Question

The owners of a shaved ice stand on the beach are trying to improve their sales forecast so that adequate staffing can be available during the summer. The owners decide to develop a simple linear regression model to predict sales based on the temperature forecast. Using the following table of historical data, determine the corresponding regression equation, and use the equation to predict sales for a day when the temperature is 88 degrees.

Sales	Temperature
120	82
150	90
160	96
130	84
70	80
165	95
175	98

If done in excel, please show steps.

Answer 1

X	Y	(x-x̅)²	(y-ȳ)²	(x-x̅)(y-ȳ)
82	120	53.08	344.90	135.31
90	150	0.51	130.61	8.16
96	160	45.08	459.18	143.88
84	130	27.94	73.47	45.31
80	70	86.22	4702.04	636.73
95	165	32.65	698.47	151.02
98	175	75.94	1327.04	317.45

	ΣX	ΣY	Σ(x-x̅)²	Σ(y-ȳ)²	Σ(x-x̅)(y-ȳ)
total sum	625	970	321.4285714	7735.7	1437.86
mean	89.29	138.57	SSxx	SSyy	SSxy

sample size ,   n =   7          
here, x̅ = Σx / n=   89.29   ,     ȳ = Σy/n =   138.57  
                  
SSxx =    Σ(x-x̅)² =    321.4286          
SSxy=   Σ(x-x̅)(y-ȳ) =   1437.9          
                  
estimated slope , ß1 = SSxy/SSxx =   1437.9   /   321.429   =   4.4733
                  
intercept,   ß0 = y̅-ß1* x̄ =   -260.8333          
                  
so, regression line is   Ŷ =   -260.833 +   4.473   *x

-----------------------------------

Predicted Y at X=   88   is                  
Ŷ =   -260.833   +   4.473   *   88   =   132.820

so, predicted sales is 132.82