Question Tag: Regression Analysis

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  • 17 Marks

QT – Nov 2018 – L1 – Q1b – Forecasting

Determine the number of errors a student would make after 280 minutes of study and the expected change in errors for a 1-minute change.

The study time in minutes and the number of errors on a mock examination paper made by ten (10) ICAG students are given below:

Student 1 2 3 4 5 6 7 8 9 10
Study Time (X) 90 100 130 150 180 200 220 300 350 400
Errors (Y) 25 28 20 20 15 12 13 10 8 6

(i) Determine how many errors a student would make in the examination if he studied for 280 minutes.

(ii) Determine the expected change in the number of errors if there is a 1-minute change in study time.

The regression equation Y=a+bXY = a + bX is calculated as follows:

b =

a =

Thus, the regression equation is:

    Y = 29.27 − 0.064X

(i) When X=280X  the number of errors is:

    Y = 29.27 − 0.064 (280) = 11.35

Therefore, a student studying for 280 minutes would make approximately 11 errors.

(ii) The expected change in the number of errors for a 1-minute change in study time is given by the regression coefficient b=−0.064b 
Thus, a 1-minute change in study time is expected to result in a 0.064 fewer errors.

 

(iii) The formula for the Pearson Product Moment Correlation Coefficient rr is:

r =

Substituting the given values:

r =

Thus, the Pearson Product Moment Correlation Coefficient is -0.9261.

(iv) The coefficient of determination is calculated as:

The coefficient of determination is 85.77%, which means 85.77% of the variation in the number of errors made can be explained by the study time.

  • 20 Marks

QT – Nov 2015 – L1 – Q3 – Forecasting

Calculate the regression line for maize demand and forecast demand for the first three months of the next year using regression analysis.

The monthly demand for maize (in hundreds of bags) for the last year in Bosua Market is shown below:

Month Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
Demand 42 43 40 44 50 47 53 49 54 57 63 60

Required:
(a) Calculate the regression line y=a+bxy = a + bx for the data. (9 Marks)

(b) Using the regression line in (a), determine the forecast for next year:
(i) January, (1 Mark)
(ii) February, (1 Mark)
(iii) March. (1 Mark)

(c) Determine the Standard Error in forecasting demand in (b). (8 Marks)

(a) ,(b) and (c)

b)

i) y=37.67 + 1.92(13) = 62.63

ii) y=37.67 + 1.92(14) = 64.55

iii) y=37.67 + 1.92(15) = 66.47

c)

Standard Error :

= 

 

 

  • 20 Marks

IMAC – Nov 2021 – L1 – Q5 – Budgeting | Forecasting

Analysis of a regression equation related to absenteeism, seasonal variation in production output, and variance analysis for overheads.

a) A large manufacturing company is investigating the cost of sickness amongst production workers who the company has employed for more than one year. The following regression equation, based on a random sample of 50 for such production workers, was derived for 2018:

y=15.6−1.2xy = 15.6 – 1.2x

where yy represents the number of days absent in a year because of sickness and xx represents the number of years’ employment with the company.

Required: i) Explain the meaning of each component of the regression equation. (2 marks)
ii) Predict the number of days of absence through sickness to be expected of an employee who has been with the company for eight years. (2 marks)
iii) Explain TWO (2) limitations or problems of using this equation in practice. (4 marks)

b) A statistician is carrying out an analysis of a company’s production output. The output varies according to the year’s season, and, from the data, she has calculated the following seasonal variations in units of production:

QUARTER 1 2 3 4
Year 1 +11.2 +23.5
Year 2 -9.8 -28.1 +12.5 +23.7
Year 3 -7.4 -26.3 +11.7

Required: i) Calculate and explain the average quarterly variation for each quarter. (5 marks)
ii) If the trend output in the 4th Quarter of Year 3 is expected to be 10,536 units, what is the forecast output? (2 marks)

c) KK Ltd operates a standard absorption costing system and has provided the following costs data in relation to its prime product, Qwikpass:

Standard Cost Card:

GH¢
Direct Material 4kg @ GH¢3/kg 12
Direct Labour 3hrs @ GH¢5/hr 15
Variable Overheads 3hrs @ GH¢3/hr 9
Fixed Overheads 3hrs @ GH¢2 6
Total Cost per Unit 42

Budgeted Units: 6,000

Actual Results:

GH¢
Units produced 6,400
Direct Materials Purchased and used 32,000kg 144,000
Direct Labour 30,720hrs 199,680
Variable Overheads 138,240
Fixed Overheads 45,000
Total costs 526,920

Required: i) Compute the Variable Overheads Expenditure Variance. (1 mark)
ii) Compute the Fixed Overheads Expenditure Variance. (2 marks)
iii) Compute the Fixed Overheads Volume Variance. (2 marks)

a)
i) We have a negative correlation here, as shown by the negative coefficient of xx in the regression line. As the number of years employed with the company rises, the number of days absent in a year through sickness falls.
y=15.6−1.2xy = 15.6 – 1.2x
The 15.6 represents the number of days of absence through sickness that an employee with zero years’ service is expected to suffer, so it is the number of days that an employee will need off through sickness in their first year of employment.
The -1.2 represents the gradient of the regression line, meaning that for each extra year’s service with the company, an employee will take 1.2 fewer days off sick per year. (2 marks)

ii) An employee who has been with the company for eight years is expected to require six days of sick leave per year. (2 marks)

iii) Limitations and problems of using the equation in practice:

  • The regression line approach presupposes a linear relationship between the two variables: a sample of 50 workers has given us quite a strong correlation, but a strict linear relationship seems unlikely.
  • A linear relationship may hold well within a small relevant range of data within which the equation may be useful in practice. But extrapolating outside the range will lead to serious inaccuracies. Thus, the equation would predict that an employee with more than 15.3/1.2 = 13 years’ service would have less than zero sick leave.
  • If we use the equation to predict the future, we will use historical data to forecast the future, which is always risky.
  • The regression line shows the expected number of days sick for a given employment period. But it is unlikely that all categories of workers will experience the same sickness pattern. The equation would be most useful if there were many employees all doing the same job in the same work conditions. (4 marks)

b)
i) Average Quarterly Variation Calculation:

QUARTER 1 2 3 4 TOTAL
Year 1 +11.2 +23.5
Year 2 -9.8 -28.1 +12.5 +23.7
Year 3 -7.4 -26.3 +11.7
Average variation -8.6 -27.2 +11.8 +23.6 -0.4
Adjust total variation to nil +0.1 +0.1 +0.1 +0.1 +0.4
Estimated seasonal variation -8.5 -27.1 +11.9 +23.7 0.0

Seasonal variations are short-term fluctuations in recorded values, due to different circumstances which affect results at different times of the year, on different days of the week, at different times of day, or whatever. For example, sales of ice cream will be higher in summer than in winter.
In this data, the highest output can be expected to be in the winter and the lowest in the summer. (5 marks)

ii) Forecast output = Trend + Seasonal variation
= 10,536 + 23.7
= 10,559.7 units (2 marks)

c)
i) Variable Overheads Expenditure Variance:

GH¢
30,720 hours should have cost (30,720 x 3) 92,160
But did cost 138,240
Variance 46,080 A

ii) Fixed Overheads Expenditure Variance:

GH¢
Budgeted Overheads (6,000 x GH¢6) 36,000
Actual Overheads 45,000
Variance 9,000 A

iii) Fixed Overheads Volume Variance:

GH¢
Budgeted production units 6,000
Actual production units 6,400
Variance (400 F @ cost/unit GH¢6) GH¢2,400 F

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