ISYE 6414 CORE EXAM TEST 2026 QUESTIONS AND
SOLUTIONS RATED A+
✔✔An ANOVA model with a single qualitative predicting variable containing k groups
will have k + 1 parameters to estimate. - ✔✔true
See 2.2 Estimation Method
We have to estimate the means of the k groups and the pooled variance estimator, s p
o o l e d 2.
✔✔In simple linear regression models, we lose three degrees of freedom when
estimating the variance because of the estimation of the three model parameters β 0 , β
1 , σ 2. - ✔✔false
See 1.2 Estimation Method
"The estimator for σ 2 is σ ^ 2, and is the sum of the squared residuals, divided by n -
2."
✔✔The pooled variance estimator, s p o o l e d 2, in ANOVA is synonymous with the
variance estimator, σ ^ 2, in simple linear regression because they both use mean
squared error (MSE) for their calculations. - ✔✔true
See 1.2 Estimation Method for simple linear regression
See 2.2 Estimation Method for ANOVA
The pooled variance estimator is, in fact, the variance estimator.
✔✔The normality assumption states that the response variable is normally distributed. -
✔✔false
See 1.8 Diagnostics
"Normality assumption: the error terms are normally distributed."
The response may or may not be normally distributed, but the error terms are assumed
to be normally distributed.
✔✔If the constant variance assumption in ANOVA does not hold, the inference on the
equality of the means will not be reliable. - ✔✔true
See 2.8 Data Example
"This is important since without a good fit, we cannot rely on the statistical inference."
Only when the model is a good fit, i.e. all model assumptions hold, can we rely on the
statistical inference.
✔✔A negative value of β 1 is consistent with an inverse relationship between the
predictor variable and the response variable. - ✔✔true
,See 1.2 Estimation Method
"A negative value of β 1 is consistent with an inverse relationship"
✔✔The p-value is a measure of the probability of rejecting the null hypothesis. -
✔✔false
See 1.5 Statistical Inference Data Example
"p-value is a measure of how rejectable the null hypothesis is... It's not the probability of
rejecting the null hypothesis, nor is it the probability that the null hypothesis is true."
✔✔We assess the constant variance assumption by plotting the error terms, ϵ i, against
fitted values. - ✔✔false
See 1.2 Estimation Method
"We use ϵ ^ i as proxies for the deviances or the error terms. We don't have the
deviances because we don't have β 0 and β 1.
✔✔With the Box-Cox transformation, when λ = 0 we do not transform the response. -
✔✔false
See 1.8 Diagnostics
When λ = 0, we transform using the normal log.
✔✔The sampling distribution of β ^ 0 is a
t-distribution
chi-squared distribution
normal distribution
None of the above - ✔✔t-distribution
See 1.4 Statistical Inference
The distribution of β 0 is normal. Since we are using a sample and not the full
population, the sampling distribution of β ^ 0 is the t-distribution.
✔✔A data point far from the mean of the x's and y's is always:
an influential point and an outlier
a leverage point but not an outlier
an outlier and a leverage point
an outlier but not a leverage point
None of the above - ✔✔None of the above
See 1.9 Outliers and Model Evaluation
, We only know that the data point is far from the mean of x's and y's. It only fits the
definition of a leverage point because the only information we know is that it is far from
the mean of the x's. So you can eliminate the answers that do not include a leverage
point. That leaves us with remaining possibilities, "a leverage point but not an outlier"
and "an outlier and a leverage point" , both of which we can eliminate. We do not have
enough information to know if it is or is not an outlier . None of the answers above fit the
criteria of it being always being a leverage point.
✔✔The following output was captured from the summary output of a simple linear
regression model that relates the duration of an eruption with the waiting time since the
previous eruption.
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -1.374016 A -1.70 0.045141 *
waiting 0.043714 0.011098 B 0.000052 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.4965 on 270 degrees of freedom
Multiple R-squared: 0.8115, Adjusted R-squared: 0.8108
F-statistic: 1162 on 1 and 270 DF, p-value: < 2.2e-16
Using the table above, what is the standard error of the intercept, labeled A, and
rounded to three decimal places?
2.336
0.808
0.806
-0.806
None of the above - ✔✔0.808
See 1.4 Statistical Inference
Std.Err = Estimate /t-value = -1.374016/-1.70 = 0.808
✔✔The following output was captured from the summary output of a simple linear
regression model that relates the duration of an eruption with the waiting time since the
previous eruption.
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -1.374016 A -1.70 0.045141 *
waiting 0.043714 0.011098 B 0.000052 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
SOLUTIONS RATED A+
✔✔An ANOVA model with a single qualitative predicting variable containing k groups
will have k + 1 parameters to estimate. - ✔✔true
See 2.2 Estimation Method
We have to estimate the means of the k groups and the pooled variance estimator, s p
o o l e d 2.
✔✔In simple linear regression models, we lose three degrees of freedom when
estimating the variance because of the estimation of the three model parameters β 0 , β
1 , σ 2. - ✔✔false
See 1.2 Estimation Method
"The estimator for σ 2 is σ ^ 2, and is the sum of the squared residuals, divided by n -
2."
✔✔The pooled variance estimator, s p o o l e d 2, in ANOVA is synonymous with the
variance estimator, σ ^ 2, in simple linear regression because they both use mean
squared error (MSE) for their calculations. - ✔✔true
See 1.2 Estimation Method for simple linear regression
See 2.2 Estimation Method for ANOVA
The pooled variance estimator is, in fact, the variance estimator.
✔✔The normality assumption states that the response variable is normally distributed. -
✔✔false
See 1.8 Diagnostics
"Normality assumption: the error terms are normally distributed."
The response may or may not be normally distributed, but the error terms are assumed
to be normally distributed.
✔✔If the constant variance assumption in ANOVA does not hold, the inference on the
equality of the means will not be reliable. - ✔✔true
See 2.8 Data Example
"This is important since without a good fit, we cannot rely on the statistical inference."
Only when the model is a good fit, i.e. all model assumptions hold, can we rely on the
statistical inference.
✔✔A negative value of β 1 is consistent with an inverse relationship between the
predictor variable and the response variable. - ✔✔true
,See 1.2 Estimation Method
"A negative value of β 1 is consistent with an inverse relationship"
✔✔The p-value is a measure of the probability of rejecting the null hypothesis. -
✔✔false
See 1.5 Statistical Inference Data Example
"p-value is a measure of how rejectable the null hypothesis is... It's not the probability of
rejecting the null hypothesis, nor is it the probability that the null hypothesis is true."
✔✔We assess the constant variance assumption by plotting the error terms, ϵ i, against
fitted values. - ✔✔false
See 1.2 Estimation Method
"We use ϵ ^ i as proxies for the deviances or the error terms. We don't have the
deviances because we don't have β 0 and β 1.
✔✔With the Box-Cox transformation, when λ = 0 we do not transform the response. -
✔✔false
See 1.8 Diagnostics
When λ = 0, we transform using the normal log.
✔✔The sampling distribution of β ^ 0 is a
t-distribution
chi-squared distribution
normal distribution
None of the above - ✔✔t-distribution
See 1.4 Statistical Inference
The distribution of β 0 is normal. Since we are using a sample and not the full
population, the sampling distribution of β ^ 0 is the t-distribution.
✔✔A data point far from the mean of the x's and y's is always:
an influential point and an outlier
a leverage point but not an outlier
an outlier and a leverage point
an outlier but not a leverage point
None of the above - ✔✔None of the above
See 1.9 Outliers and Model Evaluation
, We only know that the data point is far from the mean of x's and y's. It only fits the
definition of a leverage point because the only information we know is that it is far from
the mean of the x's. So you can eliminate the answers that do not include a leverage
point. That leaves us with remaining possibilities, "a leverage point but not an outlier"
and "an outlier and a leverage point" , both of which we can eliminate. We do not have
enough information to know if it is or is not an outlier . None of the answers above fit the
criteria of it being always being a leverage point.
✔✔The following output was captured from the summary output of a simple linear
regression model that relates the duration of an eruption with the waiting time since the
previous eruption.
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -1.374016 A -1.70 0.045141 *
waiting 0.043714 0.011098 B 0.000052 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.4965 on 270 degrees of freedom
Multiple R-squared: 0.8115, Adjusted R-squared: 0.8108
F-statistic: 1162 on 1 and 270 DF, p-value: < 2.2e-16
Using the table above, what is the standard error of the intercept, labeled A, and
rounded to three decimal places?
2.336
0.808
0.806
-0.806
None of the above - ✔✔0.808
See 1.4 Statistical Inference
Std.Err = Estimate /t-value = -1.374016/-1.70 = 0.808
✔✔The following output was captured from the summary output of a simple linear
regression model that relates the duration of an eruption with the waiting time since the
previous eruption.
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -1.374016 A -1.70 0.045141 *
waiting 0.043714 0.011098 B 0.000052 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
