Quantitative Methods in Economics - Homework

Homework 3 Required Exercises: Chapter 5: 13, 15, 17, 27, 29, 57, 69, 71 Required Problems: n: 78, mean: 22.4, sd: 4.8 .15 Density (1) An Assistant Manager has evaluated employees each month for the last twelve months. Each month an employee’s performance is scored from 1 to 5 where 5 is “exceptional” and 1 is “needs improvement.” At year end these monthly scores are summed up and reported to the Manager. The best possible score would be 60 (=12 months * 5 points). The Manager decides that the Assistant Manager has been overly harsh in scoring the 78 employees and wishes to adjust the scores. Call X the unadjusted annual employee performance score and Y the adjusted scores. .1 .05 0 20 40 X 60 (a) Describe the shape of the unadjusted scores distribution (X). (d) Suppose the Manager adjusts these by adding 5 points to each employee’s score and then also increasing the score by 25 percent. What is the equation that describes the adjustment? Comparing the histogram of these adjusted scores (to the right) with the histogram of the unadjusted scores, how does this adjustment affect the shape of the histogram? The mean? The s.d.? In your opinion, is this fair for all of the employees? Density .1 .05 0 20 40 Y 60 n: 78, mean: 33.5, sd: 7.2 .15 Density (c) Suppose the Manager increases each employee’s score by 50 percent. What is the equation that describes the adjustment? Comparing the histogram of the adjusted scores (to the right) with the histogram of the unadjusted scores, how does this adjustment affect the shape of the histogram? The mean? The s.d.? In your opinion, is this fair for all of the employees? n: 78, mean: 32.4, sd: 4.8 .15 .1 .05 0 20 40 Y 60 n: 78, mean: 34.2, sd: 6.0 .15 Density (b) Suppose the Manager adjusts these by adding 10 points to each employee’s score. What is the equation that describes the adjustment? (Use X and Y notation.) Comparing the histogram of the adjusted scores (to the right) with the histogram of the unadjusted scores, how does this adjustment affect the shape of the distribution? The mean? The s.d.? In your opinion, is this fair for all of the employees? .1 .05 0 20 40 Y 60 (e) A linear transformation is one that takes the form Y = a + bX where a and b are constants. A linear transformation does not change the shape of the histogram: if it is skewed it will remain skewed; if it is symmetric it will remain symmetric; if it is Bell shaped it will remain Bell shaped; if it is bi-modal it will remain bimodal. Are the adjustments in (b) – (d) examples of linear transformations? Would changes in units of measurement (e.g. dollars versus thousands of dollars) also be examples of linear transformations? (2) Consider the multiple-choice question below. Suppose 19 students answered (A), 6 answered (B), 19 answered (C), 45 answered (D), and 56 answered (E). What is the approximate mean and s.d. of the fraction of readings completed? What is the median and interquartile range (IQR)? Multiple-choice question: What fraction of today’s required readings did you complete? (Choose the answer that best applies). (A) none (B) less than half (C) about half (D) more than half (E) all of it Page 1 of 6 (3) Given the tabulation, find the mean, median, mode, 25th percentile, 75th percentile, IQR, and range. x | Freq. Percent Cum. -------+-------------------------------0 | 6 3.77 3.77 .1 | 7 4.40 8.18 .2 | 16 10.06 18.24 .3 | 8 5.03 23.27 .4 | 10 6.29 29.56 .5 | 12 7.55 37.11 .6 | 12 7.55 44.65 .7 | 14 8.81 53.46 .8 | 9 5.66 59.12 .9 | 10 6.29 65.41 1 | 55 34.59 100.00 -------+-------------------------------Total | 159 100.00 (4) This STATA summary is of the variable age from gasoline data posted on Professor Adonis Yatchew’s website. What is unusual about these data? About how many people in the sample are 30 years old or less? How many people are 19 years old? 20 years old? Has this sample been taken from a Normal (bell shaped) population? . summarize age, detail age ------------------------------------------------------------Percentiles Smallest 1% 24 19 5% 29 20 10% 31 20 Obs 6491 25% 37 20 Sum of Wgt. 6491 50% 46 75% 90% 95% 99% 59 66 66 66 Largest 66 66 66 66 Mean Std. Dev. 47.46033 12.4329 Variance Skewness Kurtosis 154.577 .0843854 1.859655 (5) Give a complete qualitative description of the histogram and sketch the box plot. Density .3 n = 102 .2 .1 0 -25 -20 -15 -10 -5 0 (6) Consider the information below from the first “World Happiness Report.” It’s online but you do not need to read this report to complete this question. On page 11 it states “In the Gallup World Poll respondents are asked (using fresh annual samples of 1,000 respondents aged 15 or over in each of more than 150 countries) to evaluate the quality of their lives on an 11-point ladder scale running from 0 to 10, with the bottom rung of the ladder (0) being the worst possible life for them and 10 being the best possible.” This is called the Cantril Ladder. Here is a figure showing Page 2 of 6 the distribution of world results (weighted by each country’s population level). For context, the average happiness in Canada is about 7.5, U.S. 7.3, S. Korea 5.7, India 5, and China 4.7. (a) Using the information in Figure 2.1, compute the mean, s.d., and interquartile range. Show your work and reasoning. [Answer with a quantitative analysis] (b) Does the Empirical Rule hold for this figure? Fully explain. [Answer with a quantitative analysis and 2 – 4 sentences] (c) Considering now the source of these data, identify one likely non-sampling error. Explain why you think it would be worthy of careful consideration in this particular context. [Answer with 2 – 3 sentences]