Question

2. A baseball analyst wants to determine which variables are important in predicting a team's wins in a given season. He has collected data related to wins, earned run average (ERA), runs scored, and errors. Analyze the data below: y = #wins X1 = ERA X2 = runs scored X3 = #errors bo = 84 b1 = -18 b2 = 1 b3 = -2 Sb1 = 2.1 Sb2 = .04 Sb3 = 1.1 SSR = 1500 SSE = 646 n = 30 a) Given the information above, state the multiple regression equation. b) Given R2 = SSR/SST, calculate and interpret R2 adjusted. c) Interpret the slope for errors (bs). d) Perform an F-test to determine if there is a significant relationship between the number of wins and the three independent variables. Alpha= .05. e) Perform three t-tests to determine whether each independent variable makes a significant contribution to the regression model. Alpha = .05. f) State which variable(s) contribute to the model and should be left in the model.

Answer 1

a)

The multiple linear regression equation is defined as,

$\widehat{y}=b_0+b_1x_1+b_2x_2+b_3x_3$

given:

b0 = 84, b1 = -18, b2 = 0.1, b3 = -2

hence the estimate regression equation is,

$\widehat{y}=84-18x_1+0.1x_2-2x_3$

b)

The R square value is,

$R^2=\frac{SSR}{SST}=\frac{SSR}{SSR+SSE}=\frac{1500}{1500+646}=0.6990$

The adjusted R square value is obtained using the following formula,

$\text{Adjusted }R^2=1-\left [\frac{(1-R^2)(n-1)}{(n-k-1)} \right ]$

where n = total number of observations, k - total numbers of predictors variables

$\text{Adjusted }R^2=1-\left [\frac{(1-0.6990)(30-1)}{(30-3-1)} \right ]$

$\text{Adjusted }R^2=0.6642$

The adjusted R square value is used to determine whether the added variable in the model is significant or not such that its value increases if the added variable is significant otherwise decreases.

c)

The slope coefficient of b3 = -2

Interpretation: For an increase of one error, the number of the team's wins decreases by 2.

d)

F Test

Hypothesis

The Null and Alternative Hypotheses are defined as,

$H_0:\beta_1=\beta_2=\beta_3=0$

$H_a:\text{ at least one variable is significant}$

F statistic

The F statistic is obtained using the following formula,

$F=\frac{R^2/(k-1)}{(1-R^2)/(n-k)}$

where n = total number of observations, k = total numbers of predictors variables (including the intercept)

$F=\frac{0.6990/(4-1)}{(1-0.6990)/(30-4)}$

$F=20.1238$

P-value

The p-value is obtained from the F distribution table for F = 20.1238, numerator degree of freedom = 3, and denominator degree of freedom = 26

$\text{P-value}=0.0000$

Conclusion:

Since the p-value is less than 0.05 at a 5% significance level, the null hypothesis is rejected. hence there is sufficient evidence to conclude that at least one predictor variable significantly fit the regression model.

e)

Hypothesis

The Null and Alternative Hypotheses are defined as,

$H_0:\beta=0$

$H_a:\beta\neq0$

T statistic

The t statistic is obtained using the following formula,

$t=\frac{\text{Slope Coefficient}}{\text{Standard Error}}$

P-value

The p-value is obtained from the t distribution table for the obtained t statistic for the degree of freedom = n-1=29

From the data values,

predictor variable	Estimate	Standard Error	t statistic	P-value
x1	-18	2.1	-8.5714	0.00000
x2	0.1	0.04	2.5	0.00916
x3	-2	1.1	-1.8182	0.03969

Conclusion:

predictor variable	P-value		Significance level	Whether significant? (Y/N)
x1	0.0000	<	0.05	Y
x2	0.0092	<	0.05	Y
x3	0.0397	<	0.05	Y

All the predictor variables are significant in the model.

f)

Since all the variables are significant in the model, no variable needs to be removed from the model.