Find the equation of the regression line for the given data. Then construct a scatter plot of the data and draw the regression line.

# Find the equation of the regression line for the given data. Then construct a scatter plot of the data and draw the regression line.

Asked by 3 years ago
393k points

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 3 years ago
403.4k points

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.

how did you get 161/3?

- richie 1 year ago

322/6 = 161/3

- maddy 1 year ago

Surround your text in *italics* or **bold**, to write a math equation use, for example, $x^2+2x+1=0$ or $$\beta^2-1=0$$

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