Least Squares regression questions

[Jaskaran]

I have a set of data:

1) Average height "x" vs average weight "y" of American women:

x |  58 |  59 |  60 |  61 |  62 |  63 |  64 |  65 |  66 |  67 |  68 |  69 |  70 |  71 |  72 |
y | 113 | 115 | 118 | 121 | 124 | 128 | 131 | 134 | 137 | 141 | 145 | 150 | 153 | 159 | 164 |

a) Is the line a good fit for this data? Explain.

The least squares regression is Weight = -98.23 + 3.60height..

I would say yes, this is a good fit, but the book says no. Why? Is it because it's unpractical to predict the weight of a heightless woman?

b) Define extrapolation in the context of this data...

By that, do they mean if height continues further and further, so will weight?

2) Re-express the data to strighten the plot. Give the regression line.

x | 0.375 | 0.5 | 0.75 |  1 | 1.25 | 1.5| 2 |
y |  140  |  80 |  35  | 20 |  13  | 10 | 5 |

This data when plotted shows a negative, nearly-exponential line. To straighten it up, I did a recipricol square root on the y values. This straightens the line postively, the new equation being 1/√y = 0.00243 + 0.2186x

Then, it asks me to predict y for x = 0.25. I get 4.13, but it's obviously wrong. Can anyone help me sort this out? What did I do wrong?

3) A regression equation for the range of motion of a knee versus age is range = 108 + 0.871age. If the residual of a 25 yar old man is -3.36, what was his actual range of motion?

I get 123.415, but it's wrong.

 

[marcmtlca]

for a) you should calculate \(\displaystyle R^2=\frac{\sum_{i=1}^{n}(\hat{y}_i-\overline{y})^2}{\sum_{i=1}^{n}(y_i-\overline{y})^2}\)

In your case:
\(\displaystyle \hat{y}_i=3.60x_i-98.23\)

And

\(\displaystyle \overline{y}=\frac{1}{n}\sum_{i=1}^{n}y_i\)

If this number is reasonably close to 1. then your fit is good. I'll leave it up to you to figure out why (it is important)

b) I suppose you could conclude that since the data does not fit well to a linear model then you could say that weight is not linearly correlated with height... not sure exactly what they are asking. The purpose of this whole analysis is to come up with a relation between an american woman's height and her weight.

2) Try plotting x vs log(y), might be interesting to see what pops up. Log plots are used very often in statistical analysis.

3) Residual = Range - Predicted Range = Range - (108 + 0.871*(25))=-3.36

Solve for Range