Math 207 Problems – One-variable Graphics and Numerical Summaries

1. Included below are the reported ACT scores for students in a previous Math 207 class.

29, 28, 32, 31, 30, 22, 31, 28, 31, 30, 23, 30, 24, 32, 27, 27, 29, 30, 31, 29

1. Create a stem-and-leaf plot of the ACT scores. (Note there are many stem-and-leaf plots that can be made from these data, depending on leaf unit and how the leaves are split. Choose the representation that seems to best show the distribution.)

2. Describe the distribution of the ACT scores (think about overall pattern—shape, center, and spread—and any deviations from the pattern). Would the distribution best be characterized as approximately symmetric, skewed left, or skewed right?

3. Calculate the five-number summary for the ACT scores.

4. According to the rule on page 81 of the textbook, are there any suspected outliers?

5. Suppose you found an outlier, would it be okay to simply throw it out of the data set? (That is, once an outlier is detected, what issues should be considered? This is not something the textbook directly addressed, but brainstorm on ideas within your group.)

6. Create a boxplot of the ACT scores.

2. The 21,892 residents of Retirement City have an average age of 72 years. A sample of 50 residents is polled at a local health club. The average age of people in the sample is 64 years.

1. Is the value “72” a parameter or a statistic? Is the value “64” a parameter or a statistic? (Note that we use different notation for a parameter than we do for a statistic. Typically we want to use the value of a statistic to estimate the value of the corresponding parameter.)

2. In this case, do you think the sample average is a good estimate of the population average? Why or why not (again, simply brainstorm within your group)?

3. In recent years, the distribution of scores on the SAT verbal test has looked mound-shaped with a mean score of 505 and a standard deviation of 110.

1. About what percent of the scores are below 395? (Draw a picture as part of your solution.)

2. About what percent of the scores are between 615 and 725? (Draw a picture as part of your solution.)

3. Bubba scored 405 on the verbal portion of the SAT. Is this an unusually low score? Why or why not? What does a z-value mean in words?

4. Suppose you weren’t told the distribution of scores was mound-shaped. What could you say about the percentage of scores that fell between 285 and 725?

4. Shown below is a clustered bar chart showing peanut butter preferences by sex for students in a Math 117 class. How is this graph potentially misleading (specifically when comparing female and male preferences)? What change could be made to better the graphical representation?

5. Two different sections of a statistics course took the same exam. The distributions of exam scores (separated by section) are shown in the dotplots below. The value labels along the horizontal axis are purposely left off, as you need not do any calculations to answer the following questions.

1. Is the mean score for Section A the same, bigger, or smaller than the mean score for Section B? Explain your answer.

2. Is the standard deviation of scores for Section A the same, bigger, or smaller than the standard deviation for Section B? Explain your answer.

3. In this case, how is the standard deviation a more informative measure of variability than the range is?

6. In the past, Math 207 students completed a survey on the first day of class (we didn’t have time for the survey this term). Shown below is the histogram of responses to one of the questions. Consider the following variables: hours of sleep on a typical weeknight, monetary amount (in dollars) of carried coin money, randomly selected integer from 0 to 9, and height in inches. Which of these variables do you think is depicted in the histogram? Give reasons why your answer is correct, and why the other answers are incorrect.

In problems 7 – 10, a part (or parts) of the given analysis, graph, calculation, or interpretation is incorrect. You need to determine what is incorrect and why it is incorrect.

7. A local business has 11 employees. The incomes of the employees are $19,000, $25,000, $34,000, $45,000, $27,000, $63,000, $23,000, $31,000, $42,000, $61,000, and $31,000. Bubba creates the following stem-and-leaf plot (with multiple errors):

1 | 9

2 | 5 7 3

3 | 4 1 1

4 | 5 2

6 | 3 1

8. A sample of 100 Lawrence students is selected. The GPA, height, and home state are recorded for each of the students. Bubba wants to graphically display the distributions of these variables. He decides to create stem-and-leaf plots of the distributions of GPA and height, and to create a histogram of the home states.


9. The Roller Coaster Database maintains a web site (www.rcdb.com) with data on roller coasters around the world. Some of the recorded data include whether the coaster is made of steel or wood and the maximum speed achieved by the coaster, in miles per hour. The boxplots below display the distributions of speed by type of coaster for 305 active coasters in the United States.

            Bubba makes the following statements (poor Bubba is often confused):

· The average maximum speed for steel roller coasters is 50 mph.

· The distribution of maximum speeds for steel coasters is skewed to the left.

· The median maximum speed for wooden coasters is higher than the median for steel coasters.

· From the boxplots, we can tell there are more steel roller coasters in the sample.

· A higher percentage of steel coasters have maximum speeds above 50 mph (as compared to wooden coasters).

10. Bubba plans to find the yearly income data (in dollars) for a random sample of people in the United States. He speculates that the mean income will be less than the median income. Furthermore, he thinks the best numerical summary of the data will be to present the mean and standard deviation.