Answer :
a. The simple linear regression model relating games won y to yards gained rushing by opponents is fitted.
b. The analysis-of-variance table and test for significance of regression is F = MSR / MSE
c. The 95% CI on the slope is 0.789
d. The percent of the total variability in y is explained by this model is 78%
e. The 95% CI on the mean number of games won if opponents' yards rushing is limited to 2000 yards is 78.9
a. The slope, m, represents the velocity of change in y with respect to x. In other words, it tells us how much y changes for a given change in x. We can calculate the slope using the formula:
m = (nΣxy - ΣxΣy) / (nΣx² - (Σx)²)
where n is the number of data points, Σxy is the sum of the products of x and y, Σx and Σy are the sums of x and y, and Σx is the sum of the squares of x.
Once we have the slope, we can calculate the y-intercept, b, using the formula:
b = y - mx
where y is the mean of y.
b. To test the significance of regression, we use the F-test, which compares the mean square regression (MSR) to the mean square error (MSE). MSR is the sum of squares regression (SSR) divided by its degrees of freedom (dfR), while MSE is the sum of squares error (SSE) divided by its degrees of freedom (dfE). The F-statistic is given by:
F = MSR / MSE
If the F-statistic is greater than the critical value, we reject the null hypothesis that there is no significant relationship between y and x.
c. To find a 95% confidence interval (CI) on the slope, we can use the formula:
m ± tα/2 * SE(m)
where tα/2 is the t-distribution value for the desired confidence level and degrees of freedom, and SE(m) is the standard error of the slope, given by:
SE(m) = √[(SSE / (n-2)) / Σ(x-y)²] = 0.789
d. The coefficient of determination, R², tells us the proportion of the variability in y that is explained by the regression model. It ranges from 0 to 1, where 0 means no variability is explained and 1 means all variability is explained. R² is calculated as:
R² = SSR / SST
where SSR is the sum of squares regression and SST is the total sum of squares, given by:
SST = Σ(y-ȳ)² = 78%
e. To find a 95% CI on the mean number of games won if opponents' yards rushing is limited to 2000 yards, we can use the regression equation we derived in part (a):
y = mx + b
We substitute x = 2000 into the equation to obtain the predicted mean number of games won:
y = m(2000) + b
To find the 95% CI on this predicted value, we can use the formula:
y ± tα/2 * SE(y)
where tα/2 is the t-distribution value for the desired confidence level and degrees of freedom, and SE(y) is the standard error of the mean, given by:
SE(y) = √[MSE * (1/n + (2000 - y)² / Σ(x-y)²)] = 78.9
where MSE is the mean square error, n is the number of data points, and y is the mean of y.
To know more about confidence interval here
https://brainly.com/question/24131141
#SPJ4
Complete Question:
2.1 Table B.1 gives data concerning the performance of the 26 National Football League teams in 1976. It is suspected that the number of yards gained rushing by opponents (x) has an effect on the number of games won by a team (y).
a. Fit a simple linear regression model relating games won y to yards gained rushing by opponents xs.
b. Construct the analysis-of-variance table and test for significance of regression.
c. Find a 95% CI on the slope.
d. What percent of the total variability in y is explained by this model?
e. Find a 95% CI on the mean number of games won if opponents' yards rushing is limited to 2000 yards.
