Question 1

What criterion do we use to decide on the "least squares regression" line through data?

Accepted Answer

The least squares regression line is chosen to minimize the sum of the squared vertical distances (deviations in the Y direction) between the data points and the line. This method ensures the best fit line in terms of minimizing these squared differences.

Question 2

Which of the following equations describes a line that passes through the point (15,51)?

Accepted Answer

Substitute x = 15 into each equation and check if y = 51. Only Y = 6 + 3X satisfies this condition: 51 = 6 + 3(15).

Question 3

Which of the following equations describes a line that passes through the point (4,12)?

Accepted Answer

To find the correct equation, substitute the x and y values from the point (4,12) into each equation and see which one holds true. For option A, substituting x = 4 and y = 12 gives 12 = 20 - 2(4), which simplifies to 12 = 12, making it the correct equation.

Question 4

Which of the following equation defines the line with the largest slope and Y-intercept?

Accepted Answer

The slope-intercept form of a line is Y = mX + b, where m is the slope and b is the Y-intercept. The line with the largest slope and Y-intercept is Y = 15 + 9X, which has the highest values for both the slope (9) and Y-intercept (15).

Question 5

If the linear least squares equation for a data set is Y = 12 - (0.7)X, what is the residual corresponding to a data point with a value of (8, 8)?

Accepted Answer

The residual is the difference between the observed value and the value predicted by the model. For the data point (8, 8), the predicted value using the equation Y = 12 - (0.7)X is Y = 12 - (0.7)*8 = 12 - 5.6 = 6.4. The observed value is 8, so the residual is 8 - 6.4 = 1.6.

Question 6

If the linear least squares equation for a data set is Y = 2 + (0.4)X, what is the residual corresponding to a data point with a value of (9, 4)?

Accepted Answer

The predicted Y value for X=9 is Y = 2 + (0.4)(9) = 5.6. The residual is the actual Y value minus the predicted Y value: 4 - 5.6 = -1.6.

Question 7

If the slope is -1.64, the residual mean square is 3.6, the sum of squares for X is 2.5, and the sample size is 16, what is the standard error of the slope?

Accepted Answer

The answer of If the slope is -1.64, the residual...

Question 8

If the slope is -1.64, the residual mean square is 3.6, the sum of squares for X is 2.5, and the sample size is 16, what is the 95% confidence interval of the slope?

Accepted Answer

The correct answer is A because the 95% confidence interval of the slope is -4.214 to 0.934.

Question 9

Which of the following is true for 95% confidence bands?

Accepted Answer

95% confidence bands are constructed so that if we were to take many samples and construct confidence bands for each, 95% of those bands would contain the true population regression line. This is a statement about the procedure's long-term performance, not about the data or the regression line from a single sample.

Question 10

Consider the partially completed ANOVA table showing the results of a regression analysis shown.
​
 $\begin{array}{|lcrcl|}
\hline \text { Source of } & {\text { Sum of }} & {\text { Mean }} & \
\text { variation } & \text { squares } & \text { df } & \text { squares } & \text { F-ratio } \
\text { Regression } & 305.1 & 1 & ? & ? \
\text { Residual } & 1247.3 & 24 & ? & \
\text { Total } & 1552.4 & 25 & & \
\hline
\end{array}$
-What is the F-ratio?
?

Accepted Answer

The F-ratio is calculated as the Mean Square Regression divided by the Mean Square Residual. Mean Square Regression = 305.1 / 1 = 305.1. Mean Square Residual = 1247.3 / 24 = 51.97. F-ratio = 305.1 / 51.97 ≈ 5.871.

Question 11

Consider the partially completed ANOVA table showing the results of a regression analysis shown. $\begin{array}{|lcrcl|} \hline \text { Source of } & {\text { Sum of }} & {\text { Mean }} & \ \text { variation } & \text { squares } & \text { df } & \text { squares } & \text { F-ratio } \ \text { Regression } & 305.1 & 1 & ? & ? \ \text { Residual } & 1247.3 & 24 & ? & \ \text { Total } & 1552.4 & 25 & & \ \hline \end{array}$ -What is the R² value? ?

Accepted Answer

The answer of Consider the partially completed ANOVA table showing...

Question 12

Consider the partially completed ANOVA table showing the results of a regression analysis shown.
​
 $\begin{array}{|lcrcl|}
\hline \text { Source of } & {\text { Sum of }} & {\text { Mean }} & \
\text { variation } & \text { squares } & \text { df } & \text { squares } & \text { F-ratio } \
\text { Regression } & 305.1 & 1 & ? & ? \
\text { Residual } & 1247.3 & 24 & ? & \
\text { Total } & 1552.4 & 25 & & \
\hline
\end{array}$
-What is a weakness of the ANOVA method compared to the t-test method when performing a significance test on the slope of a regression?

Accepted Answer

ANOVA tests the null hypothesis that the means of two or more groups are equal. It does not directly test against null hypothesis slopes different from 0, which is a specific capability of t-tests in the context of regression analysis. This limitation can be seen as a weakness when the objective is to assess the significance of the slope of a regression line.

Question 13

Consider the partially completed ANOVA table showing the results of a regression analysis shown.
​
 $\begin{array}{|lcrcl|}
\hline \text { Source of } & {\text { Sum of }} & {\text { Mean }} & \
\text { variation } & \text { squares } & \text { df } & \text { squares } & \text { F-ratio } \
\text { Regression } & 305.1 & 1 & ? & ? \
\text { Residual } & 1247.3 & 24 & ? & \
\text { Total } & 1552.4 & 25 & & \
\hline
\end{array}$
-If the slope is -1.64, the residual mean square is 3.6, the sum of squares for X is 2.5, and the sample size is 16, what is the t-value we obtain when doing a t-test using a hypothesized slope of zero?

Accepted Answer

The answer of Consider the partially completed ANOVA table showing...

Question 14

Consider the partially completed ANOVA table showing the results of a regression analysis shown.
​
 $\begin{array}{|lcrcl|}
\hline \text { Source of } & {\text { Sum of }} & {\text { Mean }} & \
\text { variation } & \text { squares } & \text { df } & \text { squares } & \text { F-ratio } \
\text { Regression } & 305.1 & 1 & ? & ? \
\text { Residual } & 1247.3 & 24 & ? & \
\text { Total } & 1552.4 & 25 & & \
\hline
\end{array}$
-If the slope is -1.64, the residual mean square is 3.6, the sum of squares for X is 2.5, and the sample size is 16, based on the t-test statistic using a hypothesized slope of zero, and using your table of critical t-values, which of the following P-value ranges matches the one for the t-value?

Accepted Answer

The answer of Consider the partially completed ANOVA table showing...

Question 15

Consider the partially completed ANOVA table showing the results of a regression analysis shown.
​
 $\begin{array}{|lcrcl|}
\hline \text { Source of } & {\text { Sum of }} & {\text { Mean }} & \
\text { variation } & \text { squares } & \text { df } & \text { squares } & \text { F-ratio } \
\text { Regression } & 305.1 & 1 & ? & ? \
\text { Residual } & 1247.3 & 24 & ? & \
\text { Total } & 1552.4 & 25 & & \
\hline
\end{array}$
-If the slope is -1.64, the residual mean square is 3.6, the sum of squares for X is 2.5, and the sample size is 16, what is the t-value we obtain when doing a t-test using a hypothesized slope of 1.0?

Accepted Answer

The t-value for testing a hypothesized slope (β1) can be calculated using the formula: $t = \frac{\hat{\beta}_1 - \beta_{1,0}}{SE(\hat{\beta}_1)}$, where $\hat{\beta}_1$ is the estimated slope, $\beta_{1,0}$ is the hypothesized slope, and $SE(\hat{\beta}_1)$ is the standard error of the estimated slope. Given $\hat{\beta}_1 = -1.64$, $\beta_{1,0} = 1.0$, and the residual mean square (MSE) is 3.6, we can find $SE(\hat{\beta}_1)$ using the formula $SE(\hat{\beta}_1) = \sqrt{\frac{MSE}{SSX}}$, where SSX is the sum of squares for X. Given SSX = 2.5, $SE(\hat{\beta}_1) = \sqrt{\frac{3.6}{2.5}} = \sqrt{1.44} = 1.2$. Thus, $t = \frac{-1.64 - 1.0}{1.2} = \frac{-2.64}{1.2} = -2.2$.

Question 16

Consider the partially completed ANOVA table showing the results of a regression analysis shown.
​
 $\begin{array}{|lcrcl|}
\hline \text { Source of } & {\text { Sum of }} & {\text { Mean }} & \
\text { variation } & \text { squares } & \text { df } & \text { squares } & \text { F-ratio } \
\text { Regression } & 305.1 & 1 & ? & ? \
\text { Residual } & 1247.3 & 24 & ? & \
\text { Total } & 1552.4 & 25 & & \
\hline
\end{array}$
-If the slope is -1.64, the residual mean square is 3.6, the sum of squares for X is 2.5, and the sample size is 16, based on the t-test statistic using a hypothesized slope of 1.0, and using your table of critical t-values, which of the following P-value ranges matches the one for the t-value?

Accepted Answer

The answer of Consider the partially completed ANOVA table showing...

Question 17

Which of the following is not an assumption of linear regression?

Accepted Answer

The assumption C, stating that the number of Y-values is constant across the range of X-values, is not an assumption of linear regression. Linear regression assumptions typically concern the distribution, mean, and variance of Y-values in relation to X-values, not the count of Y-values.

Question 18

If the assumptions of linear regression are met, then a residual plot will have all of the following features except which one?

Accepted Answer

The distribution of X-values does not need to be symmetric for the assumptions of linear regression to be met. The key assumptions regarding residuals are about their distribution around the regression line (homoscedasticity and no clear patterns), not the distribution of the independent variable (X-values).

Question 19

What is the effect of measurement error in the Y-values?

Accepted Answer

The answer of What is the effect of measurement error...

Question 20

The Michaelis-Menten equation is used to model what kind of nonlinear pattern in a data set?

Accepted Answer

The Michaelis-Menten equation models an asymptotic increase, describing how the rate of enzymatic reactions increases with substrate concentration and approaches a maximum rate asymptotically.

Quiz 17: Regression

What criterion do we use to decide on the "least squares regression" line through data?

Which of the following equations describes a line that passes through the point (15,51) ?

Which of the following equations describes a line that passes through the point (4,12) ?

Which of the following equation defines the line with the largest slope and Y-intercept?

If the linear least squares equation for a data set is Y = 12 - (0.7) X, what is the residual corresponding to a data point with a value of (8, 8) ?

If the linear least squares equation for a data set is Y = 2 + (0.4) X, what is the residual corresponding to a data point with a value of (9, 4) ?

If the slope is -1.64, the residual mean square is 3.6, the sum of squares for X is 2.5, and the sample size is 16, what is the standard error of the slope?

If the slope is -1.64, the residual mean square is 3.6, the sum of squares for X is 2.5, and the sample size is 16, what is the 95% confidence interval of the slope?

Which of the following is true for 95% confidence bands?

Which of the following is not an assumption of linear regression?

If the assumptions of linear regression are met, then a residual plot will have all of the following features except which one?

What is the effect of measurement error in the Y-values?

The Michaelis-Menten equation is used to model what kind of nonlinear pattern in a data set?