis; the F-statistic is 5.16 against a critical value of 3.00. Gender definitely affects wages. 2f) The restriction is acceptable, because the t-statistic of the hypothesis β2=0 is only 0.22. In this model, education closes the gender gap, because β3<0. Education is more valuable for women than for men. 2g) We can’t tell, because we can accept either the specification where β2=0 (in which case education does reduce the gender gap) or where β3=0 (in which case it doesn't). 3a) 12.3% of the sample is Hispanic-American, 12.7% is African-American. These numbers are pretty close to the true population percentages. 3b) Average wage income is $16,630 for African-Americans, $13,185 for Hispanic-Americans, $25,420 for the rest. Those gaps are quite large. 3c) There does not appear to be discrimination on the basis of this regression; neither parameter is significant at the 5% level, although the parameter on the African-American variable is significant at the 10% level. 3d) The null hypothesis β3=β4=0 has an F-statistic of 2.255, less than the critical value of 3.00, though not by a lot. This can be explained if education affects wage income (which we see it does) and African-Americans and Hispanic-Americans have less education (which they do - the average level of education is 11.6 years for African-Americans, 10.3 years for Hispanic-

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Americans, and 13.7 years for everyone else). 3e) Answers can vary, but improving educational opportunities for minorities is an obvious answer. 4a) 6.9% of the bus companies are privately operated, 93.1% are publicly operated. b) Private costs are estimated to be 3250 thousand dollars lower. The difference is not significant at the 5% level but is at the 10% level. c) Two of the other parameters (β1 and β5) are significant, the rest are not. The parameter on PM has the wrong sign - the estimate is negative and we’d expect higher prices to raise costs - but it’s not significantly different from zero, so the wrong sign is not too surprising. d) All the prices can be dropped except PL, and the F-statistic to drop them is 0.39. The cost savings of private firms are now 3582 thousand dollars, and the difference is significant at the 5% level, though not the 1% level. e) Adding a quadratic term does not change the result - its t-statistic is 1.07 and it changes the other results only slightly. 5a) Using this functional form there is no significant difference between public and private firms - the t-statistic on the private variable is -0.67. Dropping the insignificant variables log(PA) and log(PM) raises the t-statistic to -1.12 but still doesn’t make the result significant. b) The Chow test rejects the null hypothesis of common parameter values. The F-statistic is 2.71 and the critical value for 6 restrictions and 229 degrees of freedom is approximately 2.16. c) There are several acceptable final specifications for the interacted model. All retain the interaction term on RVM. d) Answers vary depending on choice of acceptable final specifications. Most should involve the fact that private*log(RVM) is negative and significant, so that private firms seem to enjoy better scale economies than public ones do. 6a) The sum of squared residuals for this regression is 332.35. 6b) The sums of squared residuals are 70.9 and 46.0 respectively, for a total of 116.9, which is slightly less than the total from the chapter (which is 118.9) but only slightly so. 6c) The F-statistic is 34.09, well above the critical value of 3.23. We reject the null hypothesis of no shift, in favor of the alternate hypothesis that the curve shifted out in 1974 and back in 1982. 6d) We can tell that a shift occurred, but can’t date it precisely - we have to make assumptions about exactly when the shift took place. 7a) 48.4% of the observations are employee-owned. b) Employee-owned firms appear to have lower net income. The effect is significant at the 5% level and quite large - net income is 47% lower for employee-owned firms. c) Including the time effects does not substantially change the results. d) The sum of squared residuals is 111.0 in the regression including the fixed effects and 118.7 in the regression without them. The F-statistic is (118.7-111.0)/7 / 111.0/102 = 1.01. The time effects are not statistically significant.

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e) This equation is perfectly multicollinear, because the fixed effects (plus constant) are perfectly correlated with the empowned variable. This occurs because employee ownership is a characteristic of firms that does not change over time; thus the fixed effects capture it perfectly. f) The SSR is 62.67 with the fixed effects and 124.9 without them. The F-statistic is 4.75, greater than the critical value of 1.70. The fixed effects are significant - there are significant differences among the firms over time. There is no particular pattern in the values of the fixed effects. g) Answers are open-ended, but one possibility is that employee-owned firms choose to pay out more of the firm’s revenues as wages, leaving less over for net income.