Answer to Question #344005 in Statistics and Probability for wiwili

Question

The local coffee shop keeps track of how much coffee they sell versus the temperature on that day, here are their figures:

Sales in Php

1731

1840

1427

1262

1444

1100

Temp in Celsius

21.4

22.9

24.1

26.1

29

31.1

Suppose you want to create a regression equation that will predict the total sales in Php using the temperature of the day in Celcius, what is the value of b? Round your answers to the nearest hundredths.

Answer 1

In order to compute the regression coefficients, the following table needs to be used:

$$\def\arraystretch{1.5} \begin{array}{c:c:c:c:c:c} & X & Y & XY & X^2 & Y^2 \\ \hline & 21.4 & 1731 & 37043.4 & 457.96 & 2996361 \\ \hdashline & 22.9 & 1840 & 42136 & 524.41 & 3385600 \\ \hdashline & 24.1 & 1427 & 34390.7 & 580.81 & 2036329 \\ \hdashline & 26.1 & 1262 & 32938.2 & 681.21 & 1592644 \\ \hdashline & 29.0 & 1444 & 41876 & 841 & 2085136 \\ \hdashline & 31.1 & 1100 & 34210 & 967.21 & 1210000 \\ \hdashline Sum= & 154.6 & 8804 & 222594.3 & 4052.6 & 13306070 \\ \hdashline \end{array}$$ $$\bar{X}=\dfrac{1}{n}\sum _{i}X_i=\dfrac{154.6}{6}$$

$$=25.766667$$

$$\bar{Y}=\dfrac{1}{n}\sum _{i}Y_i=\dfrac{8804}{6}$$

$$=1467.333333$$

$$SS_{XX}=\sum_iX_i^2-\dfrac{1}{n}(\sum _{i}X_i)^2$$

$$=4052.6-\dfrac{154.6^2}{6}=69.073333$$ $$SS_{YY}=\sum_iY_i^2-\dfrac{1}{n}(\sum _{i}Y_i)^2$$

$$=13306070-\dfrac{8804^2}{6}=387667.333333$$

$$SS_{XY}=\sum_iX_iY_i-\dfrac{1}{n}(\sum _{i}X_i)(\sum _{i}Y_i)$$

$$=222594.3-\dfrac{154.6(8804)}{6}$$

$$=−4255.433333$$

$$b=\dfrac{SS_{XY}}{SS_{XX}}=\dfrac{222594.3-\dfrac{154.6(8804)}{6}}{4052.6-\dfrac{154.6^2}{6}}$$

$$=−61.61$$