Need help with (d) part. Thank you. statistical program is recommended. Electromagnetic technologies offer effective nondestructive sensing techniques for determining characteristics of pavement. The...


Need help with (d) part. Thank you.


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 relatey = dielectric constant tox = air void (%) for 18 samples having 5% asphalt content.
















































y
4.554.494.504.474.474.454.404.344.434.434.424.404.334.444.404.264.324.34

x
4.354.795.575.205.075.795.366.405.665.906.495.706.496.376.517.886.747.08


The following R output is from a simple linear regression ofy onx.





































EstimateStd. Errort valuePr(>|t|)
(Intercept)4.8586910.05976881.283<>
AirVoid−0.0746760.009923−7.5261.21e-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
DfSum SqMean SqF valuePr(>F)
Airvoid10.0714220.07142256.6351.214e-06
Residuals160.201780.001261



(a)


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

y = −0.074676x + 4.858691





(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.




H
0: β1 ≠ 0

H
a: β1 = 0H
0: β1 = 0

H
a: β1 > 0H
0: β1 = 0

H
a: β1 <>H
0: β1 = 0

H
a: β1 ≠ 0



Find the test statistic value and find theP-value. (Round your test statistic to two decimal places and yourP-value to three decimal places.)


t=P-value=

State the conclusion in the problem context.



RejectH
0. There is a useful linear relationship between dielectric constant and air void percentage.RejectH
0. There is not a useful linear relationship between dielectric constant and air void percentage.    Fail to rejectH
0. There is not a useful linear relationship between dielectric constant and air void percentage.Fail to rejectH
0. 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 −0.061. 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.




H
0: β1 = −0.061

H
a: β1 > −0.061H
0: β1 = −0.061

H
a: β1 <>H
0: β1 ≠ −0.061

H
a: β1 = −0.061H
0: β1 = −0.061

H
a: β1 ≠ −0.061



Compute the test statistic value and find theP-value. (Round your test statistic to two decimal places and yourP-value to three decimal places.)


t=


P-value=

State the conclusion in the problem context.


RejectH
0. There is insufficient evidence to contradict the prior belief.

RejectH
0. There is sufficient evidence to contradict the prior belief.

Fail to rejectH
0. There is insufficient evidence to contradict the prior belief.

Fail to rejectH
0. There is sufficient evidence to contradict the prior belief.



Jun 02, 2022
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