APStatistics 3.02 Looking at the Data Directions: Complete the assignment. Clearly label each answer. Your answers for this assignment must include reasons; simply stating the answer without justification will earn partial credit. (32 points) 1. Below are four data sets. For each data set sketch a scatterplot (2 points each), draw a regression line (2 points each), and calculate the correlation coefficient r (2 points each). Provide a statement regarding how well the correlation coefficient, r, measures the relationship of the x and y variables. (2 points each). x1 y1 x2 y2 x3 y3 x4 y4 1 2 2 4 1 1 8 2 3 4 3 9 2 4 7 4 5 6 4 16 3 9 6 20 7 8 5 25 4 16 5 5 9 11 6 36 5 25 8 3 12 15 7 49 6 16 7 5 16 20 8 64 7 9 6 5 17 22 9 81 8 4 5 6 22 28 1 1 9 2 4 7 29 36 0 0 1 0 3 8 36 48 10 100 0 1 2 9 Set 1 (x1, y1) Insert scatterplot and regression line. Correlation coefficient: How well does the correlation coefficient, r, measures the relationship of the x and y variables? Set 2 (x2, y2) Insert scatterplot and regression line. Correlation coefficient: How well does the correlation coefficient, r, measures the relationship of the x and y variables? Set 3 (x3, y3) Insert scatterplot and regression line. Correlation coefficient: How well does the correlation coefficient, r, measures the relationship of the x and y variables? Set 4 (x4, y4) Insert scatterplot and regression line. Correlation coefficient: How well does the correlation coefficient, r, measures the relationship of the x and y variables? APStatistics 3.05 Super Bowl Ticket Prices Directions: This assignment models the steps for performing explanatory data analysis. Complete the assignment. Clearly label each answer. As popularity for the Super Bowl has increased, so have ticket prices. Let's take a look at the prices of Super Bowl tickets from 1985 to 2013 (data courtesy ofhttp://www.krem.com/sports/football/Historical-Super-Bowl-ticket-prices-189247771.html). Year 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 Ticket Price (in dollars) 325 325 400 450 500 550 650 650 800 750 750 900 1000 1000 1. Perform exploratory data analysis using year as the explanatory variable and ticket price as the response variable. Using your graphing calculator, answer the following questions: during this exercise, you will need to graph a normal probability plot, a residual plot, and the least squares regression line equation. a. Before we can consider using linear regression to model a data set, we need to check several conditions. The first: is the data quantitative? (1 point): b. Graph the scatterplot of the data: (1 point) c. The second condition we need to check before we can use linear regression is to see if the data is roughly linear. Based on the scatterplot, is our data roughly linear? (1 point): d. The third condition we need to check before we can use linear regression is to make sure we do not have outliers that would impact our regression line. Are there any outliers that would strongly affect a regression line? (1 point) e. Graph the scatterplot with the with the regression line: (1 point) f. Provide the linear regression information from your calculator (including r and r2 ) (1 point) g. Write a statement regarding the correlation between the variables in our data set in the context of the problem (3 points): h. Write a statement interpreting the coefficient of determination relative to the data set in the context of the problem (3 points): i. Write the equation of the linear regression line. Define any variables used. (2 points): j. Interpret the slope of the regression line in the context of the problem (2 points): k. Draw a graph of the normal probability plot (1 point): Based on the normal probability plot, does the data appear normal? (3 points) l. Draw a graph of the residual plot (1 point): m. Based on the residual plot, do you think a linear regression line is an appropriate model for this data? Why? (3 points) n. What is the formula for calculating a residual? Calculate the residual for the following years: 2003, 2007, and 2010. (4 points) APStatistics 4.01 Transforming Data Directions: Complete the assignment. Clearly label each answer. The last page contains a table of common transformations. (27 points) 1. Consider the following set of observations: Obs. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 input 1 2 3 4 5 6 7 8 9 10 11 12 13 14 result 1 2 3 5 8 13 21 34 55 89 144 233 377 610 a. Enter the data in L1 and L2 in your TI calculator, find the regression line, and construct a scatterplot with the regression line included. Does a line appear to be a good model for these data? Be sure to check your residuals plot. (7 points: 2 points regression line, 2 points scatter plot, 2 points for residual plot; 1 points comment) b. What is r2? (1 point) c. What type of relationship does the data appear to have (linear, logarithmic, exponential, etc.)? (1 point) d. What type of re-expression would work in this case? (1 point) e. Find the natural logarithm of the y-values. (1 point) f. Draw a scatterplot of x vs. ln y. Find the regression equation on ln y on x and include it on the graph. Does it appear to be a better fit than the fit in part (a)? Be sure to check your residuals plot. (7 points: 2 points regression line, 2 points scatter plot, 2 points for residual plot; 1 points comment) g. Write a prediction (regression) equation for your re-expressed data (2 points): h. Use the regression equation you found in part (f) to predict the value of y when x = 10.5. (2 points) i. Does your answer for part (h) seem reasonable? Why or why not? (3 points) j. Explain the importance of checking the residuals plot before re-expressing data and then again after re-expressing data. (2 points) Table of common transformations There are many types of associations that you may encounter. This table lists the most common and summarizes the way each variable may be transformed. The list is not exhaustive, there are other possible transformations. Logarithmic General algebraic equation Take the log (natural or base 10) of the response variable, y = ln x y. Exponential General algebraic equation Take the log of the explanatory variable, x. x y = ae Quadratic General algebraic equation Take the square root of the response variable, y. 2 y = ax + bx + c Power General algebraic equation Take the log of both variables. b y = ax Complex More than one type of equation may be used to describe the association. Break the data into two or more functions and app.lied the appropriate transfromations Notice that the transformations exponential, quadratic, and power look very similar. Check the coefficient of determination after transformation to see which may be the best model. APStatistics 4.04 Two -Way Tables Directions: Complete the assignment. Your answers for this assignment must include reasons; simply stating the answer without justification will earn partial credit. (18 points) 1. A researcher suspected a relationship between people's preferences in music and preference in sports. A random sample of 100 people produced the following two-way table: Favorite Music Favorite Sport Hip Hop Basketball 35 Football 13 Softball 5 Classic Rock 5 24 6 Country 3 2 7 a. Calculate the overall (marginal) distributions for the table. (2 points) b. Compute (in percents) the conditional distribution of favorite music among those who prefer football. Show the distribution in a table. (2 points) c. Briefly describe your finding in words. (2 points) d. Compute (in percents) the conditional distribution of sport among those who chose Hip Hop as their favorite music. Show the distribution in a table. (2 points) e. Briefly describe your finding in words. (2 points) 2. Let's look at the voting record of the Civil Rights Act of 1964. Northern States Voted yes Democrats 145 Republicans 138 Southern States Democrats 7 Republicans 0 Voted no Total 9 154 24 162 87 10 94 10 a. Show the percentages of Democrats and Republicans for each region that voted in favor of the act. (2 points) : b. Show the overall percentage of Republicans that voted in favor of the act and then the overall percentage of the Democrats that voted in favor of the act (2 points) c. What is the name for this apparent contradiction? (2 points) d. Explain the phenomenon. (2 points) APStatistics Unit 4 Exam More on Two Variable Relationships: Free Response Directions: Complete the assignment. Your answers for this assignment must include reasons; simply stating the answer without justification will earn partial credit. 1. A political scientist believes that there is a \"gender gap\" in American voting with women more likely to vote for the Democratic candidate. She therefore interviews a random sample of voters and records the gender of the respondents and the political party of the candidates for whom they voted in the last presidential election. (1 point each) Identify the following variables: a. Quantitative: b. Categorical: c. Explanatory: d: Response: 2. According to data from the U.S. Health Care Financing Administration, the national expenditures for drugs and other medical nondurables (in billions of dollars) for selected years from 1970 to 1997 are as follows: (Note that Year is coded: 1970 is recorded simply as 70.) Year Spent 70 8.8 80 21.6 85 37.1 87 43.2 89 50.6 90 59.9 91 65.6 92 71.2 93 75 94 77.7 95 83.4 97 108.9 a. Apply a test to show that the national expenditures for drugs and other medical nondurables are increasing exponentially. (4 points) b. Calculate the logarithms of the y-values and extend the table above to show the transformed data. (2 points) Year 70 80 85 87 89 90 91 92 93 94 95 Spent 8.8 21.6 37.1 43.2 50.6 59.9 65.6 71.2 75 77.7 83.4 97 108.9 c. Plot the transformed data. Label the axes completely. (4 points) d. You want to construct a model to predict the national drug expenditures in the near future. Perform linear regression on the transformed data and write your least squares equation. (4 points) e. Now transform your linear equation back to obtain a model for the national drug expenditures data. It should be in the form y = (constant)(10bx) Write the equation for this model. (4 points) f. Predict the national drug expenditure for the year 2015. Do you have confidence in this result? Why or why not? (4 points: 2 points for prediction, 2 points for question) 3. Foresters are interested in predicting the amount of usable lumber they can harvest from various tree species. The following data have been collected on the diameter of Ponderosa pine trees, measured at chest height, and the yield in board feet. Note that a board foot is defined as a piece of lumber 12 inches by 12 inches by 1 inch. Construct an appropriate model for these data. Then comment on the quality of your model. (12 points) Diameter 36 28 28 41 19 32 Bd Feet 192 113 88 294 28 123 22 38 25 17 31 20 25 19 39 33 17 37 23 39 51 252 56 16 141 32 86 21 231 187 22 205 57 265 4. In a study of the relationship between the amount of violence a person watches on TV and the viewer's age, 81 regular TV watchers were randomly selected and classified according to their age group and whether the were a \"low-violence\" or \"high violence\" viewer. Here is a two-way table of the results. Age Group 16-34 35-54 55 & over Totals Amount of Low 8 12 21 Violence Watched High 18 15 7 Totals a. Compute (in percents) the marginal distribution of age group for all people surveyed. (2 points) b. Construct a bar chart to show your results visually. (2 points) c. Compute (in percents) the conditional distributions of age group among \"lowviolence\" viewers. Then do the same for \"high-violence\" viewers. (4 points) d. How do these distributions differ from the marginal distribution of age group? (4 points)