B.3. Forecasting and Probability

Question 1

Assumptions about the outlook for the environment in which a company operates are called premises, and a company’s assumptions, or premises, about its future business environment must be identified as part of the planning and budgeting process.

What is the purpose of identifying premises in the planning and budgeting process?

Answer 1

To focus on assumptions that will impact the potential success of the business and avoid wasting time on irrelevant premises.

Question 2

What is a mathematical model in forecasting?

Answer 2

An equation that attempts to represent an actual situation.

Mathematical models are used to develop budgeted amounts based on identified premises, or assumptions, about the future.

Question 3

What is linear regression analysis used for?

Answer 3

To develop a mathematical equation that models the extent to which one variable (the dependent variable) is affected by one or more other variables (the independent variable[s]).

Linear regression can be used to make decisions and predict future values based on historical relationships.

Question 4

What are the two basic forecasting methods using linear regression?

Answer 4

• Time series methods
• Causal forecasting methods

Time series methods: focus on historical patterns of one dependent variable, where time is the causal factor and the independent variable.

Causal forecasting methods: look for cause-and-effect relationships between the variable being forecasted (the dependent variable) and one or more other variables (the independent variables).

Question 5

In time series analysis, what is the independent variable?

Answer 5

Time

Time is the predictor variable graphed on the x-axis, while historical data serves as the dependent variable on the y-axis.

Question 6

What is a trend pattern in time series analysis?

Answer 6

A pattern where historical data exhibit a gradual shift to a higher or lower level as time passes.

Trend patterns are used for forecasting when a long-term trend is apparent despite short-term fluctuations.

Question 7

What are the assumptions of simple linear regression analysis?

Answer 7

• Variations in the dependent variable are explained by variations in one independent variable.
• The relationship between the independent and dependent variables is linear.

A linear relationship is approximated by a straight line on a graph.

Question 8

What is the equation of a simple linear regression line?

Answer 8

ŷ = a + bx

Where ŷ is the predicted value of y, a is the constant coefficient (the y-intercept), b is the variable coefficient (the slope of the regression line), and x is the independent variable.

Question 9

What is the “line of best fit” in linear regression?

Answer 9

A line where the deviations (or residuals) between each graphed value and the regression line are minimized.

The line of best fit is used for forecasting by extrapolating into future periods.

Question 10

What is causal forecasting?

Answer 10

A method where the value being forecast (the dependent variable) is affected by one or more independent variables.

Causal forecasting requires a cause-and-effect relationship between the independent variable(s) and the dependent variable.

Question 11

What is correlation analysis used for in causal forecasting?

Answer 11

To evaluate the closeness of the relationship between two or more variables.

Correlation analysis helps determine if a linear relationship exists for causal forecasting.

Question 12

True or False:

If a dependent variable is correlated with an independent variable (there is a close relationship between them), it means the independent variable is the cause of the dependent variable.

Answer 12

False

Correlation does not necessarily imply causation. Two variables may be correlated without having a direct cause-and-effect relationship.

Question 13

What is multiple regression analysis?

Answer 13

Regression analysis involving more than one independent variable affecting the dependent variable that can be used for causal forecasting.

Multiple regression can be used for forecasting when several factors that can be expressed numerically impact the dependent variable, such as advertising expenditures, the size of the sales staff, the economy, and any number of other variables.

Question 14

In order to assume a cause-and-effect relationship between the independent variable(s) and the dependent variable in a regression analysis, what is required?

Answer 14

A reasonable basis must exist.

Correlation does not prove causation, and a linear relationship does not prove a cause-and-effect relationship.

Question 15

What are the benefits of using regression analysis in forecasting?

Answer 15

• It is a quantitative method and therefore is objective because a given data set generates specific results.
• For budgeting purposes, it can be used to compute fixed and variable portions of costs that contain both fixed and variable components (mixed costs).

Question 16

What are the limitations of using regression analysis in forecasting?

Answer 16

• It requires historical data.
• The use of historical data is questionable for future predictions if conditions change.
• In causal forecasting, the usefulness of the data generated by regression analysis depends on the appropriate choice of independent variable(s).
• The statistical relationships that can be developed using regression analysis are valid only for the range of data in the sample.

Question 17

What does the term “learning curve” refer to?

Answer 17

The concept that efficiency increases as experience with a task increases, reducing the time required for the task.

Learning curves are used in planning, budgeting, and forecasting to estimate long-term production costs.

Question 18

What is the cumulative average-time learning model?

Answer 18

A model assuming the cumulative average time per unit declines at a constant rate each time the cumulative quantity of units produced doubles.

This model can be used to estimate the total time required to produce a given number of units or the average time per unit or batch.

Question 19

What is the range for the learning curve percentage?

Answer 19

Greater than 50% and less than 100%.

If the learning curve were 100%, then no learning and no decrease in time required is taking place.

A learning curve percentage of less than or equal to 50% is impossible as it implies that the initial and subsequent units can be produced in the same or less time than is required for the initial unit, which is not possible.

Question 20

What does a learning curve percentage of 70% indicate?

Answer 20

Every time the total number of units produced doubles, the estimated cumulative production time for the doubled production increases, but to only 70% of what it would have been if no learning had taken place.

A lower percentage indicates a greater amount of learning.

Question 21

What are the two methods that can be used to estimate the total time required for all units and the average time required per unit or batch using the cumulative average-time learning model?

Answer 21

• Method 1: Calculate the estimated total time required, then divide by the total units or total batches to calculate the average time per unit or batch.
• Method 2: Calculate the estimated average time required per unit or batch, then multiply by the total units or batches to calculate the total time.

The choice of method depends on the exam question and information given.

Question 22

What is the learning curve effect?

Answer 22

It refers to the reduction in labor time per unit as workers gain experience and efficiency over time.

This concept is crucial in forecasting, budgeting, cost management, and bidding on contracts, as it allows for more accurate predictions of labor costs and production times.

Question 23

True or False:

A learning rate of 50% or lower is better than a learning rate of 70% in the cumulative average-time learning model.

Answer 23

False

A learning curve percentage of less than or equal to 50% is impossible as it implies the initial and subsequent units can be produced in the same or less time than is required for the initial unit, which is not possible.

Question 24

What is the formula for estimating the cumulative average time per unit using the cumulative average-time learning model and a given learning curve percentage?

Answer 24

Time required for the first unit × LCⁿ

LC represents the learning curve percentage (in decimal format), and n, the exponent, is the number of doublings of production.

Question 25

What is the formula for estimating the **total time required for all units produced** using the cumulative average-time learning model and a given learning curve?

Answer 25

Time required for the first unit × (2 × LC)*ⁿ* ## Footnote LC represents the learning curve percentage (in decimal format), and n, the exponent, is the number of doublings of production.

Question 26

What factors other than learning can affect the **reliability** of learning curve calculations?

Answer 26

An observed **change in productivity** might be associated with factors other than learning, such as: * Changes in the labor mix * Changes in the product mix * Other external factors affecting productivity ## Footnote If some factor or factors other than learning are affecting productivity, a learning model developed using the affected historical data will produce inaccurate estimates of labor time and cost.

Question 27

What is **probability** and how is it expressed?

Answer 27

It is a numerical measurement of the **likelihood** that an event will occur. It is expressed as a value between 0% and 100%, such as 40% or 55%, that represents the **chances** (out of 100%) that the event will occur. ## Footnote Probabilities are usually converted to their decimal equivalents, such as 0.40 or 0.55, for use in calculations.

Question 28

What is the benefit of using **learning curves** in life-cycle costing and bidding?

Answer 28

In calculating the cost of a contract, learning curve analysis can increase the probability that the **cost estimates** will be accurate over the life of the contract, leading to more accurate and more competitive bidding on projects. ## Footnote The heightened accuracy helps in planning and evaluating the financial viability of long-term projects.

Question 29

What are the **limitations** of using learning curves in forecasting?

Answer 29

* It is appropriate only for **labor-intensive** operations involving repetitive tasks where repeated trials improve performance. * The **learning rate** is assumed to be constant, whereas the decline in labor time might not be constant. * An observed change in **productivity** might be associated with factors other than learning. * For a **new product**, it may not be possible to accurately forecast the amount of improvement that may result from learning. * The company may or may not be able to realize **actual efficiencies** from learning.

Question 30

What are the two requirements of **probability**?

Answer 30

* The probability values assigned to each of the possible outcomes must be between 0.0 and 1.0 or between **0% and 100%**. * The probability values assigned to all the possible outcomes must sum to **1.0 or 100%**.

Question 31

What is a **random variable**?

Answer 31

A variable that can have any value within a **range of values** that occurs randomly and can be described using probabilities.

Question 32

What is a **discrete random variable**?

Answer 32

A variable that is a **whole number**, representing a number of items that can be counted (such as the number of items sold).

Question 33

What is a **continuous random variable**?

Answer 33

A variable that can take on **any value** whatsoever within an interval or a collection of intervals (such as 5.635 or 72.36092). It can have an **unlimited number of decimal places**, so there can be no limit to the number of different values the variable could assume. ## Footnote A continuous random variable can be a whole number, but it does not need to be a whole number.

Question 34

What is a **probability distribution**?

Answer 34

A table or an equation that links each **outcome** of a statistical experiment with its probability of occurring. ## Footnote A probability distribution for a discrete random variable can be developed by observing historical data. For continuous random variables, probability is expressed in terms of the probability that a variable will have a value within a specified interval, defined as the area under the curve on a graph called a probability density function.

Question 35

What are the **three methods** of assigning probable values?

Answer 35

* Classical method * Relative frequency method * Subjective method ## Footnote **Classical method**: assumes that each possible outcome has an equal probability of occurring. **Relative frequency method**: the use of information available that can be used to determine the probability that something will occur. **Subjective method**: assigning a probable value that expresses the decision maker's degree of belief that the outcome will occur. Used when neither the classical nor the relative frequency method can be used.

Question 36

What is **expected value**?

Answer 36

The **mean value**, also known as the average value, of all the possible outcomes that could occur over a long period of time. The expected value of a discrete random variable is calculated as the **weighted average** of all the possible values of the random variable using the probabilities of each of the outcomes as the weights.

Question 37

What are the **benefits** of expected value techniques?

Answer 37

* Expresses the most likely result of a decision in situations involving risk * Takes uncertainty into account * The calculation is simple and easily understood * Reduces information to a single outcome for easier decision making * Incorporates every identified possible outcome and its probability as determined by the decision maker

Question 38

What are the **shortcomings** of expected value techniques?

Answer 38

* More reliable as a long-run average forecast, less reliable as a short-term forecast * Accuracy depends on the probability distribution used, and probabilities assigned to the various potential outcomes are usually subjective * Gives no information about the dispersion of potential outcomes about the expected value (the variance and standard deviation are needed for that), which would give some insight into the amount of risk. * An expected value is an average of all identified possible outcomes, so it does not provide an actual outcome * Uses discrete variables rather than continuous variables, and so may not accurately model the situation * The expected value may be different from all of the potential outcomes.

Question 39

What do the **variance** and **standard deviation** of a probability distribution measure?

Answer 39

They measure how far from the expected value of the distribution (the mean) the various possible values lie, indicating the variability or **dispersion** of the possible values about the mean.

Question 40

What does the **variance** of a probability distribution represent?

Answer 40

It measures the **dispersion** of the values about their mean. ## Footnote The variance is the weighted average of the squared differences of the values of a random variable from the mean, with the probabilities of each serving as the weights. The difference from the mean of each result is important because it indicates the distance of that measurement from its expected value. The variance of a population is represented by σ² (sigma squared).

Question 41

How is standard deviation related to variance?

Answer 41

The standard deviation is the **positive square root** of the variance and is represented by σ (sigma). ## Footnote Both the variance and the standard deviation of a probability distribution provide information on how much the various values are dispersed around the mean. Whereas variance is measured in squared units, though, standard deviation is measured in the same units as the variable.

Question 42

In an analysis of a probability distribution, why is the amount of **dispersion** of the values about their mean important for **risk measurement**?

Answer 42

The greater the **dispersion of values around their mean**, the greater the **risk**, as it increases the probability that actual results will differ from their expected value.

Question 43

What is the significance of **standard deviation** being measured in the same units as the variable?

Answer 43

It provides a measure of **dispersion** that is directly comparable to the original **data units**, making it easier to interpret than the **variance**, which is in squared units.

Question 44

# True or False: A higher standard deviation of a probability distribution indicates less variability in the data.

Answer 44

False ## Footnote A higher standard deviation indicates **more** variability in the data, which indicates greater risk.

Question 45

What can be inferred about the **sales budget accuracy** between two stores with different standard deviations in the various possible sales volumes used to develop the expected values for the forecasted sales?

Answer 45

The sales budget for the store with a **lower standard deviation** in its possible sales volumes will likely be **more accurate** because its possible sales volumes **vary less**.