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.

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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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
H_a: ?₁ = ?₂ = … = ?₁₄ = 0H₀: ?₁ = ?₂ = … = ?₁₄ = 0
H_a: at least one of ?₁, ?₂, …, or ?₁₄ is not 0. H₀: ?₁ = ?₂ = … = ?₁₄ = 0
H_a: ?₁, ?₂, …, and ?₁₄ are all not 0H₀: at least one of ?₁, ?₂, …, or ?₁₄ is not 0.
H_a: ?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.

Calculate

s_e.

(Round your answer to four decimal places.)

s_e =

Interpret

s_e.