M2 Problem set and sample questions

Which of the following is an example of a time-series data set?

a. amount of labor employed yearly in a specific factory from 1990 through 2010.

b. average amount of labor employed at specific times of the day at a specific factory in 2010.

c. amount of labor employed in each factory in the U.S. in 2010.

d. All of the choices are time-series data sets.

a. amount of labor employed yearly in a specific factory from 1990 through 2010.

Explanation: Time-series data consists of observations on a single entity (the factory) over a specific time period.

In a linear regression equation Y=a+bX, the fitted or predicted value of Y is...

a. the value of Y obtained by substituting specific values of X into the sample regression equation.

b. the values of the parameters predicted by the estimators.

c. the value of X that the regression equation predicts.

d. the value of X associated with a particular value of Y.

e. the value of Y associated with a particular value of X in the sample.

a. the value of Y obtained by substituting specific values of X into the sample regression equation.

Explanation: The fitted value is the value of the dependent variable that the regression model predicts for a given value of the independent variable.

Choice variables...

a. cannot be continuous.

b. both "can only take on integer values" and "cannot be continuous".

c. determine the value of the objective function.

d. determine the constraint.

e. can only take on integer values.

c. determine the value of the objective function.

Explanation: Choice variables are the variables that a decision maker controls, and their values are used to find the optimal value of the objective function.

The method of least squares...

a. can be used to estimate the explanatory variables in a linear regression equation.

b. minimizes the distance between the population regression line and the sample regression line.

c. All of the choices are correct.

d. can be used to estimate the slope parameters of a linear equation.

d. can be used to estimate the slope parameters of a linear equation.

Explanation: The method of least squares is a mathematical technique used in regression to calculate the specific values of the slope and intercept parameters that minimize the sum of the squared errors.

A continuous choice variable...

a. is in unconstrained but not constrained problems.

b. must be continuously varied to attain the goal.

c. can take on only special values between two end points.

d. None of the choices is correct.

d. None of the choices is correct.

Explanation: A continuous choice variable can take on any value within a range and can be used in both constrained and unconstrained problems.

The function a decision maker seeks to maximize or minimize is the ________ function.

a. None of the choices is correct.

b. marginal

c. decision-making

d. objective

e. optimal

d. objective

Explanation: An objective function represents the goal of a decision maker in an optimization problem.

Whenever the additional revenue from the last unit of output exceeds the additional cost of that unit, a profit-maximizing firm should...

a. do nothing, the firm is making profits.

b. produce less in order to increase profits.

c. think about investing in another industry.

d. produce more in order to increase profits.

d. produce more in order to increase profits.

Explanation: When marginal revenue is greater than marginal cost, producing an additional unit adds more to revenue than to cost, increasing total profit.

For the equation Y=a+bX, the objective of regression analysis is to...

a. estimate the variables Y and X.

b. estimate the parameters a and b.

c. fit a straight line through the data scatter in such a way that the sum of the squared errors is minimized.

d. both "estimate the parameters a and b" and "fit a straight line through the data scatter in such a way that the sum of the squared errors is minimized".

d. both "estimate the parameters a and b" and "fit a straight line through the data scatter in such a way that the sum of the squared errors is minimized".

Explanation: The goal of regression is to both estimate the parameters and to do so by finding the line that minimizes the squared errors.

An estimator is unbiased if it produces...

a. estimates of a parameter that are close to the true parameter.

b. estimates of a parameter that are on average equal to the true parameter.

c. a parameter from the sample that equals the true parameter.

d. estimates of a parameter that are statistically significant.

e. both "estimates of a parameter that are close to the true parameter" and "estimates of a parameter that are statistically significant".

b. estimates of a parameter that are on average equal to the true parameter.

Explanation: An unbiased estimator is one where the expected value of its estimates across many samples is equal to the true population parameter.

In a regression equation, the ______ captures the effects of factors that might influence the dependent variable but aren't used as explanatory variables.

a. random error term

b. intercept

c. R-square

d. slope parameter

a. random error term

Explanation: The random error term represents the influence of all unobserved or unmeasured factors on the dependent variable.

In a linear regression equation of the form Y=a+bX, the intercept parameter a shows...

a. the amount that Y changes when X changes by one unit.

b. the value of Y when X is zero.

c. the value of X when Y is zero.

d. the amount that X changes when Y changes by one unit.

b. the value of Y when X is zero.

Explanation: The intercept is the predicted value of the dependent variable when the independent variable is zero.

Which of the following statements represents bad decision making?

a. I've put in so much time on this paper, I can't quit now.

b. My stock has dropped $10 a share so I can't afford to sell it now.

c. All of the choices are correct.

d. I've already spent 3 years in the college so I can't drop out and go to work now.

e. I've already paid for the ticket so I might as well stay to the end.

c. All of the choices are correct.

Explanation: All of these statements are examples of the sunk cost fallacy, where a decision is based on irretrievable past costs rather than on a rational assessment of future benefits and costs.

If the marginal benefits of increasing study time are less than the marginal costs, then...

a. study time should be increased.

b. no conclusion about the relative merits of more or less study time is possible.

c. there is too little study time.

d. study time should be decreased to zero.

e. study time should be decreased.

e. study time should be decreased.

Explanation: When the cost of an action exceeds its benefit (MC > MB), reducing the activity will increase your net benefit.

In making a decision about whether to increase its advertising budget the firm management should not consider...

a. the cost of the increased advertising.

b. interest payments on the firm's loan.

c. the added revenue from increased sales.

d. the added cost of producing more goods for sale.

b. interest payments on the firm's loan.

Explanation: The interest on the loan is a sunk cost that is not affected by the decision to increase the advertising budget.

The optimization rule for unconstrained optimization is to select that level of activity at which...

a. marginal benefit equals marginal cost.

b. total benefit is equal to total cost.

c. marginal benefit exceeds marginal cost.

d. total benefit is less than total cost.

a. marginal benefit equals marginal cost.

Explanation: This is the core principle of marginal analysis. The optimal level of an activity is where the additional benefit equals the additional cost.

The sample regression line...

a. is estimated by the population regression line.

b. maximizes the sum of the squared differences between the data points in a sample and the sample regression line.

c. connects the data points in a sample.

d. is used to estimate the population regression line.

e. shows the actual (or true) relation between the dependent and independent variables.

d. is used to estimate the population regression line.

Explanation: The sample regression line is our best estimate of the true but unobservable relationship between variables, which is represented by the population regression line.

In the regression model Y=a+bX+cZ, a test of the hypothesis that parameter c equals zero is...

a. an R2-test.

b. a t-test.

c. a zero-statistic.

d. an F-test.

e. a Z-test.

b. a t-test.

Explanation: A t-test is used to determine if a single regression coefficient is statistically significant.

To test whether the overall regression equation is statistically significant one uses...

a. the R2-statistic.

b. the standard error statistic.

c. the F-statistic.

d. the t-statistic.

c. the F-statistic.

Explanation: The F-statistic is used to test the overall significance of the entire regression model. It evaluates whether the independent variables as a group are useful in explaining the dependent variable.

In a linear regression equation of the form Y=a+bX, the slope parameter b shows...

a. None of the choices is correct.

b. ΔX/ΔY.

c. ΔY/ΔX.

d. ΔY/Δb.

e. ΔX/Δb.

c. ΔY/ΔX.

Explanation: The slope parameter (b) represents the change in the dependent variable (Y) for every one-unit change in the independent variable (X).

A firm is deciding whether or not to close down its plant and modernize by installing new technology. Which of the following should management ignore when making the decision?

a. cost of lost sales while the plant is closed

b. added cost of the labor needed for the new plant

c. All of the choices are correct.

d. how much the present plant cost

d. how much the present plant cost

Explanation: The initial cost of the plant is a sunk cost—it's already been paid and cannot be recovered. A rational decision should only consider future costs and benefits.

A firm will maximize profit by producing that level of output at which...

a. the additional revenue from the last unit sold exceeds the additional cost of the last unit by the largest amount.

b. the additional revenue from the last unit sold equals the additional cost of the last unit.

c. total revenue exceeds total cost by the largest amount.

d. total revenue equals total cost.

e. both b and c

e. both b and c

Explanation:

A firm's profit is the difference between total revenue and total cost. Profit is maximized when this difference is as large as possible, which is what option c states. The way to find this point is by using marginal analysis. As long as the marginal revenue (MR) from the last unit sold is greater than the marginal cost (MC) of that unit (MR > MC), producing it will add to total profit. A firm should continue producing until MR = MC. If it produces beyond that point, MC > MR, and the additional unit will reduce total profit. Therefore, the profit-maximizing condition is where marginal revenue equals marginal cost, which is what option b states.

The function a decision maker seeks to maximize or minimize is the ________ function.

a. optimal

b. decision-making

c. objective

d. marginal

e. none of the above

c. objective

Explanation:

In economics and optimization theory, the objective function represents the goal of a decision-maker. This is the function they are trying to either maximize (e.g., profit, utility) or minimize (e.g., cost, risk).

Choice variables

a. determine the value of the objective function

b. determine the constraint

c. can only take on integer values

d. cannot be continuous

e. both c and d

a. determine the value of the objective function

Explanation:

Choice variables are the variables that a decision-maker controls. By choosing different values for these variables, they can influence the outcome they are trying to optimize, which is the value of the objective function. For example, in a profit-maximization problem, the quantity to produce is a choice variable that determines total revenue and cost, and thus total profit.

For an unconstrained maximization problem

a. the decision maker seeks to maximize net benefits.

b. the decision maker seeks to maximize total benefits.

c. the decision maker does not take cost into account because there is no constraint.

d. the decision maker does not take the objective function into account because there is no constraint.

a. the decision maker seeks to maximize net benefits.

Explanation:

In an unconstrained maximization problem, the goal is to maximize the difference between total benefits and total costs (net benefits). Even without a specific budget or resource constraint, the decision-maker must still account for the costs associated with the activity to find the optimal level. For example, a firm maximizing profit without a budget constraint must still consider production costs.

When marginal cost is greater than marginal benefit at the current activity level, the decision maker can increase net benefit by decreasing the activity because

a. total benefit will rise by more than total cost will rise.

b. marginal cost is rising faster than marginal benefit is falling.

c. net benefit is upward sloping at this point.

d. total cost will fall by more than total benefit will fall.

d. total cost will fall by more than total benefit will fall.

Explanation:

If marginal cost (MC) is greater than marginal benefit (MB) (MC > MB), it means the last unit of activity added more to total cost than it added to total benefit. Therefore, to increase net benefit, you should reduce the activity. When you do, you avoid a cost (MC) that is greater than the benefit you would have received (MB), so your total cost will fall by more than your total benefit will fall, increasing your net benefit.

For a constrained minimization problem, the decision maker

a. is constrained by the specific amount of total benefits.

b. is constrained by the choice set of values for the activities.

c. seeks to minimize the cost of achieving a specific goal.

d. all of the above

e. none of the above

c. seeks to minimize the cost of achieving a specific goal.

Explanation:

A constrained minimization problem is one where a decision-maker's goal is to minimize a certain value (like cost) subject to a constraint (like a minimum output requirement or a quality standard). A classic example is a firm trying to produce a certain number of units while spending the least amount of money.

A continuous choice variable

a. must be continuously varied to attain the goal.

b. can take on only special values between two end points.

c. is in unconstrained but not constrained problems.

d. all of the above

e. none of the above

e. none of the above

Explanation:

A continuous choice variable is one that can take on any value within a given range (e.g., temperature, time, or the quantity of a liquid). It doesn't have to be an integer. It is a fundamental concept used in both constrained and unconstrained optimization problems.

For the equation Y=a+bX, the objective of regression analysis is to...

a. estimate the parameters a and b.

b. estimate the variables Y and X.

c. fit a straight line through the data scatter in such a way that the sum of the squared errors is minimized.

d. both a and c

d. both a and c

Explanation: Regression analysis aims to both estimate the unknown parameters (a and b) and to do so by finding the line of best fit that minimizes the squared differences between the actual data points and the predicted line. This process, called Ordinary Least Squares (OLS), accomplishes both goals.

In a linear regression equation of the form Y=a+bX, the slope parameter b shows...

a. ΔX/ΔY.

b. ΔY/ΔX.

c. ΔY/Δb.

d. ΔX/Δb.

e. none of the above

b. ΔY/ΔX.

Explanation: The slope parameter b represents the change in the dependent variable (Y) for a one-unit change in the independent variable (X). This is mathematically represented as ΔXΔY​.

In a linear regression equation of the form Y=a+bX, the intercept parameter a shows...

a. the value of X when Y is zero.

b. the value of Y when X is zero.

c. the amount that Y changes when X changes by one unit.

d. the amount that X changes when Y changes by one unit.

b. the value of Y when X is zero.

Explanation: The intercept a is the point where the regression line crosses the y-axis. By definition, this is the value of Y when the independent variable X is equal to zero.

In a regression equation, the ______ captures the effects of factors that might influence the dependent variable but aren't used as explanatory variables.

a. intercept

b. slope parameter

c. R-square

d. random error term

d. random error term

Explanation: The random error term (ϵ or e) is included in a regression model to account for all the unobserved factors that can influence the dependent variable. It represents the difference between the actual observed value and the value predicted by the regression line.

The sample regression line

a. shows the actual (or true) relation between the dependent and independent variables.

b. is used to estimate the population regression line.

c. connects the data points in a sample.

d. is estimated by the population regression line.

e. maximizes the sum of the squared differences between the data points in a sample and the sample regression line.

b. is used to estimate the population regression line.

Explanation: The true relationship between variables is captured by the unobservable population regression line. Since we can only observe a sample of data, we use the sample regression line as our best estimate of the true relationship.

Which of the following is an example of a time-series data set?

a. amount of labor employed in each factory in the U.S. in 2010

b. amount of labor employed yearly in a specific factory from 1990 through 2010

c. average amount of labor employed at specific times of the day at a specific factory in 2010

d. All of the above are time-series data sets.

b. amount of labor employed yearly in a specific factory from 1990 through 2010

Explanation: Time-series data consists of observations on a single subject over a period of time. Option b shows labor data for one factory over multiple years, making it a classic time series. Option a is cross-sectional data (multiple subjects at one point in time), and option c is a variation of time-series data but is not the best example given the options.

The method of least squares

a. can be used to estimate the explanatory variables in a linear regression equation.

b. can be used to estimate the slope parameters of a linear equation.

c. minimizes the distance between the population regression line and the sample regression line.

d. all of the above

b. can be used to estimate the slope parameters of a linear equation.

Explanation: The method of least squares is the mathematical technique used to calculate the slope (b) and intercept (a) of the sample regression line. Its goal is to find the line that minimizes the sum of the squared vertical distances from each data point to the line.

In a linear regression equation Y=a+bX, the fitted or predicted value of Y is...

a. the value of Y obtained by substituting specific values of X into the sample regression equation.

b. the value of X associated with a particular value of Y.

c. the value of X that the regression equation predicts.

d. the values of the parameters predicted by the estimators.

e. the value of Y associated with a particular value of X in the sample.

a. the value of Y obtained by substituting specific values of X into the sample regression equation.

Explanation: The fitted or predicted value of Y (Y^) is what the regression model estimates for a given value of X. It is calculated by plugging a specific value of X into the estimated regression equation, Y^=a+bX.