Linear Regression 307De7

1. **State the problem:** We are given a data set with time (seconds) and height (feet) values and asked to find the linear regression equation $y = mx + b$ that best fits the data, and the correlation coefficient $r$. 2. **Data points:** $$ (0,5), (1,21), (2,34), (3,31), (4,18), (5,3) $$ 3. **Formulas:** - Slope $m = \frac{n\sum xy - \sum x \sum y}{n\sum x^2 - (\sum x)^2}$ - Intercept $b = \frac{\sum y - m \sum x}{n}$ - Correlation coefficient $r = \frac{n\sum xy - \sum x \sum y}{\sqrt{(n\sum x^2 - (\sum x)^2)(n\sum y^2 - (\sum y)^2)}}$ where $n$ is the number of points. 4. **Calculate sums:** - $n = 6$ - $\sum x = 0 + 1 + 2 + 3 + 4 + 5 = 15$ - $\sum y = 5 + 21 + 34 + 31 + 18 + 3 = 112$ - $\sum x^2 = 0^2 + 1^2 + 2^2 + 3^2 + 4^2 + 5^2 = 0 + 1 + 4 + 9 + 16 + 25 = 55$ - $\sum y^2 = 5^2 + 21^2 + 34^2 + 31^2 + 18^2 + 3^2 = 25 + 441 + 1156 + 961 + 324 + 9 = 2916$ - $\sum xy = (0)(5) + (1)(21) + (2)(34) + (3)(31) + (4)(18) + (5)(3) = 0 + 21 + 68 + 93 + 72 + 15 = 269$ 5. **Calculate slope $m$:** $$ m = \frac{6 \times 269 - 15 \times 112}{6 \times 55 - 15^2} = \frac{1614 - 1680}{330 - 225} = \frac{-66}{105} = -\frac{22}{35} \approx -0.6286 $$ 6. **Calculate intercept $b$:** $$ b = \frac{112 - (-0.6286)(15)}{6} = \frac{112 + 9.429}{6} = \frac{121.429}{6} \approx 20.238 $$ 7. **Linear regression equation:** $$ y = -0.6286x + 20.238 $$ 8. **Calculate correlation coefficient $r$:** $$ r = \frac{6 \times 269 - 15 \times 112}{\sqrt{(6 \times 55 - 15^2)(6 \times 2916 - 112^2)}} = \frac{-66}{\sqrt{105 \times (17496 - 12544)}} = \frac{-66}{\sqrt{105 \times 3952}} = \frac{-66}{\sqrt{414960}} \approx \frac{-66}{644.13} \approx -0.1025 $$ 9. **Interpretation:** The slope is negative, indicating a decreasing trend, but the correlation coefficient is close to zero, indicating a weak linear relationship. **Final answers:** - Linear regression equation: $y = -0.6286x + 20.238$ - Correlation coefficient: $r \approx -0.1025$