A group of students measure the length and width of a random sample of beans. Th

Question

A group of students measure the length and width of a random sample of beans. They are interested in investigating the relationship between the length and width. Their summary statistics are displayed in the table below. All units, if applicable, are millimeters Mean width: Stdev width: Mean height Stdev height: C'orrelation coefficient: 0.8433 7.5 0.909 13.574 a) The students are interested in using the width of the beans to predict the height. Cakculate the slope of the regression equation. b) Write the equation of the best-fit line thal can be used to predict hcan hcighls. Use x lo represent widih and y to represent height. c) What fraction of the variability in bean heights can be explained by the linear model of beaa height vs width Express your answer as a decimal d) If, instead, the students are interested in using the height of the beans to predict the width. calculate the slope of this new regression equation e) Write the equation of the best-fit line that can be used to predict bean widths, Use x to represent height and y to represent width. Preview Preview

Explanation / Answer

Solution

Let x = width of the beans and y = height of the beans.

Linear regression of height on width is: y = ?0 + ?1x.

And the least square estimates of ?0 and ?1 are represented by ?0cap and ?1cap

Back-up Theory:

?0cap = ybar - ?1cap.xbar and ?1cap = r(SDy/SDx)

For Regression of width on height width : ?0cap = Mean width - ?1cap.Mean height and ?1cap = r(SDwidth/SDheight)

Part (a)

Slope of regression of height on width = (0.8433 x 1.717)/0.909 = 1.5929 ANSWER

Part (b)

The intercept (?0cap) of regression of height on width = 13.574 – (1.5929 x 7.5) = 1.62715.

Thus, equation of best-fit line is: height = 1.6273 + 1.5929width ANSWER

Part (c)

Fraction of variability in height explained by the width is represented by r2 = 0.84332

= 0.7112 ANSWER

Part (d)

Slope of regression of width on height = (0.8433 x 0.909)/1.717 = 0.4465 ANSWER

Part (e)

The intercept (?0cap) of regression of width on height = 7.5 - (0.4465 x 13.574) = 1.4392.

Thus, equation of best-fit line is: width = 1.4392 + 0.4465 height ANSWER

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