A statistical program is recommended. Electromagnetic technologies offer effective nondestructive sensing techniques for determining characteristics of pavement. The propagation of electromagnetic waves through the material depends on its dielectric properties. The following data, kindly provided by the authors of the article "Dielectric Modeling of Asphalt Mixtures and Relationship with Density,"+ was used to relate y = dielectric constant to x = air void (%) for 18 samples having 5% asphalt content. y 4.55 4.49 4.50 4.47 4.47 4.45 4.40 4.34 4.43 4.43 4.42 4.40 4.33 4.44 4.40 4.26 4.32 4.34 x 4.35 4.79 5.57 5.20 5.07 5.79 5.36 6.40 5.66 5.90 6.49 5.70 6.49 6.37 6.51 7.88 6.74 7.08 The following R output is from a simple linear regression of y on x. Estimate Std. Error 4.858691 -0.074676 y = (Intercept) AirVoid Analysis of Variance Table Response: Dielectric Df 0.059768 0.009923 Residual standard error: 0.03551 on 16 DF Multiple R-squared: 0.7797, F-statistic: 56.63 on 1 and 16 DF, t value. 81.283 -7.526 Mean Sq 0.071422 0.001261 % Pr (>|t|) <2e-16 1.21e-06 Adjusted R-squared: 0.766 p-value: 1.214e-06 F value 56.635 Airvoid 1 Sum Sq 0.071422 0.20178 Residuals 16 (a) Obtain the equation of the least squares line. (Enter your numerical values to six decimal places.) Pr (>F) 1.214e-06 Interpret the slope of the equation of the least squares line. According to the slope, a one-percentage-point increase in air void is associated with an estimated ---Select-- in dielectric constant of ---Select--- 8. (b) What percentage of observed variation in dielectric constant can be attributed to the approximate linear relationship between dielectric constant and air void? (Round your answer to two decimal places.) appropriate hypotheses using a significance level of 0.01.
A statistical program is recommended. Electromagnetic technologies offer effective nondestructive sensing techniques for determining characteristics of pavement. The propagation of electromagnetic waves through the material depends on its dielectric properties. The following data, kindly provided by the authors of the article "Dielectric Modeling of Asphalt Mixtures and Relationship with Density,"+ was used to relate y = dielectric constant to x = air void (%) for 18 samples having 5% asphalt content. y 4.55 4.49 4.50 4.47 4.47 4.45 4.40 4.34 4.43 4.43 4.42 4.40 4.33 4.44 4.40 4.26 4.32 4.34 x 4.35 4.79 5.57 5.20 5.07 5.79 5.36 6.40 5.66 5.90 6.49 5.70 6.49 6.37 6.51 7.88 6.74 7.08 The following R output is from a simple linear regression of y on x. Estimate Std. Error 4.858691 -0.074676 y = (Intercept) AirVoid Analysis of Variance Table Response: Dielectric Df 0.059768 0.009923 Residual standard error: 0.03551 on 16 DF Multiple R-squared: 0.7797, F-statistic: 56.63 on 1 and 16 DF, t value. 81.283 -7.526 Mean Sq 0.071422 0.001261 % Pr (>|t|) <2e-16 1.21e-06 Adjusted R-squared: 0.766 p-value: 1.214e-06 F value 56.635 Airvoid 1 Sum Sq 0.071422 0.20178 Residuals 16 (a) Obtain the equation of the least squares line. (Enter your numerical values to six decimal places.) Pr (>F) 1.214e-06 Interpret the slope of the equation of the least squares line. According to the slope, a one-percentage-point increase in air void is associated with an estimated ---Select-- in dielectric constant of ---Select--- 8. (b) What percentage of observed variation in dielectric constant can be attributed to the approximate linear relationship between dielectric constant and air void? (Round your answer to two decimal places.) appropriate hypotheses using a significance level of 0.01.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter7: Analytic Trigonometry
Section7.6: The Inverse Trigonometric Functions
Problem 93E
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Transcribed Image Text:A statistical program is recommended.
Electromagnetic technologies offer effective nondestructive sensing techniques for determining characteristics of pavement. The propagation of electromagnetic waves through the material depends
on its dielectric properties. The following data, kindly provided by the authors of the article "Dielectric Modeling of Asphalt Mixtures and Relationship with Density,"t was used to relate
y = dielectric constant to x = air void (%) for 18 samples having 5% asphalt content.
y 4.55 4.49 4.50 4.47 4.47 4.45 4.40 4.34 4.43 4.43 4.42 4.40 4.33 4.44 4.40 4.26 4.32 4.34
x 4.35 4.79 5.57 5.20 5.07 5.79 5.36 6.40 5.66 5.90 6.49 5.70 6.49 6.37 6.51 7.88 6.74 7.08
The following R output is from a simple linear regression of y on x.
Std. Error
Estimate
4.858691
0.059768
-0.074676
0.009923
y =
(Intercept)
AirVoid
Residual standard error: 0.03551 on 16 DF Multiple
R-squared: 0.7797,
F-statistic: 56.63 on 1 and 16 DF,
Analysis of Variance Table
Response: Dielectric
t value
81.283
-7.526
Pr (>|t|)
<2e-16
1.21e-06
Adjusted R-squared: 0.766
p-value: 1.214e-06
F value
56.635
%
Airvoid
Residuals
Df Sum Sq
1 0.071422
16 0.20178
Mean Sq
0.071422
0.001261
(a) Obtain the equation of the least squares line. (Enter your numerical values to six decimal places.)
Pr (>F)
1.214e-06
Interpret the slope of the equation of the least squares line.
According to the slope, a one-percentage-point increase in air void is associated with an estimated ---Select--- in dielectric constant of ---Select--- 8
mitted
(b) What percentage of observed variation in dielectric constant can be attributed to the approximate linear relationship between dielectric constant and air void? (Round your answer to two
decimal places.)
and air void? Carry out a test of appropriate hypotheses using a significance level of 0.01.
Transcribed Image Text:0 P1
H₂: B₁ <0
Find the test statistic value and find the P-value. (Round your test statistic to two decimal places and your P-value to three decimal places.)
t =
P-value =
State the conclusion in the problem context.
O Reject Ho. There is a useful linear relationship between dielectric constant and air void percentage.
O Reject Ho. There is not a useful linear relationship between dielectric constant and air void percentage.
Fail to reject Ho. There is not a useful linear relationship between dielectric constant and air void percentage.
O Fail to reject Ho. There is a useful linear relationship between dielectric constant and air void percentage.
d) Suppose it had previously been believed that when air void increased by 1 percent, the associated true average change in dielectric constant would be at least -0.066. Does the sample data
contradict this belief? Carry out a test of appropriate hypotheses using a significance level of 0.01.
State the appropriate null and alternative hypotheses.
Ho: B₁ = -0.066
H₂: B₁
-0.066
OHO: B₁ = -0.066
Ha: B₁
-0.066
O Ho: B₁ -0.066
H₂: B₁ = -0.066
O Ho: P₁ = -0.066
H₂: B₁-0.066
Compute the test statistic value and find the P-value. (Round your test statistic to two decimal places and your P-value to three decimal places.)
CH
t =
P-value =
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