Question

The following output summarizes the results of fitting a least-squares regression to simulated data. Because we constructed these data using a computer, we know the SRM holds and we know the parameters of the model. We chose Beta0 = 7, Beta1 = 0.5, and se = 1.5. The fit is to a sample of n = 50 cases.

 

• If Beta1 = ½ in the population, then why isn’t b1 = ½?
• Do the 95% confidence intervals for Beta0 and Beta1 contain Beta0 and Beta1 in this example?
• What’s going to change in this summary if we increase the sample size from 50 cases to 5,000 cases?

 

Answer 1

a) The values of $\beta_1$ and b₁ are not equal because b₁ is only an estimate of the population slope that will change due to sampling variation.

b) Degree of freedom = n-2 = 50-2 = 48

Critical value of t at 95% confidence interval and df = 48 is 2.01

95% confidence interval of \beta_0 is

$(6.993459 -2.01 \times 0.181933, 6.993459 + 2.01 \times 0.181933) \\ (6.993459 -0.365685, 6.993459 + 0.365685) \\ (6.627774, 7.359144)$

The 95% confidence interval for $\beta_0$ contains the value of $\beta_0$

95% confidence interval of $\beta_1$ is

$(0.5134397 -2.01 \times 0.029887, 0.5134397 + 2.01 \times 0.029887) \\ ( 0.5134397 -0.060072, 0.5134397 +0.060072) \\ (0.4533677, 0.5735117)$

The 95% confidence interval for $\beta_1$ contains the value of $\beta_1$

c) If we increase the sample size, the model will overfit the data.

SSError will decrease and SSRregression will increase

So, the value of $r^2$ will increase.

The value of se will decrease.

The values of standard errors will decrease.