Question

In: Statistics and Probability

Does knee height predict overall height? Since elderly people may have difficulty standing to have their...

Does knee height predict overall height? Since elderly people may have difficulty standing to have their heights measured, a study looked at predicting overall height (y) in centimeters, from height to the knee (x) in centimeters. Use this data to answer the following questions.

Data looking at knee height (in cm) and overall height (in cm)

Knee

Height (cm)

57.70

47.40

43.50

44.80

55.20

54.60

Overall Height (cm)

192.10

153.30

146.40

162.70

169.10

177.80
  1. What is the mean x and standard deviation sxof x?
  2. What is the mean y and standard deviation sy of y?
  3. What is the correlation r of x and y? What does the correlation mean?
  4. The least-squares regression line. (3 pts)

  1. What is the value of the slope?
  2. What is the value of the intercept?
  3. What is the equation of the least-squares regression line

  1. Someone with a knee height of 65 cm would be approximately 220 cm tall. Is this a good use of the least-squares regression line? Why or why not? (2 pts)
  2. What is the mean x and standard deviation sxof x?
  3. What is the mean y and standard deviation sy of y?
  4. What is the correlation r of x and y? What does the correlation mean?
  5. The least-squares regression line. (3 pts)

    1. What is the value of the slope?
    2. What is the value of the intercept?
    3. What is the equation of the least-squares regression line?

  9. What is the coefficient of determination r2AND what does it tell you about the least-squares regression line? (2 pts)

  11. What is the predicted height for someone with a knee height of 430? (2 pts)

  13. Someone with a knee height of 65 cm would be approximately 220 cm tall. Is this a good use of the least-squares regression line? Why or why not? (2 pts)
  14. What is the mean x and standard deviation sxof x?
  15. What is the mean y and standard deviation sy of y?
  16. What is the correlation r of x and y? What does the correlation mean?
  17. The least-squares regression line. (3 pts)

    1. What is the value of the slope?
    2. What is the value of the intercept?
    3. What is the equation of the least-squares regression line?

  21. What is the coefficient of determination r2AND what does it tell you about the least-squares regression line? (2 pts)

  23. What is the predicted height for someone with a knee height of 430? (2 pts)

  25. Someone with a knee height of 65 cm would be approximately 220 cm tall. Is this a good use of the least-squares regression line? Why or why not? (2 pts)

Solutions

Expert Solution

Knee height (X) overall height (Y) XY X2 Y2
57.7 192.1 11084.17 3329.29 36902.41
47.4 153.3 7266.42 2246.76 23500.89
43.5 146.4 6368.4 1892.25 21432.96
44.8 162.7 7288.96 2007.04 26471.29
55.2 169.1 9334.32 3047.04 28594.81
54.6 177.8 9707.88 2981.16 31612.84
Total 303.2 1001.4 51050.15 15503.54 168515.2

n = 6

a)

The equation of the least-squares regression line is given by

where

The formula for finding slope is given by

Therefore,the value of the slope = 2.45

b)

The formula for finding intercept is given by

We have

Now

= 42.93

Therefore,the value of the intercept = 42.93

c)

The equation of the least-squares regression line is

When knee height (x) =  65 cm

= 42.93 + 2.45*65

= 202.18

Overall Height = 202.18 cm

Finding mean and standard deviation of X

Mean of X = 50.53

= 6.03

Finding mean and standard deviation of Y

Mean of Y = 166.9

Finding correlation r of x and y

The formula for finding correlation coefficient is given by

= 0.89

The correlation coefficient between x and y is 0.89

It indicates a strong uphill (positive) linear relationship.

Finding coefficient of determination (R2)

coefficient of determination is the square of the correlation coefficient.

coefficient of determination = R2 = 0.892 = 0.7921

R2  of  0.7921 suggests that 79.21% of the dependent variable is predicted by the independent variable.

When  knee height (X) = 430 cm

= 42.93 + 2.45*430

= 1096.43

Overall Height = 1096.43 cm

The predicted height for someone with a knee height of 430 cm is 1096.43 cm

orchestra answered 3 years ago
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