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## Section 6.2 Linear Regression

Text of section.
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### Exercises Exercises

#### 1.

A survey of local companies collects information about marketing budgets \(X\) and revenue \(Y\) (each measures in thousands of dollars), shown below. A linear regression gives the best linear fit as \(Y = 18.28 X + 29.69\text{.}\) What is the coefficient of determination \(r^2\text{?}\)
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Table 6.2.1.
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\(x_i\) | \(y_i\) | \((y_i - \text{avg})^2\) | \(\text{pred } y_i\) | \(\text{res}^2\)
--- | --- | --- | --- | ---
200 | 4300 | 2073600 | 3686 | 377377
420 | 7700 | 3841600 | 7707 | 53
270 | 4500 | 1537600 | 4965 | 216495
380 | 7000 | 1587600 | 6976 | 572
300 | 5200 | 291600 | 5514 | 98401
sum: | 28700 | 9332000 | 28848 | 692898

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#### 2.

A sample of 100 measurements are taken and a best fit line is calculated, resulting in the data below (the final line shows the sums for each column). Find the coefficient of determination.
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Table 6.2.2.
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\(x_i\) | \(y_i\) | \(\text{pred } y_i\) | \((y - \text{avg})^2\) | \(\text{res}^2\)
--- | --- | --- | --- | ---
38.00 | 121.00 | 93.17 | 11.83 | 774.33
87.00 | 241.00 | 238.95 | 13586.23 | 4.22
30.00 | 61.00 | 69.37 | 4024.63 | 70.12
35.00 | 85.00 | 84.25 | 1555.51 | 0.57
⋮ | ⋮ | ⋮ | ⋮ | ⋮
33.00 | 108.00 | 78.30 | 270.27 | 882.19
26.00 | 32.00 | 57.47 | 8545.15 | 648.91
sum: | 12444.00 | 12444.00 | 515206.64 | 29789.80

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