Question

Jason believes that the sales of coffee at his coffee shop depend upon the weather. He has taken a sample of 6 days. Below you are given the results of the sample.

Cups of Coffee Sold	Temperature (in ⁰F)
350	50
200	60
210	70
100	80
60	90
40	100

a. Compute the least squares estimated regression equation.

b. Is there a significant relationship between the sales of coffee and temperature? Use a t test and a .05 level of significance. Be sure to state the null and alternative hypotheses.

c. Perform F test for the model with a .01 level of significance

Answer 1

X	Y	XY	X²	Y²
350	50	17500	122500	2500
200	60	12000	40000	3600
210	70	14700	44100	4900
100	80	8000	10000	6400
60	90	5400	3600	8100
40	100	4000	1600	10000
Ʃx =	Ʃy =	Ʃxy =	Ʃx² =	Ʃy² =
960	450	61600	221800	35500

Sample size, n =	6
x̅ = Ʃx/n = 960/6 =	160
y̅ = Ʃy/n = 450/6 =	75
SSxx = Ʃx² - (Ʃx)²/n = 221800 - (960)²/6 =	68200
SSyy = Ʃy² - (Ʃy)²/n = 35500 - (450)²/6 =	1750
SSxy = Ʃxy - (Ʃx)(Ʃy)/n = 61600 - (960)(450)/6 =	-10400

a)

Slope, b = SSxy/SSxx = -10400/68200 =    -0.152492669

y-intercept, a = y̅ -b* x̅ = 75 - (-0.15249)*160 =    99.39882698

Regression equation :   

ŷ = 99.3988 + (-0.1525) x  

b)

Sum of Square error, SSE = SSyy -SSxy²/SSxx = 1750 - (-10400)²/68200 = 164.0762463

Standard error, se = √(SSE/(n-2)) = √(164.07625/(6-2)) = 6.4046

Null and alternative hypothesis:

Ho: β₁ = 0 ; Ha: β₁ ≠ 0

α = 0.05

Test statistic:

t = b/(se/√SSxx) = -6.2180

df = n-2 = 4

p-value = T.DIST.2T(ABS(-6.218), 4) = 0.0034

Conclusion:

p-value < α, Reject the null hypothesis.

There is a significant relationship between the sales of coffee and temperature.

c)

Null and alternative hypothesis:

Ho: β₁ = 0 ; Ha: β₁ ≠ 0

α = 0.01

SSE = SSyy -SSxy²/SSxx = 1750 - (-10400)²/68200 = 164.08

SSR = SSxy²/SSxx = (-10400)²/68200 = 1585.9238

df1 = 1

df2 = n-2 = 6-2 = 4

Test statistic:

F = SSR/(SSE/(n-2)) = 1585.9238/(164.0762/4) = 38.6631

p-value = F.DIST.RT(38.6631, 1, 4) = 0.0034

Conclusion:

p-value < α, Reject the null hypothesis.

The model is significant.