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 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 = −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. H0: β1 ≠ 0 Ha: β1 = 0H0: β1 = 0 Ha: β1 > 0 H0: β1 = 0 Ha: β1 < 0H0: β1 = 0 Ha: 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 −0.061. 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. H0: β1 = −0.061 Ha: β1 > −0.061H0: β1 = −0.061 Ha: β1 < −0.061 H0: β1 ≠ −0.061 Ha: β1 = −0.061H0: β1 = −0.061 Ha: β1 ≠ −0.061 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.
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 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 = −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. H0: β1 ≠ 0 Ha: β1 = 0H0: β1 = 0 Ha: β1 > 0 H0: β1 = 0 Ha: β1 < 0H0: β1 = 0 Ha: 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 −0.061. 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. H0: β1 = −0.061 Ha: β1 > −0.061H0: β1 = −0.061 Ha: β1 < −0.061 H0: β1 ≠ −0.061 Ha: β1 = −0.061H0: β1 = −0.061 Ha: β1 ≠ −0.061 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.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter7: Analytic Trigonometry
Section7.6: The Inverse Trigonometric Functions
Problem 91E
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Question
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 | 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 = −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.
H0: β1 ≠ 0
Ha: β1 = 0H0: β1 = 0
Ha: β1 > 0 H0: β1 = 0
Ha: β1 < 0H0: β1 = 0
Ha: β1 ≠ 0
Ha: β1 = 0H0: β1 = 0
Ha: β1 > 0 H0: β1 = 0
Ha: β1 < 0H0: β1 = 0
Ha: 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 −0.061. 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.
H0: β1 = −0.061
Ha: β1 > −0.061H0: β1 = −0.061
Ha: β1 < −0.061 H0: β1 ≠ −0.061
Ha: β1 = −0.061H0: β1 = −0.061
Ha: β1 ≠ −0.061
Ha: β1 > −0.061H0: β1 = −0.061
Ha: β1 < −0.061 H0: β1 ≠ −0.061
Ha: β1 = −0.061H0: β1 = −0.061
Ha: β1 ≠ −0.061
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.
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