Oct. 31, 2002 Midterm Key L. Phillips
Econ 240A-5
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)
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*pand n*(1-p). 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 = = [20.85 – 20]/(6.7) = 0.85/(6.76/4.47) = 0.85/1.51=0.56, where E(= the population mean under the null hypothesis, i.e. 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
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?
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:
] = (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.
e. Which model appears to fit better, the linear one or the proportional(power function) one?
Answer: the proportional has a higher R2, but they are comparable.
Figure 5.1 UC Budget, General Fund Component Vs. CA Personal Income, both in Billions of Nominal $, 1968-69 through 2002-03, log-log scale
---------------------------------------------------------------------------------------------------------
Table 5.1 Regression of the Natural Logarithm of the UC Budget, General Fund Component, on the Natural Logarithm of CA Personal Income, both in Billions of Nominal $, 1968-69 through 2002-03