Exam 12: B: linear Regression and Correlation

Oil Quality and Price Narrative Quality of oil is measured in API gravity degrees; the higher the degrees API, the higher the quality. The table shown below was produced by an expert in the field who believes that there is a relationship between quality and price per barrel. A partial MINITAB output follows: Descriptive Statistics Covariances Degrees Price Degrees 21.281667 Price 2.026750 0.208833 Regression Analysis S = 0.1314 R-Sq = 92.46% R-Sq(adj) = 91.7% Analysis of Variance -Refer to Oil Quality and Price Narrative. Plot the residuals against the predicted values.

SAT Scores and GPA Narrative A university admissions committee was interested in examining the relationship between a student's score on the Scholastic Aptitude Test, x, and the student's grade point average, y, at the end of the student's first year of university. The committee selected a random sample of 25 students and recorded the SAT score and GPA at the end of the first year of university for each student. Use the following output that was generated using statistical software to answer the questions below: Regression Analysis The regression equation is GPA = -1.09 + 0.00349 SAT S = 0.1463 R-Sq = 91.8% R-Sq(adj) = 91.5% Analysis of Variance Correlations (Pearson) Correlation of SAT and GPA = 0.958 -Refer to SAT Scores and GPA Narrative. Find the estimated GPA at the end of the freshman year for a student who scored 1175 on the SAT exam.

Ice Cream Sales Narrative The manager of an ice cream store is interested in examining the relationship between sales of ice cream (in litres per day) and maximum temperature of the day. The vendor records the following data for a random sample of five days in the summer, where y is number of litres of ice cream sold per day and x is maximum temperature, in degrees Celsius, recorded for the day: The following summary information was computed: -Refer to Ice Cream Sales Narrative. Construct a scatterplot for the data. Do the data appear to be reasonably linear?

Delivery Time Narrative Let x be the number of pieces of furniture in a delivery truck and y be the time (in hours) it takes the delivery person to deliver all the pieces of furniture. A simple linear regression analysis related x and y where the least-squares estimates of the regression parameters are a = 1.85 and b = 0.55. -Refer to Delivery Time Narrative. Identify and interpret the y-intercept. Does this make sense?

Advertising and Money Spent Narrative A marketing analyst is studying the relationship between x = money spent on television advertising and y = increase in sales. One study reported the following data (in dollars) for a particular company. -Refer to Advertising and Money Spent Narrative. Use a statistical software package of your choice and report the regression analysis results.

Advertising and Money Spent Narrative A marketing analyst is studying the relationship between x = money spent on television advertising and y = increase in sales. One study reported the following data (in dollars) for a particular company. -Refer to Advertising and Money Spent Narrative. What is the least-squares regression line?

Weight and Height Narrative Evidence supports using a simple linear regression model to estimate a person's weight based on a person's height. Let x be a person's height (measured in cm) and y be the person's weight (measured in kg). A random sample of 11 people was selected and the following data recorded: The following output was generated using statistical software: Regression Analysis The regression equation is y = -148 + 4.18x S = 1.7698; R-Sq = 96.7%; R-Sq(adj) = 96.3% Analysis of Variance Table Unusual Observations denotes an observation with a large standardized residual. -Refer to Weight and Height Narrative. Based on the scatterplot above, does a simple linear regression model seem appropriate? Justify your answer.

Vending Machines Narrative Let x be the number of vending machines and let y be the time (in hours) it takes to stock them. The data are as follows. -Refer to Vending Machines Narrative. What is the predicted time required to stock seven vending machines?

Age of Forest and Diameter of Trees A scientist is studying the relationship between age of a forest, x, in years and the average diameter of the trees, y, in cm. One study reported the following data. -Refer to Age of Forest and Diameter of Trees. What is the equation of the least-squares regression line?

Age of Forest and Diameter of Trees A scientist is studying the relationship between age of a forest, x, in years and the average diameter of the trees, y, in cm. One study reported the following data. -Refer to Age of Forest and Diameter of Trees. Construct a scatterplot for this data, including the least-squares regression line.

Income and Education Narrative A professor of economics wants to study the relationship between income (y in $1,000s) and education (x in years). A random sample eight individuals is taken and the results are shown below. -Refer to Income and Education Narrative. Determine the standard error of estimate and describe what this statistic tells you about the regression line.

Blacktop Let x be the area (in square metres) to be covered with blacktop, and let y be the time (in minutes) it takes a construction crew to completely cover the area. The simple linear regression model relates x and y where the least-squares estimates of the regression parameters are b = 0.207 and a = 81.6. -Refer to Blacktop statement. What is the average change in time per one square metre increase in area?

Sales and Experience Narrative The general manager of a chain of furniture stores believes that experience is the most important factor in determining the level of success of a salesperson. To examine this belief, she records last month's sales (in $1000s) and the years of experience of ten randomly selected salespeople. These data are listed below. -Refer to Sales and Experience Narrative. Interpret the value of the slope of the regression line.

Age of Forest and Diameter of Trees A scientist is studying the relationship between age of a forest, x, in years and the average diameter of the trees, y, in cm. One study reported the following data. -Refer to Age of Forest and Diameter of Trees. Use a statistical software package of your choice and report the regression analysis results.

TV Game Show Revenues Narrative An ardent fan of television game shows has observed that, in general, the more educated the contestant, the less money he or she wins. To test her belief, she gathers data about the last eight winners of her favourite game show. She records their winnings in dollars and the number of years of education. The results are as follows. -Refer to TV Game Show Revenues Narrative. Estimate with 95% confidence the average winnings of all contestants who have 15 years of education.

Extra Help Sessions Narrative A study was conducted to determine the effect of extra help sessions attended on students' ability to avoid mistakes on a 20-question test. The data shown below represent the number of extra help sessions attended (x) and the average number of mistakes (y) recorded. -Refer to Extra Help Sessions Narrative. Do the data provide sufficient evidence to indicate that y and x are linearly related at the 1% level of significance?

Income and Height Narrative Do tall men earn more than short ones? An economist collected the data shown below for 25 men, where the annual income (y) in thousands of dollars and the height of the income earner (x) in cm. -Refer to Income and Height Narrative. Do the data present sufficient evidence to indicate that annual income and height of income earner are linearly related? Use the t test at the 5% level of significance.

Young Aspen Trees and Growth Narrative Let x be the number of leaves on a young aspen tree and let y be the growth of the tree (in mm). The data are as follows. -Refer to Young Aspen Trees and Growth Narrative. Use the F-test to test the hypotheses at = 0.05.

Salary and Years Narrative A company manager is interested in the relationship between x = number of years that an employee has been with the company and y = the employee's annual salary (in thousands of dollars). The following statistical software output is from a regression analysis for predicting y from x for n = 15 data points. s = 0.8081 R-sq = 97.9% R-sq(adj) = 97.8% -Refer to Salary and Years Narrative. What are the values of and the sum of squares for error?

Advertising and Money Spent Narrative A marketing analyst is studying the relationship between x = money spent on television advertising and y = increase in sales. One study reported the following data (in dollars) for a particular company. -Refer to Advertising and Money Spent Narrative. Develop a scatterplot and determine whether a linear relationship appears to provide a good fit to this data set.