PRACTICE OF STATS - 1 TERM ACCESS CODE
PRACTICE OF STATS - 1 TERM ACCESS CODE
4th Edition
ISBN: 9781319403348
Author: BALDI
Publisher: Macmillan Higher Education
Question
Book Icon
Chapter 28, Problem 28.37E

(a)

To determine

To make a scatterplot of the amount of phosphorus in the plant against nitrogen and explain does the graph suggest that a multiple regression might be appropriate .

(a)

Expert Solution
Check Mark

Answer to Problem 28.37E

Yes, the graph suggest that a multiple regression might be appropriate.

Explanation of Solution

In the question, it is given that an experiment compared the effects of adding various amounts of nitrogen fertilizers to two genotypes of tomato plants, a mild-type and a mutant variety. The percent of phosphorus in the plant, nitrogen and genotype is given in a table. Thus, the scatterplot of the amount of phosphorus in the plant against nitrogen, using different symbols for the two plant genotype is as follows:

PRACTICE OF STATS - 1 TERM ACCESS CODE, Chapter 28, Problem 28.37E , additional homework tip  1

In the scatterplot, we can see that the genotype zero is in red color and genotype one is in blue color. And the lines in the scatterplot are almost parallel so it is linear in nature and in both the lines the points are moving downwards so they are negative in relationship. Thus, as they are parallel so the graph suggest that a multiple regression might be appropriate for these data.

(b)

To determine

To use a software to obtain the estimated multiple linear regression equation when the two explanatory variables nitrogen and genotype are included and create a residual plot and explain are the conditions for multiple linear regression satisfied.

(b)

Expert Solution
Check Mark

Answer to Problem 28.37E

The estimated multiple linear regression equation is y^=0.25680.0007x1+0.2056x2 and the conditions for multiple linear regressionare satisfied.

Explanation of Solution

In the question, it is given that an experiment compared the effects of adding various amounts of nitrogen fertilizers to two genotypes of tomato plants, a mild-type and a mutant variety. The percent of phosphorus in the plant, nitrogen and genotype is given in a table. Now, we will use the Excel to obtain the estimated multiple linear regression equation when the two explanatory variables nitrogen and genotype are included and also the residual plot is constructed. We will use the option data analysis in the data tab and run the regression analysis. The result will be as:

    ANOVA
      df SS MS F Significance F
    Regression20.469160.2345876.23754.25E-13
    Residual330.101540.003077
    Total350.5707   
      Coefficients Standard Error t Stat P-value
    Intercept0.2568530.01548916.583261.43E-17
    Nitrogen-0.000710.000133-5.374616.11E-06
    Genotype0.2055560.0184911.117041.07E-12

The residual plot will be constructed as:

PRACTICE OF STATS - 1 TERM ACCESS CODE, Chapter 28, Problem 28.37E , additional homework tip  2

PRACTICE OF STATS - 1 TERM ACCESS CODE, Chapter 28, Problem 28.37E , additional homework tip  3

And the normal plot will be constructed as:

PRACTICE OF STATS - 1 TERM ACCESS CODE, Chapter 28, Problem 28.37E , additional homework tip  4

Now, the estimated multiple linear regression equation when the two explanatory variables nitrogen and genotype are included is as:

  y^=b0+b1x1+b2x2y^=0.25680.0007x1+0.2056x2

Where x2 is the indicator variable for the genotype. Now, as we look at the scatterplot we can see that the lines are almost parallel, so the data is linear in nature and the condition for linear is satisfied. And as we look at the residual plot, we cannot find any pattern in the points so the constant variance is satisfied. As we look at the normal plot we can see that the normal condition is satisfied and also the data is randomly selected so the independence is satisfied. Thus, the conditions for inferences are satisfied.

(c)

To determine

To create a new variable called interaction by multiplying the explanatory variables nitrogen and genotype and add this new variable to your regression model and provide the estimated multiple linear regression equation and create regression plot for this and discuss whether the conditions for multiple linear regression are met.

(c)

Expert Solution
Check Mark

Answer to Problem 28.37E

The conditions for multiple linear regression are met and the estimated multiple linear regression equation is y^=0.23390.0004x1+0.2515x20.0007x1x2 .

Explanation of Solution

In the question, it is given that an experiment compared the effects of adding various amounts of nitrogen fertilizers to two genotypes of tomato plants, a mild-type and a mutant variety. The percent of phosphorus in the plant, nitrogen and genotype is given in a table. And a new variable called interaction is created by multiplying the explanatory variables nitrogen and genotype. Now, we will use the Excel to obtain the estimated multiple linear regression equation when the two explanatory variables nitrogen and genotype and the interaction are included and also the residual plot is constructed. We will use the option data analysis in the data tab and run the regression analysis. The result will be as:

    ANOVA
      df SS MS F Significance F
    Regression30.492730.16424367.40796.35E-14
    Residual320.077970.002437
    Total350.5707   
      Coefficients Standard Error t Stat P-value
    Intercept0.233870.01563914.954385.42E-16
    Nitrogen-0.000350.000167-2.07150.046455
    Genotype0.2515220.02211711.372458.91E-13
    Interaction-0.000730.000236-3.110220.003912

The residual plot is as follows:

PRACTICE OF STATS - 1 TERM ACCESS CODE, Chapter 28, Problem 28.37E , additional homework tip  5

PRACTICE OF STATS - 1 TERM ACCESS CODE, Chapter 28, Problem 28.37E , additional homework tip  6

The normal plot is as follows:

PRACTICE OF STATS - 1 TERM ACCESS CODE, Chapter 28, Problem 28.37E , additional homework tip  7

Now, the estimated multiple linear regression equation when the two explanatory variables nitrogen and genotype and interaction are included is as:

  y^=b0+b1x1+b2x2+b3x1x2y^=0.23390.0004x1+0.2515x20.0007x1x2

Where x2 is the indicator variable for the genotype. Now, as we look at the scatterplot we can see that the lines are almost parallel, so the data is linear in nature and the condition for linear is satisfied. And as we look at the residual plot, we cannot find any pattern in the points so the constant variance is satisfied. As we look at the normal plot we can see that the normal condition is satisfied and also the data is randomly selected so the independence is satisfied. Thus, the conditions for inferences are satisfied.

(d)

To determine

To explain does the ANOVA table for the model with the interaction term indicate that at least one of the explanatory variables is helpful in predicting the amount of phosphorus in the plant and explain do the individual t tests indicate that all coefficients are significantly different from zero.

(d)

Expert Solution
Check Mark

Answer to Problem 28.37E

Yes, the ANOVA table for the model with the interaction term indicates that at least one of the explanatory variables is helpful in predicting the amount of phosphorus in the plant and the individual t tests indicate that all coefficients are significantly different from zero.

Explanation of Solution

In the question, it is given that an experiment compared the effects of adding various amounts of nitrogen fertilizers to two genotypes of tomato plants, a mild-type and a mutant variety. The percent of phosphorus in the plant, nitrogen and genotype is given in a table. Since in the ANOVA table in part (c) we can see that the P-value is less than the level of significance,

  P<0.05Reject H0

Thus, we can say that the ANOVA table for the model with the interaction term indicates that at least one of the explanatory variables is helpful in predicting the amount of phosphorus in the plant. And as we can see in the result of the regression analysis in part (c), we can see that all the P-values are less than the level of significance i.e.

  P<0.05Reject H0

Thus, we have sufficient evidence to conclude that the individual t tests indicate that all coefficients are significantly different from zero.

Want to see more full solutions like this?

Subscribe now to access step-by-step solutions to millions of textbook problems written by subject matter experts!
Students have asked these similar questions
I just need to know why this is wrong below: What is the test statistic W? W=5 (incorrect) and  What is the p-value of this test? (p-value < 0.001-- incorrect)   Use the Wilcoxon signed rank test to test the hypothesis that the median number of pages in the statistics books in the library from which the sample was taken is 400. A sample of 12 statistics books have the following numbers of pages pages 127 217 486 132 397 297 396 327 292 256 358 272 What is the sum of the negative ranks (W-)? 75 What is the sum of the positive ranks (W+)? 5What type of test is this?     two tailedWhat is the test statistic W? 5 These are the critical values for a 1-tailed Wilcoxon Signed Rank test for n=12 Alpha Level 0.001 0.005 0.01 0.025 0.05 0.1 0.2 Critical Value 75 70 68 64 60 56 50 What is the p-value for this test?         p-value < 0.001
ons 12. A sociologist hypothesizes that the crime rate is higher in areas with higher poverty rate and lower median income. She col- lects data on the crime rate (crimes per 100,000 residents), the poverty rate (in %), and the median income (in $1,000s) from 41 New England cities. A portion of the regression results is shown in the following table. Standard Coefficients error t stat p-value Intercept -301.62 549.71 -0.55 0.5864 Poverty 53.16 14.22 3.74 0.0006 Income 4.95 8.26 0.60 0.5526 a. b. Are the signs as expected on the slope coefficients? Predict the crime rate in an area with a poverty rate of 20% and a median income of $50,000. 3. Using data from 50 work
2. The owner of several used-car dealerships believes that the selling price of a used car can best be predicted using the car's age. He uses data on the recent selling price (in $) and age of 20 used sedans to estimate Price = Po + B₁Age + ε. A portion of the regression results is shown in the accompanying table. Standard Coefficients Intercept 21187.94 Error 733.42 t Stat p-value 28.89 1.56E-16 Age -1208.25 128.95 -9.37 2.41E-08 a. What is the estimate for B₁? Interpret this value. b. What is the sample regression equation? C. Predict the selling price of a 5-year-old sedan.
Knowledge Booster
Background pattern image
Similar questions
SEE MORE QUESTIONS
Recommended textbooks for you
Text book image
MATLAB: An Introduction with Applications
Statistics
ISBN:9781119256830
Author:Amos Gilat
Publisher:John Wiley & Sons Inc
Text book image
Probability and Statistics for Engineering and th...
Statistics
ISBN:9781305251809
Author:Jay L. Devore
Publisher:Cengage Learning
Text book image
Statistics for The Behavioral Sciences (MindTap C...
Statistics
ISBN:9781305504912
Author:Frederick J Gravetter, Larry B. Wallnau
Publisher:Cengage Learning
Text book image
Elementary Statistics: Picturing the World (7th E...
Statistics
ISBN:9780134683416
Author:Ron Larson, Betsy Farber
Publisher:PEARSON
Text book image
The Basic Practice of Statistics
Statistics
ISBN:9781319042578
Author:David S. Moore, William I. Notz, Michael A. Fligner
Publisher:W. H. Freeman
Text book image
Introduction to the Practice of Statistics
Statistics
ISBN:9781319013387
Author:David S. Moore, George P. McCabe, Bruce A. Craig
Publisher:W. H. Freeman