Asked by bhdrkn 3 years ago

Find the equation of the regression line for the given data. Then construct a scatter plot of the data and draw the regression line. (The pair of variables have a significant correlation.) Then use the regression equation to predict the value of y for each of the given x-values, if meaningful. The number of hours 6 students spent for a test and their scores on that test are shown below.

Hours spent studying, x |
1 | 2 | 3 | 4 | 4 | 5 |

Test score, y |
42 | 47 | 52 | 50 | 62 | 69 |

(a) x = 3 hours

(b) x = 3.5 hours

(c) x = 12 hours

(d) x = 4.5 hours

linear regression

scatter plot

Answered by baptiste 3 years ago

The find the line of regression use the following equations,

$$m = \frac{\bar{x}\bar{y}-\overline{xy}}{(\bar{x})^2-\overline{x^2}}$$

$$b = \bar{y}-m\bar{x}$$

$$\hat{y} = mx+b$$

now it is only a matter of finding $\bar{x}, \bar{y}, \overline{xy},\text{ and }\overline{x^2}$. So,

$$\bar{x}=\frac{1+2+3+4+4+5}{6}=\frac{19}{6}$$

$$\bar{y}=\frac{42+47+52+50+62+69}{6}=\frac{161}{3}$$

$$\overline{xy}=\frac{1(42)+2(47)+3(52)+4(50)+4(62)+5(69)}{6}=\frac{1085}{6}$$

$$\overline{x^2}=\frac{1^2+2^2+3^2+4^2+4^2+5^2}{6}=\frac{71}{6}$$

plugging these values into $m$ and $b$ gives,

$$m=\frac{(19/6)(161/3)-(1085/6)}{(19/6)^2-71/6}=\frac{392}{65}\approx 6.031$$

$$b=\frac{161}{3}-\frac{392}{65} \frac{19}{6}=\frac{2247}{65} \approx 34.569$$

and hence the line of regression is,

$$\hat{y}=6.031x+34.569$$

by plugging in the x values you will obtain the test score for any value of x. Remember that if the test score is above 100 that would be viewed as not meaningful.

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Asked: 3 years ago

how did you get 161/3?