Prehistoric pottery vessels are usually found as sherds (broken pieces) and are

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.

a=-0.206

b=0.7662

YHat=-0.206+0.7662x

(a) 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?

(b)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.)

Predictor Coef SE Coef T P Constant -0.206 2.429 -0.09 0.929 Diameter 0.7662 0.1471 4.39 0.001
S =  4.07980       R-Sq = 70.7%

Explanation / Answer

What is the P-value of the slope?

it's 0.929 . as written in the table.

R2 is the square of the correlation between the true and predicted outcomes.

so if r= cor(Yhat, y)

then r= cor(a+bX,y) = sgn(b)*cor(x,y)

now r= sqrt(0.707)=0.8408

so, sgn(b)*cor(x,y)=0.8408

therefore cor(x,y)= 0.8408 (since sgn(b)= +1 as b>0)

so, sample correlation coefficient is 0.8408~0.841(to three decimal places)

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