Exam 7: Relationships in Data

The following scatterplot displays the number of points scored by Kobe Bryant in each game of the 20062007 National Basketball Association season, as well as the number of shots that he attempted in each game: a. What are the observational units in this graph? b. Use the following summary statistics to determine the equation of the least squares line for predicting points from shots. Mean points: 31.56 SD points: 11.79 Mean shots: 22.82 SD shots: 7.41 Correlation between points and shots: .770 c. Write a sentence interpreting the slope coefficient. d. What proportion of the variability in Bryant's points is explained by the least squares line with number of shots? e. Suppose Bryant was to play one more game and take 50 shots. Which would have a greater impact on the least squares line that you calculated in part b: if he were to score 60 points in that game or if he were to score 2 points? Explain your answer.

The following scatterplot displays lung capacity (forced expiratory volume, measured in liters) vs. age (in years) for a sample of children: 1-3. Comment on the direction, strength, and form of the relationship between lung capacity and age. (Be sure to relate your comments to the context.) The following scatterplot also incorporates information about whether the child is a smoker: 4. Do smokers tend to be older or younger than nonsmokers? Explain how you can tell from the graph. 5. For children of similar age, do smokers tend to have more or less lung capacity than nonsmokers? Explain how you can tell from the graph.

Suppose for each student in a statistics class, you record how far he/she sits from the teacher during a particular class session and his/her score on that day's quiz. (No explanations are necessary for the following questions.) a. Students who sit closer to the teacher tend to score higher on the quiz. Would you expect the correlation coefficient between distance and quiz score to be positive, negative, or close to zero? b. The correlation coefficient between distance and quiz score turns out to be close to -1. Would that prove that sitting close to the teacher causes higher quiz scores? c. There is really no association between distance and quiz score. Would you expect the correlation coefficient to be positive, negative, or close to zero? d. There is really no association between distance and quiz score. Is it still possible to calculate a correlation coefficient?

Suppose you record data on these four variables for every student in your class: height, weight, gender, and handedness (whether the student is right- or left-handed). a. Give an example of one pair of these variables for which it makes sense to calculate a correlation coefficient. b. Give an example of one pair of these variables for which it does not make sense to calculate a correlation coefficient.

The following scatterplot displays the rushing yardage per game vs. number of rushes for LaDainian Tomlinson in the 2006 National Football League season. Also displayed is the least squares line. The equation of the least squares line is predicted rushing yards = -3.69 + 5.37 × number of rushes. The value of r² is .332. -Interpret the value of r² in this context.

Every month for a period of many years, a homeowner kept track of several variables related to natural gas and electricity usage in his house, before and after he added a new room to the house. The following scatterplots show the average daily natural gas usage vs. mean temperature for the month and average daily electricity usage vs. mean temperature for the month, respectively. a. Which utility's usage (natural gas or electricity) appears to be more temperature dependent for this homeowner? Explain how you can tell from the scatterplots. b. Do the scatterplots provide any reason to believe that natural gas or electricity usage (or both) are higher after the extra room is added? Explain how you can tell.

The following scatterplot displays the rushing yardage per game vs. number of rushes for LaDainian Tomlinson in the 2006 National Football League season. Also displayed is the least squares line. The equation of the least squares line is predicted rushing yards = -3.69 + 5.37 × number of rushes. The value of r² is .332. -Count how many of these 16 games have a negative residual.

A statistician found data in an advertisement that listed the price of a diamond and the number of carats in the diamond. He entered the data (for 48 diamonds) into a computer package and fit a regression model for predicting price based on carats. He obtained the following computer output: -Identify and interpret the slope coefficient.

Naturalists recorded the following data on size of islands in the West Indies and number of species of reptiles and amphibians living on the island: The goal of this study was to develop a model for predicting the number of species from the island size. a. Which is the response variable in this study? The following scatterplots show least squares lines for predicting number of species from island size, first without a transformation and then taking log (base 10) of both variables: b. Which model would you use? Explain your choice. c. Use the model that you chose to predict the number of species on an island with a size of 2000 square miles.

In a recent study of the effectiveness of popular diet plans, subjects were given a numerical score of how well they adhered to the diet, as well as having their weight loss measured. For the 93 subjects who completed the study, the correlation coefficient between adherence score and weight loss turned out to equal .533. Conduct a significance test of whether this sample correlation coefficient is significantly greater than 0. Report the null and alternative hypotheses, test statistic, and p-value. Also write a sentence summarizing your conclusion.

The following four scatterplots pertain to a study that investigated predicting the price of a wine based on its age and three variables associated with the weather during the year in which it was produced: summer temperature, winter rain, and harvest rain: A: B: C: D: Consider the following five correlation values. Four of them are calculated from these scatterplots, and the fifth does not apply to any of these graphs. For each correlation value, you are either to identify the graph that it is calculated from or say that it does not apply to any of these graphs. 1. -.750 2. -.449 3. .234 4. .452 5. .589

A sample of students at a university took a test that diagnosed their learning styles as active or reflective and also as visual or verbal. Each student received a numerical score on the active/reflective style and also a numerical score on the visual/verbal style. The sample size was 39, and the sample correlation coefficient turned out to equal .273. -State the test decision for the $\alpha$ = .10 and $\alpha$ = .05 significance levels, and summarize your conclusion in context.

A statistician found data in an advertisement that listed the price of a diamond and the number of carats in the diamond. He entered the data (for 48 diamonds) into a computer package and fit a regression model for predicting price based on carats. He obtained the following computer output: -Report the value of the standard error of the sample slope coefficient.

A sample of students at a university took a test that diagnosed their learning styles as active or reflective and also as visual or verbal. Each student received a numerical score on the active/reflective style and also a numerical score on the visual/verbal style. The sample size was 39, and the sample correlation coefficient turned out to equal .273. -Calculate the value of the test statistic.

The following data are the age (in years) and baby's birth weight (in grams) for a sample of 10 young women who gave birth as teenagers: Consider the following summary statistics: Mean age: 17.0 years SD ages: 1.49 years Mean birth weight: 3004 grams SD birth weights: 414 grams Correlation coefficient between age and birth weight: .884 -What percentage of variability in baby weights is explained by the linear relationship with mother's age?