Answer:
Given that,
We have a dataset with n=10 pairs of observations (xi, yi), and
,
,
,
, and
What is an approximate 95% confidence interval for the mean response at x0=90:
Let us define the terms as follows:
Now,
=683/10
=68.3
= 68.3
And,
=813/10
=81.3
=81.3
Therefore,
=47405-(10)(68.3)^2
Sxx=756.1
=66731-(10)(81.3)^2
Syy=634.1
=56089-(10)(68.3)(81.3)
Sxy=561.1
Now the regression equation as follows:
Where,
=561.1/756.1
=0.742097606
=0.7421 (Approximately)
=81.3-(0.742097606)(62.3)
=30.6147335
=30.6147(Approximately)
Therefore,
y=30.6147+0.7421x
The variance of the error term ():
=217.7090332/8
=27.21362915
=27.2136
Let be the mean response when X0=90,
=30.6147335+(0.742097606)(90)
=97.40351804
Therefore,
=4.435051786
=4.4351
The mean response ,
The 95% CI:
=1-CI
CI=0.95
= 1-0.95
=0.05
Then,
=97.40351804 10.22722942
=[87.17628861, 107.6307475]
=[87.1763, 107.6308]
Hence, the 95% confidence interval for the mean response at X0=90 is,
=[87.1763, 107.6308].
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