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2018/2019

SPRING

BS2506

INFERENTIAL STATISTICS, STATISTICAL MODELLING & SURVEY METHODS

1.     a)   Explain the advantages and disadvantages of non-parametric statistical tests.

(10 marks)

b)   A manufacturer and distributor of mini computers has three compensation plans for sales people: commission, fixed salary and commission + salary. Random samples of previous month’s sales were selected from the company’s file and are given as:

Commission

+ Salary

165

98

130

210

195

187

240

120

115

90

126

107

155

80


140

156

220

112

104

235

(i)        Use a non-parametric method with α =0.01 and test to determine whether the three population sales are identical.

(ii)       The p-value for the test in Part (i) is 0.057, interpret its meaning.

(23 marks)

2       a) i) What is the nature of multicollinearity and what are its practical consequences?

ii) How can you detect and deal with multicollinearity

(12 marks)

b) A company has developed a regression model relating their Sales (in £10,000) with four independent variables. These variables are:

X1:   Price per unit (in £)

X2 :   Competitors price (in £)

X3  : Advertising expenditure (in £1000)

X4  : 1 if cans used, 0 if bottles used

The regression was performed on a sample of 25 observations and part of the regression results are shown below.

Variables in equation

Variable

ˆ

SE( ˆ )

Constant

555.0

130.0

X1

-58.52

20.43

X2

22.5

10.50

X3

2.50

1.10

X3

- 14.60

10.42

For this model:

SSR = Regression Sum of Squares = 505675

SST = Total Sum of Squares = 805675

i)   Use the above results to write the regression model and interpret the meaning of the coefficient, X4 .

ii)      Explain what happens if you would add another variable in the model for type of container used as (X5 = 0 if cans used, 1 if bottles used).

iii)     If the manufacturer uses can containers , his price is £1.25, advertising  £20,000 and his competitors price is £1.50, what is your estimate of his sales.

(9 marks)

c)  At α=0.05 level of significance,

i)       Conduct a test to determine whether there is a significant relationship between sales and the four explanatory variables

ii)       Determine  which  of  the  explanatory  variables  have  significant  regression coefficients. Which variable(s) would you consider eliminating?

(12 marks)

Q3 (a)     (i)    State the assumptions behind the classical linear regression model and explain

briefly what each means and how to check them.

(ii)    For each of the following models, outline the method you would use to estimate

the parameters.

Y = 0  +  1 X1 +  2 X2  +  3 X1(2)  +  4 X2(2) +  5X1X2

Y = 0 XX                                                                                    (15 marks)

b)          Data on a sample of 28 firms in the clothing industry gave the following cost function:

Ŷ =134.6+57.97X - 11.029X2 +1.143X3                              R2 = 0.88         (Model 2)

Where Y represent the total cost (in £1000s) and X represent the quantity of output (in thousands).

i)   Estimate the total cost from both models when quantity of output is 10,000.

ii)   At the 1% level of significance, test to determine whether Model 2 is significant.

iii)  At the 1% level of significance, test to see whether Model 2 is superior to Model 1.

iv)  Determine the adjusted coefficient of determination for Model 2. Explain the difference between R2 and the adjusted R2 .

(18 marks)

Propose a time series regression model for quarterly data ( with 16 observations) that will account for both the linear trend and seasonal variations in the data.

From your proposed model, write down the forecast for each quarter of Year 5.

Explain, with the help of a diagram, what the quarterly dummy variables do.       (12 marks)

The following data represent the annual revenues (in billions of pounds) of a company over the past 15 years.

Year

Revenues

1

5.2

2

5.4

3

6.0

4

7.0

5

8.0

6

9.7

7

10.2

8

10.8

9

10.2

10

10.7

11

10.7

12

11.5

13

13.4

14

17.0

15

18.0

Forecast the revenues for the next two years using:

i)         Moving average, with K=3

ii)        Exponential smoothing with α= 1

iii)       Holt  model with α = 0.9 and   y =0.7 (You may use L14 =16.77,  T14 = 2.89)

iv)       The estimated linear trend equation

Yt = 3.84 + 0.80t,                                t=1,2 …… 15

v)        The estimated quadratic trend model

Yt = 5.20 + 0.32t + 0.03t2                            t =1,2…… . 15

vi)       The estimated exponential trend model

Yt = 5.03e(0.08t)                                                        t =1,2…….15

(14 marks)

For the six forecasting methods in (b), the respective mean square errors (MSE) are:

MSE (Moving average) = 5.72

MSE (Exponential smoothing) = 2.03

MSE (Holt) = 1.20

MSE (Linear Trend Model) = 1.73

MSE (Quadratic trend model) = 1.40

MSE(Exponential trend model) = 1.34

Which of the six methods would you select for the purpose of forecasting? Discuss.

(7 marks)

Q5       a)        You are given a questionnaire to review about alcohol consumption habits as part

of a pilot study.  Some of the questions asked are as follows. How might you improve them?

i)   What is your age?             18 to 24          25 to 30           30 to 44          45 to 60

ii)  Which of the following best describes your alcohol consumption frequency? Every day       Once or twice a week             Once or twice a month

iii) Would you describe yourself as a problem drinker?

iv) Have you been inebriated in the last week?

b)  What is non-response error and when might it be a particular problem?

(12 marks)

(7 marks)

c)   Suggest one way in which one could analyse returns from a questionnaire-based survey to assess whether non-response error might pose a problem?                       (14 marks)