Midterm Key Econ 240A 1 Oct 31 2002 L Phillips Answer all five queations 1 15 points An investor believes that on a day when the Dow Jones Industrial Average DJIA increases the probability that the NASDAQ also increases is 77 If the investor believes that there is a 60 probability that the DJIA will increase tomorrow what is the probability that the NASDAQ will increase as well Answer Prob NASDAQ up DJIA up 0 77 Prob DJIA up tomorrow 0 60 Prob NASDAQ up Prob NASDAQ up DJIA up Prob DJIA up tomorrow 0 77 0 60 0 462 2 15 points The boss assigns you to choose an internet service provider ISP You want an ISP that is large enough so that a customer caller seldom receives a busy signal With the ISP you choose a customer caller encounters a busy signal only 8 of the time Your office makes 60 calls per week a What is the probability that callers from your office do not encounter any busy signals in a week Note Full credit for the correct numerical expression for the probability You do not have to calculate the numerical value of this expression in part a Answer each call is a Bernoulli event with a success being no busy signal with probability 0 92 and a failure being a busy signal with probability 0 08 Using the binomial distribution the probability of 60 successes in 60 calls is P k 60 60 60 0 0 92 60 0 08 0 1 0 92 60 1 0 92 60 This is a satisfactory answer for this exam LnP k 60 60 ln 0 92 5 0028965 So exp 5 002895 0 0067 Prob k 60 0 01 Oct 31 2002 Midterm Key Econ 240A 2 L Phillips b Suppose you want to approximate an answer for this probability Provide a numerical approximate value for the probability that callers from your office do not encounter a busy signal in a given week Answer Using the normal approximation the E k n p 60 0 92 55 2 The VAR k n p 1 p 60 0 92 0 08 4 416 The standard deviation is the square root of 4 41 which equals 2 1 The probability of getting less than 60 successes would be the probability of falling below a standardized z of Z 60 55 2 2 1 2 29 From Table 3 in Appendix B the area in the upper tail of the normal distribution for z 2 29 is 0 011 so the chances that no one in your office gets a busy signal are pretty low c Is this approximation pretty good Explain Answer yes the criteria for the approximation are n p 5 and n 1 p 5 The first is 60 0 92 and is clearly satisfied the second is 60 0 08 4 8 and is close 3 15 points Americans tend to eat too much fast food A doctor claims that the average Californian is more than 20 pounds overweight To check his claim you take a random sample of 20 Californians and weigh them The difference between their observed weight and their ideal weight follows 16 23 18 41 22 18 23 19 22 15 18 35 16 15 17 19 23 15 16 26 The mean of this sample is 20 85 and the standard deviation of this sample is 6 76 a From this sample data do you think the doctor s claim is true Answer The null hypothesis is that the population mean is equal to 20 and the alternative hypothesis is that the population mean is 20 Since the population mean is not known use Student s t distribution The t statistic is t x E x s n 20 85 20 6 7 20 0 85 6 76 4 47 0 85 1 51 0 56 where E x the population mean under the null hypothesis i e Oct 31 2002 Midterm Key Econ 240A 3 L Phillips 20 So do not reject the null For 19 degrees of freedom i e n 1 at the 5 level the critical t statistic is 1 73 so the calculated t statistic is well below the critical t for a 5 type I error 4 15 points The following graph plots the UC budget the component funded by the state against California Personal Income both in billions of nominal dollars A linear trendline has been fitted to the data a How would you describe the goodness of fit Answer It is fairly good in that R2 is 95 but there are some sizeable errors in the 80 s and 90 s before and after the recession of 1991 b Would you use only this trendline to predict next year s UC budget Answer It would be a good idea to calculate several forecasts for example one from problem 5 below as well c From the information provided about past experience if California Personal Income goes up by a 100 billion next year how much would you expect the UC budget to increase Answer From the slope of 0 0027 about 0 27 billion or 270 million Midterm Key Econ 240A 4 Oct 31 2002 L Phillips UC Budget General Fund Component Vs CA Personal Income Both in Billions of Nominal 1968 69 through 2002 2003 4 3 5 y 0 0027x 0 1972 R2 0 9513 3 UC Budget 2 5 2 1 5 1 0 5 0 0 200 400 600 800 1000 1200 CA Personal Income Figure 4 1 UC Budget General Fund Component Vs CA Personal Income both in Billions of Nominal Dollars 1968 69 through 2002 2003 5 15 points In the figure below the UC Budget General Fund Component is plotted against California Personal Income both in billions of nominal dollars from fiscal year 1968 69 through 2002 03 on a log log scale In the table that follows the results of regressing the natural logarithm of the UC Budget against the natural logarithm of California Personal Income follows a Is this regression statistically significant Explain Answer yes the F statistic of 1598 is highly significant b Interpret the estimated slope coefficient Answer it is the elasticity of the UC Budget to CA personal Income indicating for every 10 increase in CA personal Income the UC Budget goes up 8 78 c Is this slope coefficient significantly different from zero Midterm Key Econ 240A 5 Oct 31 2002 L Phillips Answer Yes the t statistic from Table 5 1 of 40 is highly significant d Is this slope significantly less than one Answer the null hypothesis is that the slope is one and the alternative hypothesis is that the slope is less than one The t statistic is E 0 878 1 0 0 0219 0 122 0 0219 5 57 where the expected value of the estimated slope is 1 under the null hypothesis So the elasticity is significantly less than one From Table 4 in Appendix B for 33 n 2 degrees of freedom the critical t statistic is 1 70 since the t distribution is symmetric …
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