Question

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	t value	Pr(>|t|)
(Intercept)	4.858691	0.059768	81.283	<2e-16
AirVoid	−0.074676	0.009923	−7.526	1.21e-06
Residual standard error: 0.03551 on 16 DF Multiple
R-squared: 0.7797,			Adjusted R-squared: 0.766
F-statistic: 56.63 on 1 and 16 DF,			p-value: 1.214e-06

Analysis of Variance Table

Response: Dielectric
	Df	Sum Sq	Mean Sq	F value	Pr(>F)
Airvoid	1	0.071422	0.071422	56.635	1.214e-06
Residuals	16	0.20178	0.001261

(a)

Obtain the equation of the least squares line. (Enter your numerical values to six decimal places.)

y =  

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--- increase decrease in dielectric constant of  ---Select--- 0.7797 4.858691 81.283 0.03551 0.074676 .

(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.)

%

(c)

Does there appear to be a useful linear relationship between dielectric constant and air void? Carry out a test of appropriate hypotheses using a significance level of 0.01.

State the appropriate null and alternative hypotheses.

H₀: β₁ = 0
H_a: β₁ > 0H₀: β₁ = 0
H_a: β₁ < 0     H₀: β₁ ≠ 0
H_a: β₁ = 0H₀: β₁ = 0
H_a: β₁ ≠ 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.

Reject H₀. There is a useful linear relationship between dielectric constant and air void percentage.Reject H₀. There is not a useful linear relationship between dielectric constant and air void percentage.     Fail to reject H₀. There is not a useful linear relationship between dielectric constant and air void percentage.Fail to reject H₀. 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 −0.064. 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.

H₀: β₁ = −0.064
H_a: β₁ > −0.064H₀: β₁ = −0.064
H_a: β₁ < −0.064     H₀: β₁ ≠ −0.064
H_a: β₁ = −0.064H₀: β₁ = −0.064
H_a: β₁ ≠ −0.064

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.)

t= P-value =

State the conclusion in the problem context.

Reject H₀. There is insufficient evidence to contradict the prior belief.Reject H₀. There is sufficient evidence to contradict the prior belief.     Fail to reject H₀. There is insufficient evidence to contradict the prior belief.Fail to reject H₀. There is sufficient evidence to contradict the prior belief.

Answer 1

(A)

The equation of the least squares line

             

According to the slope, a one-percentage-point increase in air void is associated with an estimated decrease in dielectric constant of 0.074676.

(B)

       As per summary, the coefficient of determination is

                                           R-squared = 0.7797

Then, 77.97 percentage of observed variation in dielectric constant can be attributed to the approximate linear relationship between dielectric constant and air void.

(C)

The appropriate null and alternative hypotheses :

H₀: β₁ = 0
H_a: β₁ ≠ 0

The test statistic value and the P-value: As per summary of regression

t = −7.53

P-value = 0.00000121

i.e., P-value = 0.000

The conclusion in the problem context:

Reject H₀. There is not a useful linear relationship between dielectric constant and air void percentage.

because p < 0.01

(D)

The appropriate null and alternative hypotheses:

H₀: β₁ = −0.064

H_a: β₁ < −0.064  

The test statistic value and the P-value:

  

  

then,

The conclusion in the problem context:

Reject H₀. There is insufficient evidence to contradict the prior belief.