Question 1

If one was estimating a simple regression of Earnings (Y) on Height of individuals (X), and got a coefficient on the Height variable of 30, what would the coefficient on Height be if you added 3 inches to every individual in the sample but kept their earnings the same?
A) 30
B) Above 30
C) Below 30
D) Not enough information.

Accepted Answer

The coefficient on Height would still be 30 because a linear regression model is constant across all units of measurement. Adding 3 inches to every individual would simply shift the data points horizontally without changing the slope or coefficient of the regression line.

Question 2

For the simple linear regression, Y = b + mX, which variable is considered the intercept?
A) Y
B) b
C) m
D) X

Accepted Answer

In the equation Y = b + mX, the variable "b" represents the intercept or the value of Y when X is equal to zero.

Question 3

In a dichotomous regression all of the following conditions must hold except for what?
A) The sum of residuals for the treated group must equal zero.
B) The sum of residuals for the untreated group must equal zero.
C) An equal number of positive and negative residuals.
D) The regression line must go through the mean of the outcomes for the treated.

Accepted Answer

In a dichotomous regression, it is not a requirement for there to be an equal number of positive and negative residuals. The distribution of residuals can vary, and their balance is not a condition for the model's validity.

Question 4

How many regression parameters will be estimated in a simple linear regression model?
A) 1
B) 2
C) 3
D) 1 + number of observations

Accepted Answer

In a simple linear regression model, which is represented by the equation y = mx + b, there are two parameters to be estimated: the slope (m) and the intercept (b).

Question 5

If we wish to use a regression line to determine the effect of multiple treatment levels , why can't we just plot the average outcome for each treatment level and "connect the dots?"
A) Connecting the dots generally will not form a line.
B) Using averages for each treatment level will generally create bias.
C) This will cause us to have too few equations to solve for our unknown parameters.
D) We cannot use a regression line to measure the effects of multiple treatments.

Accepted Answer

Connecting the dots formed by plotting the average outcome for each treatment level does not guarantee a linear relationship, which is a fundamental assumption in linear regression analysis. This method oversimplifies the data and ignores the variability within each treatment level, potentially leading to incorrect conclusions about the relationship between variables.

Question 6

If one was estimating a simple regression of Earnings (Y) on Height of individuals (X), and got a coefficient on the Height variable of 30, what would the intercept be if you added 3 inches to every individual in the sample but kept their earnings the same?
A) 30
B) Above 30, but not enough information to tell exactly.
C) Below 30, but not enough information to tell exactly.
D) None of these choices are correct.

Accepted Answer

The intercept in a regression equation represents the expected value of the dependent variable (Y, which is Earnings in this case) when all independent variables (X, which is Height in this case) are equal to zero. Adding 3 inches to every individual's height without changing their earnings would shift the regression line horizontally, not vertically. This means the slope (coefficient on the Height variable) would remain the same, but the intercept would change to reflect the new relationship between height and earnings. However, without knowing the original intercept or the specific distribution of heights and earnings, we cannot determine the exact new value of the intercept, only that it would change to maintain the same earnings at the new heights. Therefore, the correct answer is that none of the provided choices accurately describe the change in the intercept.

Question 7

The difference between the observed outcome and the corresponding point on the regression line for a given observation is a:
A) regression line prediction.
B) heteroskedasticity.
C) residual.
D) mean squared error.

Accepted Answer

The difference between the observed outcome and the corresponding point on the regression line is the residual, which is the measure of the distance between the observed values and the values predicted by the regression model.

Question 8

The process of using a function to describe the relationship among variables is known as:
A) regression analysis.
B) big data.
C) business analytics.
D) cross validation.

Accepted Answer

Regression analysis is a statistical method used to examine the relationship between one or more independent variables and a dependent variable. It involves using a function or equation to describe the relationship among the variables, and helps to predict the value of the dependent variable based on the values of the independent variables. Big data, business analytics, and cross validation are related concepts but not specific techniques for describing relationships among variables.

Question 9

A treatment that only undergoes two statuses - treated and untreated-is known as this kind of treatment?
A) Biased
B) Random assignment
C) Dichotomous
D) Attenuated

Accepted Answer

A dichotomous treatment is a treatment that only has two statuses, treated and untreated.

Question 10

In the dichotomous regression if one was to replace the regression line prediction of the outcome means (for treated/untreated) with medians, which of the following conditions would hold?
A) The sum of all the residuals would equal zero.
B) The sum of the residuals for the treated observations would equal zero.
C) The slope of the regression line would be positive.
D) The number of strictly positive residuals would equal the number of strictly negative residuals.

Accepted Answer

When using medians in regression analysis, the method aims to minimize the absolute deviations rather than the squared deviations. This leads to a situation where the number of positive residuals is equal to the number of negative residuals, ensuring that the median is the central point of the distribution of residuals. This property does not necessarily ensure that the sum of residuals equals zero (as in mean regression), nor does it directly affect the slope of the regression line or guarantee that the sum of residuals for a specific group (treated or untreated) equals zero.

Question 11

For a dichotomous treatment regression (X = 1 or 0), the mean outcome for the treated group (X = 1) is 35 and the mean outcome for the untreated group is 67. What will the slope of the regression line be?
A) 67
B) 35
C) 67/2 = 33.5
D) -32

Accepted Answer

For a dichotomous treatment regression, the regression equation is: Y = b0 + b1X. Here, X = 1 for the treated group, X = 0 for the untreated group.

Given the mean outcome for the treated group = 35 and the mean outcome for the untreated group = 67.

Hence, b0 is the mean outcome for the untreated group when X=0 = 67.

Therefore, b1 can be calculated using the following formula:
b1 = (mean outcome for treated group - mean outcome for untreated group)/ (X=1 - X=0) = (35-67)/(1-0) = -32

Therefore, the slope of the regression line will be -32.

Question 12

In a dichotomous regression which condition must hold?
A) For the treated observations, an equal number of positive and negative residuals.
B) For the untreated observations, an equal number of positive and negative residuals.
C) Across all of the observations, an equal number positive and negative residuals.
D) The sum of the residuals for treated and untreated groups must be equal.

Accepted Answer

The answer of In a dichotomous regression which condition must...

Question 13

Which of the following treatments is a multi-level treatment?
A) Receiving a cancer drug or a placebo
B) A product is advertised or it isn't
C) Employees receive bonus levels based on years of service
D) None of the answers is correct.

Accepted Answer

Option C is a multi-level treatment because the employees receive different bonus levels based on their years of service, making it a treatment with multiple levels or categories. Option A is a binary treatment where the subjects either receive the drug or placebo. Option B is also a binary treatment where the subjects are exposed or not exposed to the advertisement. Therefore, options A and B are not multi-level treatments.

Question 14

Suppose the regression line to describe the relationship between Y and a dichotomous treatment (X = 1 for treated, = 0 untreated) is given by Y = 4 + 3X. Suppose that one of the observations that was treated was observed to have an outcome, Y = 8. For this observation, what is the residual?
A) 0
B) 7
C) 1
D) -1

Accepted Answer

The answer of Suppose the regression line to describe the...

Question 15

In the simple linear regression case, (for Y and X) with a multi-level treatment X, the line to be estimated is given by: A) Y = b + mX B) Y = mX C) Y = mX² D) None of the answers is correct.

Accepted Answer

In simple linear regression, the line to be estimated for Y and X is given by Y = b + mX. This equation allows us to estimate the relationship between Y and X by calculating the slope (m) and y-intercept (b) of the line of best fit. The presence of a multi-level treatment X does not change the formulation of the equation.

Question 16

In relating a variable X to a variable Y with the regression line Y = b + mX, what value would be reported for Y when X = 0?
A) b
B) b + m
C) b ± m
D) m

Accepted Answer

When X = 0, the regression line becomes Y = b + m(0), which simplifies to Y = b. Therefore, the value reported for Y when X = 0 is b.

Question 17

Why is Line B a better fit for the data in this graph? 
A) For Line B, the average residual (difference between the actual Profits and point on the line) is zero.
B) For Line B, the residuals (difference between the actual Profits and point on the line) are uncorrelated with Price.
C) Line B's slope is less steep than the slope of Line A.
D) Line B's intercept is smaller than Line A's intercept.

Accepted Answer

For a line to be a better fit for the data, it should have uncorrelated residuals with the predictor variable (Price in this case). This means that the residuals should be spread evenly around the line, without any pattern or trend. Line B satisfies this criterion, as the residuals are scattered randomly around the line without any clear pattern, indicating that it is a better fit for the data than Line A. The other options may or may not be true and do not necessarily determine the fitness of the line for the data.

Question 18

Suppose your lead analyst runs a simple regression of Profits (Y) on price (X). You know that the average profit in the sample was $1,000 and the average price was $25. If your analyst reports that the intercept from the simple regression is 900, what can you infer about the estimated slope?
A) The estimated slope is 4.
B) The estimated slope is -4.
C) The estimated slope is 2.
D) None of these choices are correct.

Accepted Answer

The intercept is the value of Y when X is 0. So, if the intercept is 900, then the estimated profit when the price is 0 is $900. The slope is the change in Y for a one-unit change in X. So, if the slope is 4, then the estimated profit increases by $4 for each $1 increase in price.

Question 19

Why is Line A a better fit for the data in this graph? 
A) For Line B, the average error (difference between the actual Profits and point on the line) is zero.
B) For Line A, the errors (difference between the actual Profits and point on the line) are uncorrelated with Price.
C) For Line A, the average error (difference between the actual Profits and point on the line) is zero.
D) Line B's intercept is smaller than Line A's intercept.

Accepted Answer

Line A has a lower average error, meaning it has a better overall fit to the data. Option B is incorrect because the errors for Line A appear to have a slight negative correlation with price. Option A is incorrect because having an average error of zero does not necessarily mean the line is the best fit. Option D is incorrect because the intercepts alone do not determine which line is a better fit for the data.

Question 20

When a treatment can be administered in more than one quantity it is known as a:
A) randomly assigned treatment.
B) dichotomous treatment.
C) multi-level treatment.
D) multiple regression.

Accepted Answer

A multi-level treatment is a treatment that can be administered in more than one quantity or level. This allows for more flexibility in the treatment and can be especially useful in clinical trials where different doses or durations of treatment need to be tested. Randomly assigned treatment refers to treatments that are assigned randomly to different groups, such as in a randomized controlled trial. Dichotomous treatment refers to a treatment that only has two possible levels or options. Multiple regression is a statistical analysis method used to determine the relationship between multiple variables.

Quiz 5: Linear Regression As a Fundamental Descriptive Tool

If one was estimating a simple regression of Earnings (Y) on Height of individuals (X) , and got a coefficient on the Height variable of 30, what would the coefficient on Height be if you added 3 inches to every individual in the sample but kept their earnings the same?

For the simple linear regression, Y = b + mX, which variable is considered the intercept?

In a dichotomous regression all of the following conditions must hold except for what?

How many regression parameters will be estimated in a simple linear regression model?

If we wish to use a regression line to determine the effect of multiple treatment levels , why can't we just plot the average outcome for each treatment level and "connect the dots?"

If one was estimating a simple regression of Earnings (Y) on Height of individuals (X) , and got a coefficient on the Height variable of 30, what would the intercept be if you added 3 inches to every individual in the sample but kept their earnings the same?

The difference between the observed outcome and the corresponding point on the regression line for a given observation is a:

The process of using a function to describe the relationship among variables is known as:

A treatment that only undergoes two statuses - treated and untreated-is known as this kind of treatment?

In the dichotomous regression if one was to replace the regression line prediction of the outcome means (for treated/untreated) with medians, which of the following conditions would hold?

For a dichotomous treatment regression (X = 1 or 0) , the mean outcome for the treated group (X = 1) is 35 and the mean outcome for the untreated group is 67. What will the slope of the regression line be?

In a dichotomous regression which condition must hold?

Which of the following treatments is a multi-level treatment?

Suppose the regression line to describe the relationship between Y and a dichotomous treatment (X = 1 for treated, = 0 untreated) is given by Y = 4 + 3X. Suppose that one of the observations that was treated was observed to have an outcome, Y = 8. For this observation, what is the residual?

In the simple linear regression case, (for Y and X) with a multi-level treatment X, the line to be estimated is given by:

In relating a variable X to a variable Y with the regression line Y = b + mX, what value would be reported for Y when X = 0?

Why is Line B a better fit for the data in this graph?

Suppose your lead analyst runs a simple regression of Profits (Y) on price (X) . You know that the average profit in the sample was $1,000 and the average price was $25. If your analyst reports that the intercept from the simple regression is 900, what can you infer about the estimated slope?

Why is Line A a better fit for the data in this graph?

When a treatment can be administered in more than one quantity it is known as a: