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HI6007 Statistics For Business Decision

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Question 1 

Below you are given the examination scores of 20 students  

52

99

92

86

84

63

72

76

95

88

92

58

65

79

80

90

75

74

56

99

 
a. Construct a frequency distribution, cumulative frequency distribution, relative frequency distribution, cumulative relative frequency distribution and percent frequency distribution for the data set using a class widthof
 
b. Construct a histogram showing the percent frequency distribution of the examination scores. Comment on the shape ofthe  

Question

Shown below is a portion of a computer output for a regression analysis relating supply (Y in thousands of units) and unit price (X in thousands of dollars).

ANOVA

 

 

 

df

SS

Regression

1

354.689

Residual

39

7035.262

 

Coefficients

Standard Error

Intercept

54.076

2.358

X

0.029

0.021

 

a.

What has been the sample size for this problem? 

b.

Determine whether or not supply and unit price are related. Use α = 0.05. 

c.

Compute the coefficient of determination and fully interpret its meaning. Be very specific.

 

d.

Compute the coefficient of correlation and explain the relationship between supply and unit price. 

e.

Predict the supply (in units) when the unit price is $50,000. 

 

Question 3

Allied Corporation wants to increase the productivity of its line workers. Four different programs have been suggested to help increase productivity. Twenty employees, making up a sample, have been randomly assigned to one of the four programs and their output for a day's work has been recorded. You are given the results below  

Program A

Program B

Program C

Program D

150

150

185

175

130

120

220

150

120

135

190

120

180

160

180

130

145

110

175

175

 

a.

Construct an ANOVA table.

b.

As the statistical consultant to Allied, what would you advise them? Use a .05 level of significance. 


Question 4

A company has recorded data on the weekly sales for its product (y), the unit price of the competitor's product (x1), and advertising expenditures (x2). The data resulting from a random sample of 7 weeks follows. Use Excel's Regression Tool to answer the following

 

Week

Price

Advertising

Sales

 

1

.33

5

20

 

2

.25

2

14

 

3

.44

7

22

 

4

.40

9

21

 

5

.35

4

16

 

6

.39

8

19

 

7

.29

9

15

a.

What is the estimated regression equation? Show the regression output. 

b.

Determine whether the model is significant overall. Use α = 0.10. 

c.

Determine if competitor’s price and advertising is individually significantly related to sales. Use α = 0.10. 

 

d.

 

e.

Based on your answer to part (c), drop any insignificant independent variable(s) and re-estimate the model. What is the new estimated regression equation? 

 

Interpret the slope coefficient(s) of the model from part (d).

 

 

Answer 1

a. Frequency descriptions
 
 

 

 

 

 

 

 

Examination Score

Frequency distribution

Cumulative Frequency

Relative Frequency

Cumulative relative Frequency

Percent Frequency

50-60

3

3

0.15

15%

15%

 

60-70

2

5

0.10

25%

10%

 

70-80

6

11

0.30

55%

30%

 

80-90

4

15

0.20

75%

20%

 

90-100

5

20

0.25

100%

25%

 

Total

20

 

 

 

 

 

b. Construction of the histogram

 

The examination score between the range of 70-80 is responsible for 30% of the total scores. This means the range 70-80 is the modal class range.

Answer 2

  1. Suppose the sample size is denoted by n and the number of regressions by k. by observing the regression output availed, k is given as 1. Being that

       We can obtain , this gives us the value of n as 41

       The sample size is therefore obtained as 41.

  1. For x and y to be correlated, the coefficient of x must be a value other than zero.

For this reason, will test the hypothesis that the coefficient

Our hypothesis is therefore defined as

vs

The value of

The rejection rule is; reject the null hypothesis whenever the p value is less than the value of alpha.

Now we compute the p value

SSR

354.689

SSE

7035.262

df Regression

1

df Error

39

F statistic

1.96622

p value

0.16876

The value is obtained as 0.16876, which is greater than alpha which is 0.05. we thereby fail to reject the null hypothesis.

We thereby conclude that there is sufficient evidence from the sample to prove that the coefficient of x is 0, this indicates that the values of x and y are not correlated.

  1. The coefficient of determination will be obtained as 0 given that the two variables (x and y0 are independent of each other.
  2. The coefficient of correlation is obtained as 0 given that the coefficient of determination is 0. This shows indecency among the two variables.
  3. The output given can be used to derive the linear equation relating x and y as;

Fromm this equation we can substitute the value of x with 50000 and obtain the number of units as

 

Answer 3

  1. Construction of the ANOVA table

Anova: Single Factor

 

 

 

 

 

SUMMARY

 

 

 

 

 

 

Groups

Count

Sum

Average

Variance

 

 

Program A

5

725

145

525

 

 

Program B

5

675

135

425

 

 

Program C

5

950

190

312.5

 

 

Program D

5

750

150

637.5

 

 

ANOVA

 

 

 

 

 

 

Source of Variation

SS

df

MS

F

P-value

F crit

Between Groups

8750

3

2916.666667

6.140351

0.00557

3.238872

Within Groups

7600

16

475

 

 

 

Total

16350

19

 

 

 

 

 

  1. Discussion and Recommendation

Suppose

The task needs the conduction of a hypothesis

 vs at least one of the means is difference.

The

The p value is obtained from the excel output as 0.00557, compared to the value of

Alpha  this falls in the rejection region hence we reject the

Null hypothesis and conclude that the evidence indicates at least one of the means is

Difference.

The Advice to the Allied corporation is that there is a statistical difference among the

Outputs that arise from the workers under the difference. The firm therefore need to

specialise in the program that generates optimum output. This is specifically program

C.

Answer 4

  1. Regression equation

Display of the multiple regression output

SUMMARY OUTPUT

 

 

 

 

 

 

 

Regression Statistics

 

 

 

 

 

 

 

Multiple R

0.877814

 

 

 

 

 

 

 

R Square

0.770558

 

 

 

 

 

 

 

Adjusted R Square

0.655837

 

 

 

 

 

 

 

Standard Error

1.83741

 

 

 

 

 

 

 

Observations

7

 

 

 

 

 

 

 

ANOVA

 

 

 

 

 

 

 

 

 

df

SS

MS

F

Significance F

 

 

 

Regression

2

45.35284447

22.67642

6.716801

0.052644

 

 

 

Residual

4

13.50429839

3.376075

 

 

 

 

 

Total

6

58.85714286

 

 

 

 

 

 

 

Coefficients

Standard Error

t Stat

P-value

Lower 95%

Upper 95%

Lower 90.0%

Upper 90.0%

Intercept

3.597615

4.052243833

0.887808

0.424805

-7.65322

14.84845

-5.04115

12.23638

Price

41.32002

13.33736254

3.098065

0.036289

4.289567

78.35048

12.88681

69.75324

Advertising

0.013242

0.327591655

0.040422

0.969694

-0.8963

0.922782

-0.68513

0.711617

The output can be used to derive the linear equation:

  1. The significance of the model is given as 0.0526 which is less than the value of alpha (0.10). the conclusion is thus the model is statistically significant.
  2. The competitors price p value is obtained from the model as 0.0362, this is lower than the alpha value hence the price can be stated to be statistically significant to the sales values.

Meanwhile the alpha value for advertising is given from the model as 0. l9697 which is higher than the value of alpha. Advertising is therefore not statistically significant to the sales value.

  1. Revised regression model

SUMMARY OUTPUT

 

 

 

 

 

 

 

 

Regression Statistics

 

 

 

 

 

 

 

Multiple R

0.877760967

 

 

 

 

 

 

 

R Square

0.770464315

 

 

 

 

 

 

 

Adjusted R Square

0.724557178

 

 

 

 

 

 

 

Standard Error

1.643764862

 

 

 

 

 

 

 

Observations

7

 

 

 

 

 

 

 

ANOVA

 

 

 

 

 

 

 

 

 

df

SS

MS

F

Significance F

 

 

 

Regression

1

45.34732824

45.34732824

16.7831053

0.009384894

 

 

 

Residual

5

13.50981461

2.701962923

 

 

 

 

 

Total

6

58.85714286

 

 

 

 

 

 

 

Coefficients

Standard Error

t Stat

P-value

Lower 95%

Upper 95%

Lower 90.0%

Upper 90.0%

Intercept

3.581788441

3.608215389

0.992675895

0.366447177

-5.693424497

12.85700138

-3.688940109

10.85251699

Price

41.60305344

10.15521323

4.096718846

0.009384894

15.49824676

67.70786011

21.13980753

62.06629934

This model generates the linear equation;

  1. The model has a slope of 41.60 which indicates that a unit increase in the price of the competitors goods raises the sales volume by 41.60 units.
 

References

Andrew, G., 2008. Variance, analysis of. The new Palgrave dictionary of economics. 2nd ed. Basingstoke, Hampshire New York: Palgrave Macmillan.

Bailey, R. A., 2008. Design of Comparative Experiments, s.l.: Cambridge University Press.

Klaus, H. & & Oscar, K., 2008). . Design and Analysis of Experiments. I and II ed. s.l.:Wiley.

Scott, J. A., 2012. Illusions in Regression Analysis. International Journal of Forecasting, 28(3), p. 689.

Willem, W., Baets, D. & Luc, B., 2008. ROC analysis in ordinal regression learning. Pattern Recognition Letters, Volume 29, p. 1–9.

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