Exam 15: Multiple Regression

SCENARIO 15-7-B You are the CEO of a dairy company. You are planning to expand milk production by purchasing additional cows, lands and hiring more workers. From the existing 50 farms owned by the company, you have collected data on total milk production (in liters), the number of milking cows, land size (in acres) and the number of laborers. The data are shown below and also available in the Excel file Scenario15-7-DataB.XLSX. You believe that the number of milking cows $\left( X _ { 1 } \right)$ , land size $\left( X _ { 2 } \right)$ and the number of laborers $\left( X _ { 3 } \right)$ are the best predictors for total milk production on any given farm. -Referring to Scenario 15-7-B, the model that includes only X1 and X3 should be selected using the adjusted $r ^ { 2 }$ statistic.

Which of the following is used to find a "best" model?

In stepwise regression, an independent variable is not allowed to be removed from the model once it has entered into the model.

SCENARIO 15-4 The superintendent of a school district wanted to predict the percentage of students passing a sixth- grade proficiency test. She obtained the data on percentage of students passing the proficiency test (% Passing), daily mean of the percentage of students attending class (% Attendance), mean teacher salary in dollars (Salaries), and instructional spending per pupil in dollars (Spending) of 47 schools in the state. -Referring to Scenario 15-4, which of the following predictors should first be dropped to remove collinearity? a) $X _ { 1 }$ b) $X _ { 2 }$ c) $X _ { 3 }$ d) None of the above

SCENARIO 15-4 The superintendent of a school district wanted to predict the percentage of students passing a sixth- grade proficiency test. She obtained the data on percentage of students passing the proficiency test (% Passing), daily mean of the percentage of students attending class (% Attendance), mean teacher salary in dollars (Salaries), and instructional spending per pupil in dollars (Spending) of 47 schools in the state. -Referring to Scenario 15-4, what are, respectively, the values of the variance inflationary factor of the 3 predictors?

SCENARIO 15-7-B You are the CEO of a dairy company. You are planning to expand milk production by purchasing additional cows, lands and hiring more workers. From the existing 50 farms owned by the company, you have collected data on total milk production (in liters), the number of milking cows, land size (in acres) and the number of laborers. The data are shown below and also available in the Excel file Scenario15-7-DataB.XLSX. You believe that the number of milking cows $\left( X _ { 1 } \right)$ , land size $\left( X _ { 2 } \right)$ and the number of laborers $\left( X _ { 3 } \right)$ are the best predictors for total milk production on any given farm. -Referring to Scenario 15-7-B, the value of the t test statistic for testing whether the quadratic term for land size is statistically significant after you have performed a multiple regression that includes the quadratic terms for the number of milking cows, land size and the number of laborers is _____.

SCENARIO 15-3 A chemist employed by a pharmaceutical firm has developed a muscle relaxant. She took a sample of 14 people suffering from extreme muscle constriction. She gave each a vial containing a dose (X) of the drug and recorded the time to relief (Y) measured in seconds for each. She fit a curvilinear model to this data. The results obtained by Microsoft Excel follow -Referring to Scenario 15-3, suppose the chemist decides to use a t test to determine if there is a significant difference between a linear model and a curvilinear model that includes a linear term. The p-value of the test statistic for the contribution of the curvilinear term is ________.

SCENARIO 15-7-A You are the CEO of a dairy company. You are planning to expand milk production by purchasing additional cows, lands and hiring more workers. From the existing 50 farms owned by the company, you have collected data on total milk production (in liters), the number of milking cows, land size (in acres) and the number of laborers. The data are shown below and also available in the Excel file Scenario15-7-DataA.XLSX. You believe that the number of milking cows $\left( X _ { 1 } \right)$ , land size $\left( X _ { 2 } \right)$ and the number of laborers $\left( X _ { 3 } \right)$ are the best predictors for total milk production on any given farm. -Referring to Scenario 15-7-A, the model that includes only $X _ { 2 } \text { and } X _ { 3 }$ should be among the appropriate models using the Mallow's Cp statistic.

A high value of R2 significantly above 0 in multiple regression accompanied by insignificant t-values on all parameter estimates very often indicates a high correlation between independent variables in the model.

SCENARIO 15-7-B You are the CEO of a dairy company. You are planning to expand milk production by purchasing additional cows, lands and hiring more workers. From the existing 50 farms owned by the company, you have collected data on total milk production (in liters), the number of milking cows, land size (in acres) and the number of laborers. The data are shown below and also available in the Excel file Scenario15-7-DataB.XLSX. You believe that the number of milking cows $\left( X _ { 1 } \right)$ , land size $\left( X _ { 2 } \right)$ and the number of laborers $\left( X _ { 3 } \right)$ are the best predictors for total milk production on any given farm. -Referring to Scenario 15-7-B, the value of the t test statistic for testing whether the quadratic term for the number of laborers is statistically significant after you have performed a multiple regression that includes the quadratic terms for the number of milking cows, land size and the number of laborers is ______.

The goals of model building are to find a good model with the fewest independent variables that is easier to interpret and has lower probability of collinearity.

SCENARIO 15-3 A chemist employed by a pharmaceutical firm has developed a muscle relaxant. She took a sample of 14 people suffering from extreme muscle constriction. She gave each a vial containing a dose (X) of the drug and recorded the time to relief (Y) measured in seconds for each. She fit a curvilinear model to this data. The results obtained by Microsoft Excel follow -Referring to Scenario 15-3, suppose the chemist decides to use a t test to determine if there is a significant difference between a linear model and a curvilinear model that includes a linear term. If she used a level of significance of 0.05, she would decide that the linear model is sufficient.

SCENARIO 15-7-B You are the CEO of a dairy company. You are planning to expand milk production by purchasing additional cows, lands and hiring more workers. From the existing 50 farms owned by the company, you have collected data on total milk production (in liters), the number of milking cows, land size (in acres) and the number of laborers. The data are shown below and also available in the Excel file Scenario15-7-DataB.XLSX. You believe that the number of milking cows $\left( X _ { 1 } \right)$ , land size $\left( X _ { 2 } \right)$ and the number of laborers $\left( X _ { 3 } \right)$ are the best predictors for total milk production on any given farm. -Referring to Scenario 15-7-B, the model that includes only $X _ { 1 } \text { and } X _ { 2 }$ should be among the appropriate models using the Mallow's Cp statistic.

SCENARIO 15-7-A You are the CEO of a dairy company. You are planning to expand milk production by purchasing additional cows, lands and hiring more workers. From the existing 50 farms owned by the company, you have collected data on total milk production (in liters), the number of milking cows, land size (in acres) and the number of laborers. The data are shown below and also available in the Excel file Scenario15-7-DataA.XLSX. You believe that the number of milking cows $\left( X _ { 1 } \right)$ , land size $\left( X _ { 2 } \right)$ and the number of laborers $\left( X _ { 3 } \right)$ are the best predictors for total milk production on any given farm. -Referring to Scenario 15-7-A, the model that includes $X _ { 1 } , X _ { 2 } \text { and } X _ { 3 }$ should be among the appropriate models using the Mallow's Cp statistic.

One of the consequences of collinearity in multiple regression is inflated standard errors in some or all of the estimated slope coefficients.

SCENARIO 15-7-A You are the CEO of a dairy company. You are planning to expand milk production by purchasing additional cows, lands and hiring more workers. From the existing 50 farms owned by the company, you have collected data on total milk production (in liters), the number of milking cows, land size (in acres) and the number of laborers. The data are shown below and also available in the Excel file Scenario15-7-DataA.XLSX. You believe that the number of milking cows $\left( X _ { 1 } \right)$ , land size $\left( X _ { 2 } \right)$ and the number of laborers $\left( X _ { 3 } \right)$ are the best predictors for total milk production on any given farm. -Referring to Scenario 15-7-A, there is sufficient evidence to conclude that the quadratic term for land size is statistically significant at the 10% level of significance after you have performed a multiple regression that includes the quadratic terms for the number of milking cows, land size and the number of laborers.

SCENARIO 15-1 A certain type of rare gem serves as a status symbol for many of its owners. In theory, for low prices, the demand increases and it decreases as the price of the gem increases. However, experts hypothesize that when the gem is valued at very high prices, the demand increases with price due to the status owners believe they gain in obtaining the gem. Thus, the model proposed to best explain the demand for the gem by its price is the quadratic model: $Y = \beta _ { 0 } + \beta _ { 1 } X + \beta _ { 2 } X ^ { 2 } + \varepsilon$ where Y = demand (in thousands) and X = retail price per carat. This model was fit to data collected for a sample of 12 rare gems of this type. A portion of the computer analysis obtained from Microsoft Excel is shown below: -Referring to Scenario 15-1, what is the p-value associated with the test statistic for testing whether there is an upward curvature in the response curve relating the demand (Y) and the price (X)?

Collinearity is present when there is a high degree of correlation between the dependent variable and any of the independent variables.

SCENARIO 15-7-B You are the CEO of a dairy company. You are planning to expand milk production by purchasing additional cows, lands and hiring more workers. From the existing 50 farms owned by the company, you have collected data on total milk production (in liters), the number of milking cows, land size (in acres) and the number of laborers. The data are shown below and also available in the Excel file Scenario15-7-DataB.XLSX. You believe that the number of milking cows $\left( X _ { 1 } \right)$ , land size $\left( X _ { 2 } \right)$ and the number of laborers $\left( X _ { 3 } \right)$ are the best predictors for total milk production on any given farm. -Referring to Scenario 15-7-B, the p value of the t test statistic for testing whether the quadratic term for the number of milking cows is statistically significant after you have performed a multiple regression that includes the quadratic terms for the number of milking cows, land size and the number of laborers is _____.

SCENARIO 15-3 A chemist employed by a pharmaceutical firm has developed a muscle relaxant. She took a sample of 14 people suffering from extreme muscle constriction. She gave each a vial containing a dose (X) of the drug and recorded the time to relief (Y) measured in seconds for each. She fit a curvilinear model to this data. The results obtained by Microsoft Excel follow -Referring to Scenario 15-3, suppose the chemist decides to use a t test to determine if the linear term is significant. Using a level of significance of 0.05, she would decide that the linear term is significant.