You are given the following results from a regression model.

Calculate the sum of squared errors (SSE).

• -35
• -5
• 5
• 35
• 46

Determine which of the following statements is/are true for a simple linear relationship,

• If [math]\epsilon = 0[/math], the 95% confidence interval is equal to the 95% prediction interval.
• The prediction interval is always at least as wide as the confidence interval.
• The prediction interval quantifies the possible range for [math]\operatorname{E}(y | x).[/math]

• I only
• II only
• III only
• I, II, and III
• The correct answer is not given by (A), (B), (C), or (D).

From an investigation of the residuals of fitting a linear regression by ordinary least squares it is clear that the spread of the residuals increases as the predicted values increase. Observed values of the dependent variable range from 0 to 100. Determine which of the following statements is/are true with regard to transforming the dependent variable to make the variance of the residuals more constant.

• Taking the logarithm of one plus the value of the dependent variable may make the variance of the residuals more constant.
• A square root transformation may make the variance of the residuals more constant.
• A logit transformation may make the variance of the residuals more constant.

• None
• I and II only
• I and III only
• II and III only
• The correct answer is not given by (A), (B), (C), or (D).

The regression model is [math]y =\beta_0 + \beta_1x + \epsilon.[/math] There are six observations.

The summary statistics are:

Calculate the least squares estimate of [math]\beta_1[/math].

• 0.1
• 0.3
• 0.5
• 0.7
• 0.9

For a simple linear regression model the sum of squares of the residuals is

and the [math]R^2[/math] statistic is 0.64.

Calculate the total sum of squares (TSS) for this model.

• 605.94
• 638.89
• 690.77
• 701.59
• 750.87

Sarah performs a regression of the return on a mutual fund (y) on four predictors plus an intercept. She uses monthly returns over 105 months. Her software calculates the F statistic for the regression as F = 20.0, but then it quits working before it calculates the value of [math]R^2[/math] . While she waits on hold with the help desk, she tries to calculate [math]R^2[/math] from the F-statistic.

Determine which of the following statements about the attempted calculation is true.

• There is insufficient information, but it could be calculated if she had the value of the residual sum of squares (RSS).
• There is insufficient information, but it could be calculated if she had the value of the total sum of squares (TSS) and RSS.
• [math]R^2 = 0.44 [/math]
• [math]R^2 = 0.56 [/math]
• [math]R^2 = 0.80 [/math]

Two actuaries are analyzing dental claims for a group of n = 100 participants. The predictor variable is sex, with 0 and 1 as possible values.

Actuary 1 uses the following regression model:

Actuary 2 uses the following regression model:

The residual sum of squares for the regression of Actuary 2 is 250,000 and the total sum of squares is 490,000.

Calculate the F-statistic to test whether the model of Actuary 2 is a significant improvement over the model of Actuary 1.

• 92
• 93
• 94
• 95
• 96

You are given the following summary statistics:

Determine the equation of the regression line, using the least squares method.

• [math]y=1.97 + 0.25x [/math]
• [math]y =0.78 + 0.59x [/math]
• [math] y = 0.57 + 0.65 xy 0.39 + 0.70 x [/math]
• [math]y = 0.39 + 0.70x [/math]
• The correct answer is not given by (A), (B), (C), or (D).

Trish runs a regression on a data set of n observations. She then calculates a 95% confidence interval [math](t, u)[/math] on [math]y[/math] for a given set of predictors. She also calculates a 95% prediction interval [math](v, w)[/math] on [math]y[/math] for the same set of predictors.

Determine which of the following must be true.

• [math]\lim_{n \rightarrow \infty} (u-t) = 0[/math]
• [math]\lim_{n \rightarrow \infty} (w-v) = 0[/math]
• [math]w-v \gt u-t[/math]

• None
• I and II only
• I and III only
• II and III only
• The correct answer is not given by (A), (B), (C), or (D).

Determine which of the following statements is NOT true about the equation

• [math]\beta_0[/math] is the expected value of [math]Y[/math] .
• [math]\beta_1 [/math] is the average increase in [math]Y[/math] associated with a one-unit increase in [math]X[/math].
• The error term, [math]\epsilon[/math] , is typically assumed to be independent of [math]X[/math].
• The equation defines the population regression line.
• The method of least squares is commonly used to estimate the coefficients [math]\beta_0[/math] and [math]\beta_1[/math].