Stat 11 Spring 2011 - Homework 1

(1) Developing countries often have a relatively low average age. For instance, Belize and Chad each have a mean age of about 24, while the US has a mean age of about 34. Would you expect a histogram of the age of the population of a developing country to look like (a), (b), or (c)? Explain briefly.

(2) It has been said that, "The average stockholder is a 53-year old white woman." In what sense(s) is the word "average" being used here? Explain briefly.

(3) An old joke is that a certain math professor left Swarthmore to go to Haverford, thereby improving the average quality of both departments. If this statement is true, which college must have a better math department? Why?

(4) A team of seven lightweight male rowers has an average weight of 158.2 pounds. The team plans to race in an eight-man boat, so another rower is needed. If the team needs to maintain an average (mean) weight below 160 pounds, what is the maximum that this additional rower can weigh?

(5) Suppose the Philadelphia 76ers basketball team trades for star player LeBron James and signs him to a contract paying an annual salary of $40,000,000. To make room for this contract under the salary cap (the maximum allowable total salary for a team in the league), the 76ers release all their other players and replace them with 11 players from the Swarthmore freshman basketball team, paying them each the league minimum salary of $19.99.

(5a) What is the mean salary of the team? How many players earn less than the mean? What is the median salary of the team?

(5b) The next season, James gets a raise increasing his salary to $50,000,000, while the other players stay at the same salary. How does this affect the mean salary of the team? How does it affect the median?

(6) A prof gives a quiz to her section. There are 10 questions on the quiz and no partial credit is given. After grading the papers, the prof writes down for each student the number of questions the student got right and the number wrong. The average number of right answers is 6.3 with an SD of 2.2. What are the average and SD of the number of wrong answers? Or can you not determine these from the information given?

(7) You roll a standard die three times and then calculate the SD for these rolls. What is the largest possible SD? What is the smallest possible SD? Explain briefly.

(8) Among entering students at a certain college, women averaged 680 on the verbal SAT with an SD of 50. Men averaged 600, but had the same SD of 50. Considering all the students together, would the SD of verbal SAT scores likely be less than 50, exactly 50, or larger than 50? (Hint: Don't try to do this on a calculator or by plugging in numbers into a formula. Instead, sketch a picture of the two distributions.)

(9a) Professor Scully was alarmed to see that her class averaged only 60 points (out of 100) on the midterm, with a SD of 12 points. To make the results look better, she decides to add 12 points to each student's score. What is the new mean and SD after adding 12 points?

(9b) In a different course, Professor Mulder's class also averaged only 60 points on the midterm, with a SD of 12 points. To make the results look better, he decides to multiply each student's score by 1.2. What is the new mean and SD after multiplying by 1.2?

(9c) Why do you think most teachers curve by adding a fixed number to each student's score, rather than multiplying everyone's score by a fixed number?

(10) Below are five of the scores from an exam in a large class. The first row gives the original scores (out of 100 points), and the second row shows the standardized scores. Fill in the three blanks. Show your work. (Hint: think about the difference between the first two scores.)

(11) Data Desk. First, copy Data Desk from the data-software server and install it on your computer. For details on how to do this, consult with Restech (x6222). You should try this early, as difficulties inevitably arise. (Or, use one of the public campus computers, all of which should have Data Desk installed.) Then, download the dataset from http://www.swarthmore.edu/NatSci/swang1/Stat11/survey.dsk.zip. This should download a compressed (.zip) file; when you double-click on it, it should expand into a Data Desk datafile named survey.dsk. If it does not expand, you may need to download Stuffit Expander (use google to find it; it's a free download).

(11a) In the survey on the first day of class, you were asked to predict your score on the first midterm. Make a histogram of these data. To do this, open the datafile and click on the midterm icon and choose Plot > Histograms. Data Desk tries to pick a scale for the histogram, but its guess may not look right.

To change the scale, click on the HyperView menu (the triangle in the top left of the histogram window) and choose Plot Scale.... In the window that appears, try 5 for the bar width and 2 for the number of bars per tick interval. Copy the resulting histogram (click on the title bar of the window and choose Edit > Copy Window), and paste it into a word processing document. Is the histogram roughly right-skewed, symmetric, or left-skewed?

(11b) Calculate the mean, median, and SD of these data. To do this, click on the midterm icon and choose Calc > Summaries > Reports. (If the mean, median, and SD don't appear in the resulting window, click on the HyperView menu and choose Select Summary Statistics... to select the appropriate summary statistics.) Which average is larger, the mean or the median? Is this what you would have expected from your answer in part (a)?

(11c) Who tends to predict higher scores, men or women? Investigate by making boxplots for each gender. To do this, select midterm as the Y variable (option-click on a Mac or alt-click on Windows) and select gender as the X variable (shift-click). Then choose Plot > Boxplot Y by X. Are there substantial differences between how men and women predict their midterm scores, or are both genders about the same?

What to hand in: copy and paste the results for each part (i.e., the histogram, the summary statistics window, and the boxplot) into your word processing document, and write your answers in the same document. Then print out the entire document and hand it in.