Direction: Answer all the questions and submit your exercise in PDF. Make sure to include your R scripts or computer codes as an appendix to your submission. Please comment on each procedure so I would know what you are doing (or intend to do). Submission is via E-BART. (20 points) 1. Think of the situation of rolling two dice and let M denote the sum of the number of dots on the two dice. (So M is a number between 1 and 12.) a. In a table, list all of the possible outcomes for the random variable M together with its probability distribution and cumulative probability distribution. (5 points) b. Calculate the expected value and the standard deviation for M. (5 points) c. Using R, plot the probability and cumulative probability distribution. Label it properly. Looking at the plot of the probability distribution, you would notice that it resembles a normal distribution. Should you be able to use the standard normal distribution to calculate probabilities of events? Why or why not? (10 points) (20 points) 2. Using R/R-Studio, plot the following (overlay each on the other): i. Standard normal distribution ii. t density for 2 degrees of freedom iii. t density for 4 degrees of freedom iv. t density for 25 degrees of freedom Label your plot properly. (10 points) Describe the trend that you observe. Compare the Student t distribution with the standard normal distribution as degrees of freedom increases. (10 points) (20 points) 3. Math SAT scores (Y) are normally distributed with a mean of 500 and a standard deviation of 100. An evening school advertises that it can improve students' scores by roughly a third of a standard deviation, or 30 points, if they attend a course which runs over several weeks. (A similar claim is made for attending a verbal SAT course.) The statistician for a consumer protection agency suspects that the courses are not effective. She views the situation as follows: 𝐻0 ∶ = 500 𝑣𝑠. 𝐻1 ∶ = 530. a. Using R/R-studio, sketch the two distributions under the null hypothesis and the alternative hypothesis. (5 points) b. The consumer protection agency wants to evaluate this claim by sending 50 students to attend classes. One of the students becomes sick during the course and drops out. What is the distribution of the average score of the remaining 49 students under the null, and under the alternative hypothesis? (5 points) c. Assume that after graduating from the course, the 49 participants take the SAT test and score an average of 520. Is this convincing evidence that the school has fallen short of its claim? What is the p-value for such a score under the null hypothesis? (10 points) 4. The news-magazine The Economist regularly publishes data on the so called Big Mac index and exchange rates between countries. The data for 30 countries from the April 29, 2000 issue is listed below: Price of Actual Exchange Rate Country Currency Big Mac per U.S. dollar Indonesia Rupiah 14,500 7,945 Italy Lira 4,500 2,088 South Korea Won 3,000 1,108 Chile Peso 1,260 514 Spain Peseta 375 179 Hungary Forint 339 279 Japan Yen 294 106 Taiwan Dollar 70 30.6 Thailand Baht 55 38.0 Czech Rep. Crown 54.37 39.1 Russia Ruble 39.50 28.5 Denmark Crown 24.75 8.04 Sweden Crown 24.0 8.84 Mexico Peso 20.9 9.41 France Franc 18.5 .07 Israel Shekel 14.5 4.05 China Yuan 9.90 8.28 South Africa Rand 9.0 6.72 Switzerland Franc 5.90 1.70 Poland Zloty 5.50 4.30 Germany Mark 4.99 2.11 Malaysia Dollar 4.52 3.80 New Zealand Dollar 3.40 2.01 Singapore Dollar 3.20 1.70 Brazil Real 2.95 1.79 Canada Dollar 2.85 1.47 Australia Dollar 2.59 1.68 Argentina Peso 2.50 1.00 Britain Pound 1.90 0.63 United States Dollar 2.51 The concept of purchasing power parity or PPP ("the idea that similar foreign and domestic goods … should have the same price in terms of the same currency," Abel, A. and B. Bernanke, Macroeconomics, 4th edition, Boston: Addison Wesley, 476) suggests that the ratio of the Big Mac priced in the local currency to the U.S. dollar price should equal the exchange rate between the two countries. a) Enter the data into R. (Alternatively, you can copy and paste the data into Excel and load into R as a data frame). Calculate the predicted exchange rate per U.S. dollar by dividing the price of a Big Mac in local currency by the U.S. price of a Big Mac ($2.51). (5 points) b) Run a regression of the actual exchange rate on the predicted exchange rate. If purchasing power parity held, what would you expect the slope and the intercept of the regression to be? Is the value of the slope and the intercept "far" from the values you would expect to hold under PPP? (10 points) c) Plot the actual exchange rate against the predicted exchange rate. Include the 45- degree line in your graph. Which observations might cause the slope and the intercept to differ from one and zero, respectively? (5 points) (20 points) 5. You have obtained a sample of 14,925 individuals from the Current Population Survey (CPS) and are interested in the relationship between average hourly earnings and years of education. The regression yields the following result: 𝑎ℎ𝑒 ̂ = −4.58 + 1.71 × 𝑒𝑑𝑢𝑐 , 𝑅2 = 0.182, 𝑆𝐸𝑅 = 9.30 where 𝑎ℎ𝑒 ̂ and 𝑒𝑑𝑢𝑐 are measured in dollars and years, respectively. a. Interpret the results (5 points) i. Interpret the coefficients and the regression 𝑅2. ii. The average years of education in this sample is 13.5 years. What is mean of average hourly earnings in the sample? b. Is the effect of education on earnings large? Justify your answer using other possible information. (5 points) c. Why should education matter in the determination of earnings? Do the results suggest that there is a guarantee for average hourly earnings to rise for everyone as they receive an additional year of education? Do you think that the relationship between education and average hourly earnings is linear? d. Interpret the measure SER. What is its unit of measurement? (5 points)