Consider the following 10 observations on the response variable y and the explan

Question

Consider the following 10 observations on the response variable y and the explanatory variable x.

Estimate the linear regression model, y = 0 + 1x + . (Round your answers to 4 decimal places.)

Estimate the quadratic regression model, y = 0 + 1x + 2x2 + . (Negative values should be indicated by a minus sign. Round your answers to 4 decimal places.)

Use the appropriate numerical measure to justify which model fits the data best.

Given the best-fitting model, predict y for x = 4, 8, and 12. (Round intermediate coefficient values to 4 decimal places and final answers to 2 decimal places.)

Find x at which the quadratic equation reaches a minimum or maximum. (Round intermediate coefficient values to 4 decimal places and final answer to 2 decimal places.)

The (max or min) of the quadratic equation is achieved at x = ___ units.

y 13.82 19.06 16.67 13.30 11.77 13.64 18.30 20.78 13.02 16.13 x 6 6 5 3 3 12 10 8 5 11

Explanation / Answer

From Excel

a) linear regression model, y = 0 + 1x +

The regression model is y hat=13.3087+0.3391(x)

a 2) quadratic regression model, y = 0 + 1x + 2x2 + .

From excel

y = 1.7656 + 4.0966x - 0.2528x2

b)

The linear regression model, y = 0 + 1x +

and R² = 0.1317

quadratic regression model, y = 0 + 1x + 2x2 + .

R² = 0.5844

From above R2, quadratic regression model R2 is high hence quadratic regression model is best fits for the given data.

C) Given the best-fitting model, predict y for x = 4, 8, and 12.

The fitted regression model is

y = 1.7656 + 4.0966x - 0.2528x2

when x=4

y = 1.7656 + 4.0966x - 0.2528x2

y= 14.1072

when x=8

y = 1.7656 + 4.0966x - 0.2528x2

y=18.3592

when x=12

y = 1.7656 + 4.0966x - 0.2528x2

y=14.5216

SUMMARY OUTPUT Regression Statistics Multiple R 0.362961 R Square 0.131741 Adjusted R Square 0.023208 Standard Error 2.967199 Observations 10 ANOVA df SS MS F Significance F Regression 1 10.68694 10.68694 1.213837 0.302613 Residual 8 70.43415 8.804268 Total 9 81.12109 Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0% Intercept 13.30872 2.322175 5.731144 0.000438 7.953774 18.66366 7.953774 18.66366 x 0.339171 0.30785 1.101743 0.302613 -0.37073 1.049074 -0.37073 1.049074

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