Elements of Statistics (Math 106) - Exam 1

Elements of Statistics (Math 106) - Exam 1 Name

Fall 2007 - Brad Hartlaub

Directions: Please answer all of the questions below. The point values for each problem are indicated in parentheses. Partial credit will be awarded if you show your work. Be careful not to spend too much time on any one part of a question.

1. Every 17 years, swarms of cicadas emerge from the ground in the eastern United States, live for about six weeks, then die. (There are several “broods,” so we experience cicada eruptions more often than every 17 years.) There are so many cicadas that their dead bodies can serve as fertilizer and increase plant growth. In an experiment, a researcher added 10 cicadas under some plants in a natural plot of American bellflowers on the forest floor, leaving other plants undisturbed. One of the response variables was the size of seeds produced by the plants. The seed masses in milligrams for 39 cicada plants and 33 undisturbed (control) plants are in the file p:\data\math\stats\cicada.mtw.

a. Find the mean and standard deviation of the seed masses for both groups. (6) b. Provide the five-number summary for the seed masses for plants in the cicada group. (5) c. Determine whether there are any outliers in the cicada group. If there are outliers, identify them. (5) d. Describe the distribution of seed mass for plants in the cicada group. (5) e. Suppose you were asked to remove both the largest and smallest masses for the cicada group. (Don’t waste time removing these observations.) Why does removing these two observations reduce s ? Why does it have little effect on x ? (10) f. Do the data support the idea that dead cicadas can serve as fertilizer? Explain. (10)

2. Individuals with low bone density (osteoporosis) have a high risk of broken bones (fractures). Physicians who are concerned about low bone density in patients can refer them for specialized testing. Currently, the most common method for testing bone density is dual-energy X-ray absorptiometry (DXA). A patient who undergoes a DXA test usually gets bone density results in grams per square centimeter (g/cm2) and in standardized units. Judy, who is 25 years old, has her bone density measured using DXA. Her results indicate a bone density in the hip of 948 g/cm2 and a standardized score of z = -1.45 . In the reference population of 25-year-old women like Judy, the mean bone density in the hip is 956 g/cm2.

a. Judy has not taken a statistics class in a few years. Explain to her in simple language what the standardized score tells her about her bone density. (10) b. Use the information provided to calculate the standard deviation of bone density in the reference population. (5) c. What is Judy’s bone density score (not standardized)? (5) d. What is the chance that a 25-year-old woman will have a standardized score above -1.45 ? (5) This topic was not covered yet. e. What is the chance that a 25-year-old woman will have a standardized score between -1.45 and 1.45? (5) This topic was not covered yet. f. Identify the standardized bone density scores for the 10% of 25-year-old women who are most at risk for broken bones. (5) This topic was not covered yet. g. One of Judy’s friends, Mary, has the bone density in her hip measured using DXA. Mary is 35 years old. Her bone density is also reported as 948 g/cm2, but her standardized score is z = 0.5 . The mean bone density in the hip for the reference population of 35-year-old women is 944 g/cm2. Whose bones are healthier: Judy’s or Mary’s? Justify your answer. (10) This topic was not covered yet.

3. Manatees are large, gentle sea creatures that live along the Florida coast. Many manatees are killed or injured by powerboats. The data on powerboat registrations (in thousands) and the number of manatees killed by boats in Florida from 1977 to 2006 are in the file p:\data\math\stats\manatee.mtw.

a. Is the association between powerboat registrations and year positive or negative? Explain. (5) b. Is a histogram the most appropriate graph for examining the relationship between the number of manatees killed and year? Explain. (5) c. Would you be willing to use the Normal distribution as a model for the number of manatees killed during this time period? Explain. (5) d. Is there any association between the number of power boat registrations and the number of manatees killed? Explain. (5) e. What statistic summarizes the strength of the linear association between the number of powerboat registrations and the number of manatees killed? Calculate this statistic. (5) f. Find the least squares line for predicting the number of manatees killed from the number of powerboat registrations. (5) g. Are you happy with the fit of your linear model in part (f)? Be sure to comment on r 2 and the residuals. (10) h. What is the predicted number of manatees killed if the number of powerboat registrations is 983,000? (3)

4. For years, scientists have suspected that a chemical in many household deodorizing products may cause short term lung problems. A 2006 study by scientists at the National Institutes of Health say they’ve found that people with relatively high blood concentrations of the substance —1,4-dichlorobenzene, an organic chemical—show signs of slightly reduced lung function. The study followed 953 Americans, average age 37, for six years.

a. From the above description of this study, would you say it was an experiment or an observational study? Justify your answer. (6) b. The number of subjects in this study is relatively large—almost 1,000 people. Does the large sample size lend credence to the notion that having household deodorizing products around is dangerous to your lungs? Justify your answer. (6) c. A newspaper article announcing these results stated, “Other studies have had similar findings, which suggests that chronic exposure to a chemical in air fresheners can cause lung problems.” Based on this NIH study, can we conclude that exposure to products that contain 1,4-dichlorobenzene causes lung problems? Why or why not? (6)