Question

In: Statistics and Probability

Suppose a doctor measures the​ height, x, and head​ circumference, y, of 8 children and obtains...

Suppose a doctor measures the​ height, x, and head​ circumference, y, of 8 children and obtains the data below. The correlation coefficient is 0.866 and the least squares regression line is ModifyingAbove y with caret equals 0.204 x plus 11.835. Complete parts ​(a) and ​(b) below. ​Height, x 27.25 25.75 26.50 25.25 28.00 26.50 25.75 27.00 Head​ Circumference, y 17.5 17.1 17.1 16.9 17.5 17.4 17.2 17.3 ​(a) Compute the coefficient of​ determination, Rsquared. Rsquaredequals nothing​% ​(Round to one decimal place as​ needed.) ​(b) Interpret the coefficient of determination and comment on the adequacy of the linear model. A. Rsquared of the variation in head circumference is not explained by the​ least-squares regression equation. The linear model appears to be not appropriate. B. Rsquared of the variation in head circumference is explained by the​ least-squares regression equation. The linear model appears to be appropriate. C. Rsquared of the variation in head circumference is explained by the​ least-squares regression equation. The linear model appears to be not appropriate. D. Rsquared of the variation in head circumference is not explained by the​ least-squares regression equation. The linear model appears to be appropriate.

Solutions

Expert Solution

GIVEN:

The data which displays the​ height, x, and head​ circumference, y, of 8 children:

Height (x) Head circumference (y)
27.25 17.5
25.75 17.1
26.50 17.1
25.25 16.9
28.00 17.5
26.50 17.4
25.75 17.2
27.00 17.3

FORMULA:

Let us first calculate the correlation coefficient using the formula:

The coefficient of determination is the square of correlation coefficient.

CALCULATION:

Let us first compute

27.25 17.5 742.56 306.25 476.88
25.75 17.1 663.06 292.41 440.33
26.50 17.1 702.25 292.41 453.15
25.25 16.9 637.56 285.61 426.73
28.00 17.5 784 306.25 490
26.50 17.4 702.25 302.76 461.1
25.75 17.2 663.06 295.84 442.9
27.00 17.3 729 299.29 467.1

The correlation coefficient is

  

  

The coefficient of determination is the square of correlation coefficient.

Thus the coefficient of determination is %.

(b) INTERPRETATION: Option (B): Rsquared of the variation in head circumference is explained by the​ least-squares regression equation. The linear model appears to be appropriate.

Thus 77% of total variation in dependent variable "Head circumference" is explained by the​ least-squares regression equation. The linear model appears to be appropriate.

orchestra answered 1 year ago
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