Question

Find and graph a linear regression equation that models the data.

  Temperature (^oC)

No. of Ice cream sold

26

345

27

322

28

357

29

423

30

435

Answer 1

$$\def\arraystretch{1.5} \begin{array}{c:c:c:c:c:c} & X & Y & XY & X^2 & Y^2\\ \hline & 26 & 345 & 8970 & 676 & 119025\\ & 27 & 322 & 8694 & 729 & 103684\\ & 28 & 357 & 9996 & 784 & 127449\\ & 29 & 423 & 12267 & 841 & 178929\\ & 30 & 435 & 13050 & 900 & 189225\\ Sum= & 140 & 1882 & 52977 & 3930 & 718312\\ \end{array}$$ $$\bar{X}=\dfrac{1}{n}\sum_iX_i=\dfrac{140}{5}=28$$

$$\bar{Y}=\dfrac{1}{n}\sum_iY_i=\dfrac{1882}{5}=376.4$$

$$SS_{XX}=\sum_i(X_i-\bar{X})^2=10$$

$$SS_{YY}=\sum_i(Y_i-\bar{Y})^2=9927.2$$

$$SS_{XY}=\sum_i(X_i-\bar{X})(Y_i-\bar{Y})=281$$

$$m=slope=\dfrac{SS_{XY}}{SS_{XX}}=\dfrac{281}{10}=28.1$$

$$n=\bar{Y}-m\bar{X}$$

$$=376.4-28.1(28)$$

$$=-410.4$$

Therefore, we find that the regression equation is:

$$Y=-410.4+28.1X$$

Correlation coefficient:

$$r=\dfrac{SS_{XY}}{\sqrt{SS_{XX}}\sqrt{SS_{YY}}}$$

$$=\dfrac{281}{\sqrt{10}\sqrt{9927.2}}=0.8918523$$

We have strong positive correlation.

The regression equation is:

$$Y=-410.4+28.1X$$