Answer to Question #344004 in Statistics and Probability for wiwili

Question

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

Sales in Php

1690

1786

1391

1555

1111

1423

Temp in Celsius

21.1

24

25.4

26.9

29.6

30.3

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 a? Round your answers to the nearest hundredths.

Answer 1

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

$$\def\arraystretch{1.5} \begin{array}{c:c:c:c:c:c} & X & Y & XY & X^2 & Y^2 \\ \hline & 21.1 & 1690 & 35659 & 445.21 & 2856100 \\ \hdashline & 24.0 & 1786 & 42864 & 576 & 3189796 \\ \hdashline & 25.4 & 1391 & 35331.4 & 645.16 & 1934881 \\ \hdashline & 26.9 & 1555 & 41829.5 & 723.61 & 2418025 \\ \hdashline & 29.6 & 1111 & 32885.6 & 876.16 & 1234321 \\ \hdashline & 30.3 & 1423 & 43116.9 & 918.09 & 2024929 \\ \hdashline Sum= & 157.3 & 8956 & 231686.4 & 4184.23 & 13658052 \\ \hdashline \end{array}$$

$$\bar{X}=\dfrac{1}{n}\sum _{i}X_i=\dfrac{157.3}{6}$$

$$=26.216667$$

$$\bar{Y}=\dfrac{1}{n}\sum _{i}Y_i=\dfrac{8956}{6}$$

$$=1492.666667$$

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

$$=4184.23-\dfrac{157.3^2}{6}=60.348333$$

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

$$=13658052-\dfrac{8956^2}{6}=289729.333333$$

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

$$=231686.4-\dfrac{157.3(8956)}{6}=−3110.066667$$

$$b=\dfrac{SS_{XY}}{SS_{XX}}=\dfrac{231686.4-\dfrac{157.3(8956)}{6}}{4184.23-\dfrac{157.3^2}{6}}$$

$$=-51.53$$

$$a=\bar{Y}-b\bar{X}=2843.75$$