Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 12: Q52E. (page 746)

Consider the following data that fit the quadratic modelE(y)=β0+β1x+β2x2:

a. Construct a scatterplot for this data. Give the prediction equation and calculate R2based on the model above.

b. Interpret the value ofR2.

c. Justify whether the overall model is significant at the 1% significance level if the data result into a p-value of 0.000514.

Short Answer

Expert verified

a. Scatter plot, prediction equation is y=8.541667-2.25357x+0.386905x2and value of R2calculated here is 0.9516

b. The value ofR2here is 0.9516 which is a high value denoting that almost 95% of the variation in the variables is explained by the model. This means that the model is a good fit for the data.

c. At 1% significance level, it can be concluded that β1≠β2≠0

Step by step solution

01

Scatterplot for the data

X

Y

0

8

1

7.3

2

5.6

3

5.9

4

5.2

5

6.5

6

8.8

7

12.1

SUMMARY OUTPUT

















Regression Statistics








Multiple R

0.975509








R Square

0.951618








Adjusted R Square

0.932265








Standard Error

0.586556








Observations

8

















ANOVA









df

SS

MS

F

Significance F




Regression

2

33.83476

16.91738

49.17163

0.000515




Residual

5

1.720238

0.344048






Total

7

35.555













Coefficients

Standard Error

t Stat

P-value

Lower 95%

Upper 95%

Lower 95.0%

Upper 95.0%

Intercept

8.541667

0.49366

17.30272

1.18E-05

7.272673

9.810661

7.272673

9.810661

X

-2.25357

0.329452

-6.84036

0.001019

-3.10046

-1.40669

-3.10046

-1.40669

X^2

0.386905

0.045254

8.549672

0.000361

0.270576

0.503233

0.270576

0.503233

On solving the second-order model equation in excel, the output generated is attached here. The prediction equation, therefore isy=8.541667-2.25357x+0.386905x2

The value ofR2 calculated here is 0.9516.

02

Interpretation of R2

The value ofR2 here is 0.9516 which is a high value denoting that almost 95% of the variation in the variables is explained by the model. This means that the model is a good fit for the data.

03

Goodness of fit

H0:β1=β2=0

Ha:At least one of the parametersβ1 orβ2 is non zero

Here, F test statistic=SSEn-(k+1)=1.7202385=0.3440

H0is rejected if p-value < α. For α=0.01, since 0.000514 < 0.01

Sufficient evidence to rejectH0 at 95% confidence interval.

Therefore,β1≠β2≠0.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Forecasting movie revenues with Twitter. Refer to the IEEE International Conference on Web Intelligence and Intelligent Agent Technology (2010) study on using the volume of chatter on Twitter.com to forecast movie box office revenue, Exercise 12.10 (p. 723). The researchers modelled a movie’s opening weekend box office revenue (y) as a function of tweet rate (x1 ) and ratio of positive to negative tweets (x2) using a first-order model.

a) Write the equation of an interaction model for E(y) as a function of x1 and x2 .

b) In terms of theβ in the model, part a, what is the change in revenue (y) for every 1-tweet increase in the tweet rate (x1 ) , holding PN-ratio (x2)constant at a value of 2.5?

c) In terms of the in the model, part a, what is the change in revenue (y) for every 1-tweet increase in the tweet rate (x1 ) , holding PN-ratio (x2)constant at a value of 5.0?

d) In terms of theβ in the model, part a, what is the change in revenue (y) for every 1-unit increase in the PN-ratio (x2) , holding tweet rate (x1 )constant at a value of 100?

e) Give the null hypothesis for testing whether tweet rate (x1 ) and PN-ratio (x2) interact to affect revenue (y).

Question: Consumer behavior while waiting in line. The Journal of Consumer Research (November 2003) published a study of consumer behavior while waiting in a queue. A sample of n = 148 college students was asked to imagine that they were waiting in line at a post office to mail a package and that the estimated waiting time is 10 minutes or less. After a 10-minute wait, students were asked about their level of negative feelings (annoyed, anxious) on a scale of 1 (strongly disagree) to 9 (strongly agree). Before answering, however, the students were informed about how many people were ahead of them and behind them in the line. The researchers used regression to relate negative feelings score (y) to number ahead in line (x1) and number behind in line (x2).

a.The researchers fit an interaction model to the data. Write the hypothesized equation of this model.

b. In the words of the problem, explain what it means to say that “x1 and x2 interact to affect y.”

c. A t-test for the interaction β in the model resulted in a p-value greater than 0.25. Interpret this result.

d. From their analysis, the researchers concluded that “the greater the number of people ahead, the higher the negative feeling score” and “the greater the number of people behind, the lower the negative feeling score.” Use this information to determine the signs of β1 and β2 in the model.

Suppose you fit the model y =β0+β1x1+β1x22+β3x2+β4x1x2+εto n = 25 data points with the following results:

β^0=1.26,β^1= -2.43,β^2=0.05,β^3=0.62,β^4=1.81sβ^1=1.21,sβ^2=0.16,sβ^3=0.26, sβ^4=1.49SSE=0.41 and R2=0.83

  1. Is there sufficient evidence to conclude that at least one of the parameters b1, b2, b3, or b4 is nonzero? Test using a = .05.

  2. Test H0: β1 = 0 against Ha: β1 < 0. Use α = .05.

  3. Test H0: β2 = 0 against Ha: β2 > 0. Use α = .05.

  4. Test H0: β3 = 0 against Ha: β3 ≠ 0. Use α = .05.

When a multiple regression model is used for estimating the mean of the dependent variable and for predicting a new value of y, which will be narrower—the confidence interval for the mean or the prediction interval for the new y-value? Why?

Forecasting movie revenues with Twitter. Refer to the IEEE International Conference on Web Intelligence and Intelligent Agent Technology (2010) study on using the volume of chatter on Twitter.com to forecast movie box office revenue, Exercise 11.27 (p. 657). Recall that opening weekend box office revenue data (in millions of dollars) were collected for a sample of 24 recent movies. In addition to each movie’s tweet rate, i.e., the average number of tweets referring to the movie per hour 1 week prior to the movie’s release, the researchers also computed the ratio of positive to negative tweets (called the PN-ratio).

a) Give the equation of a first-order model relating revenue (y)to both tweet rate(x1)and PN-ratio(x2).

b) Which b in the model, part a, represents the change in revenue(y)for every 1-tweet increase in the tweet rate(x1), holding PN-ratio(x2)constant?

c) Which b in the model, part a, represents the change in revenue (y)for every 1-unit increase in the PN-ratio(x2), holding tweet rate(x1)constant?

d) The following coefficients were reported:R2=0.945andRa2=0.940. Give a practical interpretation for bothR2andRa2.

e) Conduct a test of the null hypothesis, H0;β1=β2=0. Useα=0.05.

f) The researchers reported the p-values for testing,H0;β1=0andH0;β2=0 as both less than .0001. Interpret these results (use).
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Logic and Functions

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free