Question

Consider the set of ordered pairs shown below. Assuming that the regression equation is y = 4.277 +0.085x and the SSE = 5.064, construct a 95% confidence interval for x = 5.

Calculate the upper and lower limits of the confidence interval. 

UCL = _______  

LCL=  _______  

(Round to three decimal places as needed)

Answer 1

ΣX = 19
ΣY = 23
ΣX * Y = 89
ΣX2 = 91
ΣY2 = 111

Standard Error of Estimate S = √ ( Σ (Y - Ŷ ) / n - 2) = √(5.0638 / 3) = 1.299

Sxx =Σ (Xi - X̅ )2 =    18.8
Syy = Σ( Yi - Y̅ )2 =    5.2
Sxy = Σ (Xi - X̅ ) * (Yi - Y̅) =    6.88
X̅ = Σ( Xi / n ) = 19/5 = 3.8
Y̅ = Σ( Yi / n ) = 23/5 = 4.6

Confidence Interval of $\mu_{Y|X = 5}$
Ŷ = 4.2766 + 0.0851 X
Ŷ = 4.702
$se[\hat{\mu_{0}}] \pm t_{\alpha/2} \times \sqrt{[\frac{1}{n} + \frac{(x_{0} - \bar{x})^2}{S_{xx}}]\times \hat{\sigma}^2}$
t(α/2) = t(0.05/2) =3.182

95% confidence interval is ( 2.528 < $\mu_{Y|X = 5}$ < 6.877 )