Question

A 10-year study conducted by the American Heart Association provided data on how age, systolic blood pressure relate to the risk of strokes. Data from a portion of this study follow. Risk is interpreted as the probability (times 100) that a person will have a stroke over the next 10-year period. Find the F test statistic for the significance of the regression model (Round to 2 decimals).

Answer 1

Answer :

1) We have to find  adjusted R-squared

The adjusted R-squared is a modified version of R-squared that has been adjusted for the number of predictors in the model.

R-Squared coefficient (R^2)=	0.5370.537
Sample Size (n)=	2020
Number of Predictors (k)=	2

We need to compute the Adjusted R-Squared coefficient. Given that we know R^2R2, we can get directly the Adjusted R-Squared coefficient, using the following formula:

Adj. R^2 = 1- [ ((1- R ^2 ) ( n-1 )) / n-k-1 ]

=0.4825

Therefore, based on the sample data provided, it is found that the adjusted R-squared coefficient is Adj. R^2 = 0.4825 , which indicates that approximately 48.25% of the variation in the dependent variable is explained by this linear regression model.

2)  

SSTotal = Total Sums of Squares

SSA = Explanatory Variable A's Sums of Squares

  Also called sums of squares for the treatment, regression, or model.

SSE = Error (Residual) Sums of Squares

Also called the sums of squares for the residuals.

SSTotal = SSA + SSE.

• MSA = SSA/(J-1), which estimates the variance of the group means around the grand mean.
•  MSError = SSError/(N-J), which estimates the variation of the errors around the group means

These results are put together using a ratio to define the ANOVA F-statistic (also called the F-ratio) as

F = MSA/MSError.

Source	DF	Sums of Squares	Mean Squares	F-ratio	P-value
Variable A	J-1	SSA	MSA = SSA/(J-1)	F=MSA/MSE	Right tail of F(J-1,N-J)
Residuals	N-J	SSE	MSE =SSE/(N-J)
Total	N-1	SSTotal

Now ,

we know , R square = 0.537 , SSE = 1940.696

R square = 1- ( SSE / SST )

0.537 = 1 - ( 1940.696 / SST )

SST =4191.5680

and

SSTotal = SSA + SSE.

4191.5680 =   SSA +   1940.696

SSA = 2250.872

Now ,

  MSA = SSA/(J-1)

= 2250.872 / 2

= 1125 . 436

&

MSError = SSError/(N-J)

= 1940.696 / 17

= 114.1585

Hence ,

F   = MSA / MSError

= 1125 . 436 /  114.1585

= 9.8585

Hence F statistic for significance of regression model is 9.85

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