A statistical program is recommended. The accompanying data resulted from a study of the relationship between y = brightness of finished paper and the independent variables x1 = hydrogen peroxide (% by weight), x2 = sodium hydroxide (% by weight), x3 = silicate (% by weight), and x4 = process temperature. † x1 x2 x3 x4 y 0.2 0.2 1.5 145 83.9 0.4 0.2 1.5 145 84.9 0.2 0.4 1.5 145 83.4 0.4 0.4 1.5 145 84.2 0.2 0.2 3.5 145 83.8 0.4 0.2 3.5 145 84.7 0.2 0.4 3.5 145 84.0 0.4 0.4 3.5 145 84.8 0.2 0.2 1.5 175 84.5 0.4 0.2 1.5 175 86.0 0.2 0.4 1.5 175 82.6 0.4 0.4 1.5 175 85.1 0.2 0.2 3.5 175 84.5 0.4 0.2 3.5 175 86.0 0.2 0.4 3.5 175 84.0 0.4 0.4 3.5 175 85.4 x1 x2 x3 x4 y 0.1 0.3 2.5 160 82.9 0.5 0.3 2.5 160 85.5 0.3 0.1 2.5 160 85.2 0.3 0.5 2.5 160 84.5 0.3 0.3 0.5 160 84.7 0.3 0.3 4.5 160 85.0 0.3 0.3 2.5 130 84.9 0.3 0.3 2.5 190 84.0 0.3 0.3 2.5 160 84.5 0.3 0.3 2.5 160 84.7 0.3 0.3 2.5 160 84.6 0.3 0.3 2.5 160 84.9 0.3 0.3 2.5 160 84.9 0.3 0.3 2.5 160 84.5 0.3 0.3 2.5 160 84.6 (a) Find the estimated regression equation for the model that includes all independent variables, all quadratic terms, and all interaction terms. (Round your numerical values to five decimal places.) ŷ = (b) Using a 0.05 significance level, perform the model utility test. State the null and alternative hypotheses. H0: ?1, ?2, …, and ?14 are all not 0 Ha: ?1 = ?2 = … = ?14 = 0H0: ?1 = ?2 = … = ?14 = 0 Ha: at least one of ?1, ?2, …, or ?14 is not 0. H0: ?1 = ?2 = … = ?14 = 0 Ha: ?1, ?2, …, and ?14 are all not 0H0: at least one of ?1, ?2, …, or ?14 is not 0. Ha: ?1 = ?2 = … = ?14 = 0 Calculate the test statistic. (Round your answer to two decimal places.) F = Use technology to calculate the P-value. (Round your answer to four decimal places.) P-value = What can you conclude? Fail to reject H0. We do not have convincing evidence that the multiple regression model is useful and cannot conclude that at least one of ?1, ?2, …, or ?14 is not 0.Reject H0. We have convincing evidence that the multiple regression model is useful and can conclude that at least one of ?1, ?2, …, or ?14 is not 0. Fail to reject H0. We do not have convincing evidence that the multiple regression model is useful and cannot conclude that ?1, ?2, …, and ?14 are all not 0.Reject H0. We have convincing evidence that the multiple regression model is useful and can conclude that ?1, ?2, …, and ?14 are all not 0. (c) Calculate SSResid. (Round your answer to four decimal places.) SSResid = Interpret SSResid. This is the interquartile range of the deviations of the brightness values in the sample from the values predicted by the estimated regression equation.This is the probability of observing a value of the F-statistic at least as extreme as the observed F-statistic when ?1, ?2, …, and ?14 are all 0. This is a typical deviation of a brightness value in the sample from the value predicted by the estimated regression equation.This is the sum of the squares of the deviations of the actual values from the values predicted by the fitted model.This tells us the proportion of the observed variation in brightness that can be explained by the fitted model. Calculate R2. (Round your answer to three decimal places.) R2 = Interpret R2. This is the interquartile range of the deviations of the brightness values in the sample from the values predicted by the estimated regression equation.This is the probability of observing a value of the F-statistic at least as extreme as the observed F-statistic when ?1, ?2, …, and ?14 are all 0. This is a typical deviation of a brightness value in the sample from the value predicted by the estimated regression equation.This is the sum of the squares of the deviations of the actual values from the values predicted by the fitted model.This tells us the proportion of the observed variation in brightness that can be explained by the fitted model. Calculate se. (Round your answer to four decimal places.) se = Interpret se.
A statistical program is recommended. The accompanying data resulted from a study of the relationship between y = brightness of finished paper and the independent variables x1 = hydrogen peroxide (% by weight), x2 = sodium hydroxide (% by weight), x3 = silicate (% by weight), and x4 = process temperature. † x1 x2 x3 x4 y 0.2 0.2 1.5 145 83.9 0.4 0.2 1.5 145 84.9 0.2 0.4 1.5 145 83.4 0.4 0.4 1.5 145 84.2 0.2 0.2 3.5 145 83.8 0.4 0.2 3.5 145 84.7 0.2 0.4 3.5 145 84.0 0.4 0.4 3.5 145 84.8 0.2 0.2 1.5 175 84.5 0.4 0.2 1.5 175 86.0 0.2 0.4 1.5 175 82.6 0.4 0.4 1.5 175 85.1 0.2 0.2 3.5 175 84.5 0.4 0.2 3.5 175 86.0 0.2 0.4 3.5 175 84.0 0.4 0.4 3.5 175 85.4 x1 x2 x3 x4 y 0.1 0.3 2.5 160 82.9 0.5 0.3 2.5 160 85.5 0.3 0.1 2.5 160 85.2 0.3 0.5 2.5 160 84.5 0.3 0.3 0.5 160 84.7 0.3 0.3 4.5 160 85.0 0.3 0.3 2.5 130 84.9 0.3 0.3 2.5 190 84.0 0.3 0.3 2.5 160 84.5 0.3 0.3 2.5 160 84.7 0.3 0.3 2.5 160 84.6 0.3 0.3 2.5 160 84.9 0.3 0.3 2.5 160 84.9 0.3 0.3 2.5 160 84.5 0.3 0.3 2.5 160 84.6 (a) Find the estimated regression equation for the model that includes all independent variables, all quadratic terms, and all interaction terms. (Round your numerical values to five decimal places.) ŷ = (b) Using a 0.05 significance level, perform the model utility test. State the null and alternative hypotheses. H0: ?1, ?2, …, and ?14 are all not 0 Ha: ?1 = ?2 = … = ?14 = 0H0: ?1 = ?2 = … = ?14 = 0 Ha: at least one of ?1, ?2, …, or ?14 is not 0. H0: ?1 = ?2 = … = ?14 = 0 Ha: ?1, ?2, …, and ?14 are all not 0H0: at least one of ?1, ?2, …, or ?14 is not 0. Ha: ?1 = ?2 = … = ?14 = 0 Calculate the test statistic. (Round your answer to two decimal places.) F = Use technology to calculate the P-value. (Round your answer to four decimal places.) P-value = What can you conclude? Fail to reject H0. We do not have convincing evidence that the multiple regression model is useful and cannot conclude that at least one of ?1, ?2, …, or ?14 is not 0.Reject H0. We have convincing evidence that the multiple regression model is useful and can conclude that at least one of ?1, ?2, …, or ?14 is not 0. Fail to reject H0. We do not have convincing evidence that the multiple regression model is useful and cannot conclude that ?1, ?2, …, and ?14 are all not 0.Reject H0. We have convincing evidence that the multiple regression model is useful and can conclude that ?1, ?2, …, and ?14 are all not 0. (c) Calculate SSResid. (Round your answer to four decimal places.) SSResid = Interpret SSResid. This is the interquartile range of the deviations of the brightness values in the sample from the values predicted by the estimated regression equation.This is the probability of observing a value of the F-statistic at least as extreme as the observed F-statistic when ?1, ?2, …, and ?14 are all 0. This is a typical deviation of a brightness value in the sample from the value predicted by the estimated regression equation.This is the sum of the squares of the deviations of the actual values from the values predicted by the fitted model.This tells us the proportion of the observed variation in brightness that can be explained by the fitted model. Calculate R2. (Round your answer to three decimal places.) R2 = Interpret R2. This is the interquartile range of the deviations of the brightness values in the sample from the values predicted by the estimated regression equation.This is the probability of observing a value of the F-statistic at least as extreme as the observed F-statistic when ?1, ?2, …, and ?14 are all 0. This is a typical deviation of a brightness value in the sample from the value predicted by the estimated regression equation.This is the sum of the squares of the deviations of the actual values from the values predicted by the fitted model.This tells us the proportion of the observed variation in brightness that can be explained by the fitted model. Calculate se. (Round your answer to four decimal places.) se = Interpret se.
Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018
18th Edition
ISBN:9780079039897
Author:Carter
Publisher:Carter
Chapter4: Equations Of Linear Functions
Section4.5: Correlation And Causation
Problem 2AGP
Related questions
Question
A statistical program is recommended.
The accompanying data resulted from a study of the relationship between y = brightness of finished paper and the independent variables
x1 = hydrogen peroxide
(% by weight),
x2 = sodium hydroxide
(% by weight),
x3 = silicate
(% by weight), and
x4 = process temperature.
†| x1 | x2 | x3 | x4 | y |
|---|---|---|---|---|
| 0.2 | 0.2 | 1.5 | 145 | 83.9 |
| 0.4 | 0.2 | 1.5 | 145 | 84.9 |
| 0.2 | 0.4 | 1.5 | 145 | 83.4 |
| 0.4 | 0.4 | 1.5 | 145 | 84.2 |
| 0.2 | 0.2 | 3.5 | 145 | 83.8 |
| 0.4 | 0.2 | 3.5 | 145 | 84.7 |
| 0.2 | 0.4 | 3.5 | 145 | 84.0 |
| 0.4 | 0.4 | 3.5 | 145 | 84.8 |
| 0.2 | 0.2 | 1.5 | 175 | 84.5 |
| 0.4 | 0.2 | 1.5 | 175 | 86.0 |
| 0.2 | 0.4 | 1.5 | 175 | 82.6 |
| 0.4 | 0.4 | 1.5 | 175 | 85.1 |
| 0.2 | 0.2 | 3.5 | 175 | 84.5 |
| 0.4 | 0.2 | 3.5 | 175 | 86.0 |
| 0.2 | 0.4 | 3.5 | 175 | 84.0 |
| 0.4 | 0.4 | 3.5 | 175 | 85.4 |
| x1 | x2 | x3 | x4 | y |
|---|---|---|---|---|
| 0.1 | 0.3 | 2.5 | 160 | 82.9 |
| 0.5 | 0.3 | 2.5 | 160 | 85.5 |
| 0.3 | 0.1 | 2.5 | 160 | 85.2 |
| 0.3 | 0.5 | 2.5 | 160 | 84.5 |
| 0.3 | 0.3 | 0.5 | 160 | 84.7 |
| 0.3 | 0.3 | 4.5 | 160 | 85.0 |
| 0.3 | 0.3 | 2.5 | 130 | 84.9 |
| 0.3 | 0.3 | 2.5 | 190 | 84.0 |
| 0.3 | 0.3 | 2.5 | 160 | 84.5 |
| 0.3 | 0.3 | 2.5 | 160 | 84.7 |
| 0.3 | 0.3 | 2.5 | 160 | 84.6 |
| 0.3 | 0.3 | 2.5 | 160 | 84.9 |
| 0.3 | 0.3 | 2.5 | 160 | 84.9 |
| 0.3 | 0.3 | 2.5 | 160 | 84.5 |
| 0.3 | 0.3 | 2.5 | 160 | 84.6 |
(a)
Find the estimated regression equation for the model that includes all independent variables, all quadratic terms, and all interaction terms. (Round your numerical values to five decimal places.)
ŷ =
(b)
Using a 0.05 significance level, perform the model utility test.
State the null and alternative hypotheses.
H0: ?1, ?2, …, and ?14 are all not 0
Ha: ?1 = ?2 = … = ?14 = 0H0: ?1 = ?2 = … = ?14 = 0
Ha: at least one of ?1, ?2, …, or ?14 is not 0. H0: ?1 = ?2 = … = ?14 = 0
Ha: ?1, ?2, …, and ?14 are all not 0H0: at least one of ?1, ?2, …, or ?14 is not 0.
Ha: ?1 = ?2 = … = ?14 = 0
Ha: ?1 = ?2 = … = ?14 = 0H0: ?1 = ?2 = … = ?14 = 0
Ha: at least one of ?1, ?2, …, or ?14 is not 0. H0: ?1 = ?2 = … = ?14 = 0
Ha: ?1, ?2, …, and ?14 are all not 0H0: at least one of ?1, ?2, …, or ?14 is not 0.
Ha: ?1 = ?2 = … = ?14 = 0
Calculate the test statistic. (Round your answer to two decimal places.)
F =
Use technology to calculate the P-value. (Round your answer to four decimal places.)
P-value =
What can you conclude?
Fail to reject H0. We do not have convincing evidence that the multiple regression model is useful and cannot conclude that at least one of ?1, ?2, …, or ?14 is not 0.Reject H0. We have convincing evidence that the multiple regression model is useful and can conclude that at least one of ?1, ?2, …, or ?14 is not 0. Fail to reject H0. We do not have convincing evidence that the multiple regression model is useful and cannot conclude that ?1, ?2, …, and ?14 are all not 0.Reject H0. We have convincing evidence that the multiple regression model is useful and can conclude that ?1, ?2, …, and ?14 are all not 0.
(c)
Calculate SSResid. (Round your answer to four decimal places.)
SSResid =
Interpret SSResid.
This is the interquartile range of the deviations of the brightness values in the sample from the values predicted by the estimated regression equation.This is the probability of observing a value of the F-statistic at least as extreme as the observed F-statistic when ?1, ?2, …, and ?14 are all 0. This is a typical deviation of a brightness value in the sample from the value predicted by the estimated regression equation.This is the sum of the squares of the deviations of the actual values from the values predicted by the fitted model.This tells us the proportion of the observed variation in brightness that can be explained by the fitted model.
Calculate
R2.
(Round your answer to three decimal places.)R2 =
Interpret
R2.
This is the interquartile range of the deviations of the brightness values in the sample from the values predicted by the estimated regression equation.This is the probability of observing a value of the F-statistic at least as extreme as the observed F-statistic when ?1, ?2, …, and ?14 are all 0. This is a typical deviation of a brightness value in the sample from the value predicted by the estimated regression equation.This is the sum of the squares of the deviations of the actual values from the values predicted by the fitted model.This tells us the proportion of the observed variation in brightness that can be explained by the fitted model.
Calculate
se.
(Round your answer to four decimal places.)se =
Interpret
se.
This is the interquartile range of the deviations of the brightness values in the sample from the values predicted by the estimated regression equation.This is the probability of observing a value of the F-statistic at least as extreme as the observed F-statistic when ?1, ?2, …, and ?14 are all 0. This is a typical deviation of a brightness value in the sample from the value predicted by the estimated regression equation.This is the sum of the squares of the deviations of the actual values from the values predicted by the fitted model.This tells us the proportion of the observed variation in brightness that can be explained by the fitted model.
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