A Complete Example

This section explains linear regression, from presenting the data to using scatter plots to identify the linear pattern. It then fits a linear model using least squares estimation and addresses statistical inferences on correlation coefficient and slope parameter.



Σx=14.2, Σy=49.6, Σxy=91.73, Σx^2=26.3, Σy^2=333.86.

SS_{xx}=6.136, SS_{xy}=21.298, SS_{yy}=87.844.

 \overline x =1.42,  \overline y =4.96.

\hat β_1=3.47, \hat β_0=0.03.



r = 0.9174, r^2 = 0.8416.

df=8, T = 6.518.

The 95% confidence interval for β_1 is: (2.24,4.70).

At x_p=2, the 95% confidence interval for E(y) is (5.77,8.17).

At x_p=2, the 95% prediction interval for y is (3.73,10.21).

3. The positively correlated trend seems less profound than that in each of the previous plots.

5. The regression line: \hat y=3.3426x+138.7692. Coefficient of Correlation: r = 0.9431. Coefficient of Determination: r^2 = 0.8894. SSE=283.2473. s_e=1.9305. A 95% confidence interval for β_1: (3-.0733,3.6120). Test Statistic for H_0:β_1=0: T = 24.7209. At x_p=10, \hat y=172.1956; a 95% confidence interval for the mean value of y is: (171.5577,172.8335); and a 95% prediction interval for an individual value of y is: (168.2974,176.0938).