Math 121
Exam 1
February 26, 1998

Please write in the blue books provided. When calculations are asked for, show the details of your work. When interpretations or explanations are called for, be clear and concise. You may use a calculator but may not use Minitab on any part of the exam. Please note the point value on each problem and budget your time accordingly; there are a total of ten problems.

1. (5 pts.) Identify the cases (observational units) in:
a) the "televisions and life expectancy" data that you analyzed in class.
b) the "toy advertising" data that you analyzed for homework.

2. (15 pts.) The following dotplot displays the distribution of weights of the members of the 1996 U.S. Men's Olympic Rowing Team:

a) Estimate the value of the median of the distribution as accurately as you can from this plot.
b) Would the mean would be greater than or less than the median for these data? Explain briefly.
c) Write a paragraph describing key features of the distribution.

3. (5 pts.) Construct a hypothetical example of ten exam scores so that the inter-quartile range equals zero and the mean is less than the median.

4. (10 pts.) In addition to the two scoring methods for tennis that you analyzed in a homework problem, a third scoring method called "handicap" scoring was analyzed by a Dickinson College student researcher. He simulated 100 games played with this scoring method and recorded the length (measured by number of points played) of each game. The results for these 100 games are tallied in the table:

 points in game 1 2 3 4 5 6 7 tally (count) 3 4 12 18 28 25 10

a) Determine the median of these game lengths.
b) Determine the mode of these game lengths.
c) Determine the range of these game lengths.

5. (15 pts.) The following data are the weights in grams of 35 male house sparrows that survived a severe winter storm and of the 24 male sparrows that perished in the same storm:

survived:

23.2  23.6  23.7  23.8  23.9  24.1  24.2  24.3  24.3  24.5  24.6  24.7

24.7  24.8  24.9  25.4  25.6  25.7  25.7  25.7  25.7  25.9  26.2  26.2

26.3  26.3  26.5  26.6  26.7  26.7  26.9  26.9  27.0  27.9  28.0

perished:

24.6  24.6  24.9  25.0  25.0  25.1  25.5  25.6  25.6  25.8  25.9  26.0

26.0  26.0  26.0  26.1  26.5  26.5  27.1  27.5  27.6  28.3  28.3  31.1

a) Determine the five-number summary of the weights of the 35 sparrows that survived. (Note that these are already in order.)

The five-number summary of the weights of sparrows that perished is:

 Minimum Lower quartile Median Upper quartile Maximum 24.6 25.3 26.0 26.8 31.1

b) Use this information to conduct the outlier test for the weights of the sparrows that perished.
c) Comment briefly on whether the data provide evidence that sparrows which survived the storm tended to weigh more than those that perished.

6. (5 pts.) Suppose that scores on this exam follow a symmetric, mound-shaped distribution with mean 75 and standard deviation 8.
a) What can you say about the proportion of students who score between 67 and 83 on the exam?
b) What can you say about the proportion of students who score above 91 on the exam? Explain briefly.

7. (5 pts.) The midrange of a distribution of data is defined to be (minimum + maximum) / 2. The midhinge of a distribution of data is defined to be (lower quartile + upper quartile) / 2. Which of these is resistant to outliers? Explain briefly.

8. (20 pts.) The following table lists the average temperature of a month and the amount of the electricity bill for that month:

 month temp bill month temp bill Apr-91 51 \$41.69 Jun-92 66 \$40.89 May-91 61 \$42.64 Jul-92 72 \$40.89 Jun-91 74 \$36.62 Aug-92 72 \$41.39 Jul-91 77 \$40.70 Sep-92 70 \$38.31 Aug-91 78 \$38.49 Oct-92 * * Sep-91 74 \$37.88 Nov-92 45 \$43.82 Oct-91 59 \$35.94 Dec-92 39 \$44.41 Nov-91 48 \$39.34 Jan-93 35 \$46.24 Dec-91 44 \$49.66 Feb-93 * * Jan-92 34 \$55.49 Mar-93 30 \$50.80 Feb-92 32 \$47.81 Apr-93 49 \$47.64 Mar-92 41 \$44.43 May-93 * * Apr-92 43 \$48.87 Jun-93 68 \$38.70 May-92 57 \$39.48 Jul-93 78 \$47.47

The regression line for predicting the bill from the temperature is: bill = 55.1 - 0.214 temp. A scatterplot of the data with the regression line drawn in follows:

a) Use the regression line to predict the electric bill for a month with an average temperature of 50 degrees.
b) Estimate as accurately as you can form the scatterplot the proportion of variability in electric bills that is explained by the regression line with average temperature.
d) Without doing any calculations, identify the month with the largest positive residual. Explain your answer.

9. (10 pts.) It is a demonstrable fact that the sum of the residuals from a regression line must equal zero.
a) Does it follow from this fact that the mean of the residuals must equal zero? Explain briefly.
b) Does it follow from this fact that the median of the residuals must equal zero? Explain briefly.

10. (10 pts.) Supose that a company has just fired a total of 300 employees and that the gender breakdown is as represented in the following table:

 overall retained fired total men 300 200 500 women 400 100 500 total 700 300 500

When the employees are further classified according to whether their position is professional or clerical, the breakdowns are as represented in the following tables:

 professional retained fired total men 255 195 450 women 25 25 50 total 280 220 500 clerical retained fired total men 45 5 50 women 375 75 420 total 420 80 500

a) Consider just the clerical employees for the moment. Calculate the proportion of clerical men who were fired and the proportion of clerical women who were fired. Which is higher?
b) It turns out in this case that men have a higher proportion of being fired overall, but men have a lower proportion of being fired among both professional and clerical employees. Write a few sentences explaining why this reversal occurs, basing your explanation on the data provided in the tables.