Chapter 9 Flashcards
What is a scatter diagram/scatter graph
A Graph where y axis = dependent variable and x axis = independent variable (aka we change x to observe change in y)
-It is a good way of determining the relationship between two variables
WHat is the formula for a line of best fit?
Y= a + bX
Y= dependent variable
a = the y intercept
b= gradient of the line of best fit (steeper = higher)
X = the independent variable
What are the two main techniques for estimating the dependent variable?
1) High/Low method
2) Linear regression
What is the High/low method? (boring definition)
Official terminology
-A method of estimating cost behaviour by comparing the total costs associated with two different levels of output.
The difference in costs is assumed to be caused by variable costs increasing, allowing unit variable cost to be calculated.
-Following from this, since total cost is know, the fixed cost can be derived
What is high/low method (steps)
Step 1: Select the periods with the highest and lowest output and calculate the difference between them
—FInd Difference in Output and Difference in Total Costs
Step 2: Estimate the variable cost (VC) per unit
VC per unit = change in total cost/change in output
Step 3 Estimate the fixed cost (FC) by substitution
Total cost = Fixed Cost + Total Variable Cost
TC = FC + VC per unit*output aka Y =a +bx
–Find FC
Our cost relationship can then be summarised as:
TC = {FC} + {VCpU} x no of units
which can be used to forecast a future cost
What are the downfalls of the high/low method
ignoring all the middle data - which may scew actual line of best fit
What is another term for Linear regression
method of least squares?
What is linear regression used for?
Using linear regression it is possible to find the ‘line of best fit’ between two variables.
What is the line of best fit?
The line of best fit estimates the value of a and b in the linear equation
Y = a + bX
Where Y=total costs
X = output
a = y intercept = fixed costs
b = gradient = variable cost per unit
Using regression analysis, how can a be estimated
a = Y* - bX*
where Y * = the mean of Y values
where X * = the mean of X values
Using regression analysis, how can b be estimated?
b = n SUMxy - (SUMx)(SUMy)/
n SUMx^2 -(SUMx)^2
where n = the number of pairs of data
What is the correlation coefficient?
The CC indicates the strength of the realtionship between the variables
aka (how close the line of best fit is to the actual variables)
symbol = r - always inbetween -1 and +1.
The sign of r indicates the gradient of the ‘best fit’ line.
-the closer the cc is to +-1, the closer the line of best fir is to the actual variables
1= perfectly straight line
What is the correlation coefficient formula?
Need to know
The linear function you want to present as your answer?
Total cost = N + NX
What is the coefficient of determination?
The CD indicates the proportion of change in the y (dep)(aka costs) variable which is explained by change in the x (indep)(aka output) variable
r^2
The CD will always be somewhere between 0 and +1
in simple terms: It indicates that N% of the change in the total cost is explained by the change in the output
and hence (100-N)% of the change is due to other factors
What are the limitations of linear regression?
1) Assumes a linear relationship between x and y
2) It only measures the relationship between the two variables d
In reality the dependent variables could be affected by many independent variables
3) Only those forecasts within the data set tend to be reliable.
The equation should not be used for extrapolation outside the data set.
4) Historical data is not necessarilt a good indicator of future events
What should spearmans rank correlation coefficient be used
R
-When you want to measure the order or rankings of data rather than their actual values
Its value can range from -1 to +1
A RCC close to +1 demonstrates a very strong relationship between the order of results of one distribution with the order of results from anothe distribution
What is the formula for speamans rank correlation?
Formula for rank correlation is given on the formulae sheet
d = difference in order between a pair of rankings
n = the number of pairs of data
NB: If items are ranked equally, then the average position is used
aka if 2nd and 3rd are equal - they would both be ranked 2.5
What is a time series?
TS - a set of data observed over a period of time
e.g. sales patterns by day, by week, by month, by year
WHat is a time series forecast?
It identifies four components from a series of past figures recorded over time and then uses them to forecast the future.
These components are
-Tren
-Seasonal Variation
-Cyclical Variation
-Random Variation
TS forecast: What is the trend component?
The underlying long-term movement over time in historic data. This can be determined by a process of moving averages.
general upward trend in house prices over the last five years
TS forecast: What is Seasonal variation
short-term periodic fluctuations in historic data. This can be determined by using either the additive or the proportional models
e.g. ice cream higher in summer than winter
TS forecast: What is seasonal variation?
longer-term fluctuations caused by such factors as economic activity or economic cycles
TS forecast: What is random variation?
unpredictable fluctuations such as an act of nature
What are the two models that we can use to determine the forecast time series?
The additive model
-The multiplicative (or proportional) model
What is the additive model?
USed where the components are assumed to add together to give the time series.
The components of the time series are quoted in actual (absolute) terms/values
**EXAM example
trend eqn for forecasting sales (A) for quarter (B) is A = N +NB
Q3 has a seasonal adj factor of +1.6 using additive model. What is the time series forecast for Q3
TS (sales forecast for Q3 = N+N(3) +1.6
What is the multiplicative model
used where the components are considered as multiplying together to give the TS
The components of the TS are quoted as a % or fraction of the trend
TS = T*S
What is the difference between the additive model and the multiplicative model?
The additive model assumes that the seasonal variations are independent of the trend
The multiplicative model assumes the seasonal variations are linked to the size of the trend
What are the three ways of determining the trend in the above data?
1) Preparing a graph
2) Moving averages
3) Linear regression
Preparing a graph method
Using the data to construct a time-series graph
- can put a line of best fit
extrapolate going forward
add seasonality but addition or multiplicative model
Moving averages?
step 1: Decide how many periods to include in the average
-four-quarter range or 7 day range etc.
step 2: work out the averages
-any seasonal distortions should be eliminated, leaving us with an estimation of the underlying trend
step 3 centre moving averages
step 4 seasonal variation
step 5 forecast
1st column: Per question
2nd - moving average
3rd Centred moving average (T)
Line up data with Q3/Q4 etc by making smaller averages between two
4th Seasonal variation
-found by TS-T or TS/T
Moving averages: how to calculate average quarterly increase?
use centred moving average
number of jumps = N
total gap/N = average quarterly increase
Add this onto the final value again and again to forecast the centred moving average
Then factor in seasonal allowance - average Q1 values and add/multiply to centreed moving average for total sales (TS)
Linear regression technique
Year |quarter |period(x) |sales (y) |(Xy) |X2)
nb period of Year 2 Q1 = 5 etc
linear regression to produce
y= N+Nx
These figures then need to be adjusted for seasonal variations either using the additive or multiplicative models
What are the limitations of forecasting?
-Assumes the historic trends will continue into the future (for TS analysis)
-Becomes less reliable if you go outside the ranges of existing observations
-Seasonal variations may not be consistent into the future
-Assumes there is an identifiable relationship
