Question

Prehistoric pottery vessels are usually found as sherds (broken pieces) and are carefully reconstructed if enough sherds can be found. An archaeological study provides data relating x = body diameter in centimeters and y = height in centimeters of prehistoric vessels reconstructed from sherds found at a prehistoric site. The following Minitab printout provides an analysis of the data.

Predictor	Coef	SE Coef	T	P
Constant	-0.224	2.429	-0.09	0.929
Diameter	0.7608	0.1471	5.29	0.003
S =  4.07980       R-Sq = 87.5%

(a) Minitab calls the explanatory variable the predictor variable. Which is the predictor variable, the diameter of the pot or the height?

diameterheight     

(b) For the least-squares line ŷ = a + bx, what is the value of the constant a? What is the value of the slope b? (Note: The slope is the coefficient of the predictor variable.) Write the equation of the least-squares line.

a =

b =

ŷ =  +  x

(c) The P-value for a two-tailed test corresponding to each coefficient is listed under P. The t value corresponding to the coefficient is listed under T. What is the P-value of the slope?


What are the hypotheses for a two-tailed test of β = 0?

H₀: β = 0; H₁: β < 0H₀: β = 0; H₁: β > 0     H₀: β ≠ 0; H₁: β = 0H₀: β < 0; H₁: β = 0H₀: β = 0; H₁: β ≠ 0

Based on the P-value in the printout, do we reject or fail to reject the null hypothesis for α = 0.01?

Reject the null hypothesis. There is sufficient evidence that β differs from 0.Fail to reject the null hypothesis. There is insufficient evidence that β differs from 0.     Fail to reject the null hypothesis. There is sufficient evidence that β differs from 0.Reject the null hypothesis. There is insufficient evidence that β differs from 0.

(d) Recall that the t value and resulting P-value of the slope b equal the t value and resulting P-value of the corresponding correlation coefficient r. To find the value of the sample correlation coefficient r, take the square root of the "R-Sq" value shown in the display. What is the value of r? (Round your answer to three decimal places.)


Consider a two-tailed test for r. Based on the P-value shown in the Minitab display, is the correlation coefficient significant at the 1% level of significance?

No, the correlation coefficient is not significant at the α = 0.01 level because the P-value ≤ α.Yes, the correlation coefficient is significant at the α = 0.01 level because the P-value > α.     Yes, the correlation coefficient is significant at the α = 0.01 level because the P-value ≤ α.No, the correlation coefficient is not significant at the α = 0.01 level because the P-value > α.

Answer 1

(A) Using the data table, it is clear that the variable under the predictor column is diameter

So, the explanatory or independent variable is diameter

(B) We have the equation of line y = a +bx

where a is intercept and b is the slope

From the data table, value of intercept a = -0.224 and value of slope b = 0.7608

So, we get

a = -0.224

b = 0.7608

this gives us y = -0.224 + 0.7608x

(C) P value for the slope under the P column is given as p = 0.003

So, the p value corresponding to slope or b is 0.003

For two tailed hypothesis,we assume alternate hypothesis to be either less than or greater than the 0 value

So, correct set of hypotheses is and

Based on the p value 0.003, we will reject the null hypothesis at 0.01 level of significance because the p value is less than 0.01, which makes is significant.

We can conclude that there is sufficient evidence to support the statement that beta differs from 0

option A is correct for conclusion

(d) we have r squared = 87.5% or 87.5/100 = 0.875

taking square root, we get

r = (rounded to 3 decimals)

P value shown in the Minitab display is 0.003 which is significant at 1% level of significance because it is less than 0.01. Thus, we can say that the correlation coefficient is significant at 0.01 level of significance because the p value is significant.

option C is correct "yes, the correlation coefficient is significant at alpha = 0.01 level because the p value is "