Exam 12: B: linear Regression and Correlation

Soft Drink Sales Narrative A soft drink vendor, set up near a beach for the summer, was interested in examining the relationship between sales of soft drinks (in litres per day) and maximum temperature of the day. The vendor records the following data for a random sample of eight days in the summer, where y = number of litres of soft drinks sold per day and x = maximum temperature, in degrees Celsius, recorded for the day: The following summary information was computed: In addition, the following partial output was generated using statistical software: The regression equation is sales = -28.460 + 1.7372 temp, MSE = 0.496767 -Refer to Soft Drink Sales Narrative. Find a 95% confidence interval for some value of y (sales) to be observed in the future when the maximum temperature is C.

Circumference and Age Narrative Evidence supports using a simple linear regression model to estimate the circumference of a pine tree based on its age. Let x be the age of the pine tree (measured in years) and y be the circumference (measured in cm). A random sample of 11 mature pine trees 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 = 4.025 R-Sq = 96.4% R-Sq(adj) = 96.0% Analysis of Variance Table Unusual Observations R denotes an observation with a large standardized residual -Refer to Circumference and Age Narrative. Consider the following residual plot of the residuals versus the fitted values. What conclusion can be drawn from the plot?

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. Determine the coefficient of determination and discuss what its value tells you about the two variables.

Willie Nelson Concert Narrative At a recent Willie Nelson concert, a survey was conducted that asked a random sample of 20 people their age and how many concerts they have attended since the first of the year. The following data were collected: An Excel output follows: -Refer to Willie Nelson Concert Narrative. Does it appear that the errors are normally distributed? Explain.

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. Find and interpret the correlation coefficient.

Sleep Deprivation Narrative A study was conducted to determine the effects of sleep deprivation on people's ability to solve s. The amount of sleep deprivation varied with 8, 12, 16, 20, and 24 hours without sleep. A total of ten subjects participated in the study, two at each sleep deprivation level. After his or her specified sleep deprivation period, each subject was administered a set of simple addition s, and the number of errors was recorded. These results were obtained: -Refer to Sleep Deprivation Narrative. Use the prediction equation to predict the number of errors for a person who has not slept for ten hours.

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. Estimate the monthly sales for a salesperson with 16 years of experience.

Circumference and Age Narrative Evidence supports using a simple linear regression model to estimate the circumference of a pine tree based on its age. Let x be the age of the pine tree (measured in years) and y be the circumference (measured in cm). A random sample of 11 mature pine trees 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 = 4.025 R-Sq = 96.4% R-Sq(adj) = 96.0% Analysis of Variance Table Unusual Observations R denotes an observation with a large standardized residual -Refer to Circumference and Age Narrative. Consider the following normal probability plot of the residuals. What conclusion can be drawn from the plot? Justify your answer.

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. Is the simple linear regression model useful for predicting the diameter of the trees from a given age of the forest? Use the t-test at = 0.05.

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. Determine the least-squares estimates of the regression parameters, and the least-squares regression line.

Wind Velocity and Windmills Narrative A scientist is studying the relationship between wind velocity (x in km/h) and DC output of a windmill (y). The following MINITAB output is from a regression analysis for predicting y from x. s = 0.2435 R-sq = 88.3% R-sq(adj) = 87.3% Analysis of Variance -Refer to Wind Velocity and Windmills Narrative. What is 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. Calculate the Pearson correlation coefficient. What sign does it have? Why?

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 is the least-squares regression equation?

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. Use the predicted values and the actual values of y to calculate the residuals.

Antibiotic Potency Narrative An experiment was conducted to observe the effect of an increase in temperature on the potency of an antibiotic. Three 25 gram portions of the antibiotic were stored for equal lengths of time at each of these temperatures: C, C, C, and C. The potency readings observed at each temperature of the experimental period are listed here: -Refer to Antibiotic Potency Narrative. Plot the points and graph the line as a check on your calculations for the previous question.

Microwave Sales Narrative A microwave oven manufacturer has collected the data shown below on number of units sold (y) in the thousands of dollars and the number of ads (x) placed during the month. -Refer to Microwave Sales Narrative. Compute a 95% confidence interval for the average number of units sold in all months with 140 ads.

Antibiotic Potency Narrative An experiment was conducted to observe the effect of an increase in temperature on the potency of an antibiotic. Three 25 gram portions of the antibiotic were stored for equal lengths of time at each of these temperatures: C, C, C, and C. The potency readings observed at each temperature of the experimental period are listed here: -Refer to Antibiotic Potency Narrative. Estimate the change in potency for a one-unit change in temperature. Use a 95% confidence interval.

Microwave Sales Narrative A microwave oven manufacturer has collected the data shown below on number of units sold (y) in the thousands of dollars and the number of ads (x) placed during the month. -Refer to Microwave Sales Narrative. Compute a point estimate of number of units sold if there are 140 ads.

Microwave Sales Narrative A microwave oven manufacturer has collected the data shown below on number of units sold (y) in the thousands of dollars and the number of ads (x) placed during the month. -Refer to Microwave Sales Narrative. Calculate the preliminary sums of squares and cross products,

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. Determine the least-squares regression line.