HI6007 Statistics For Business Decisions

Below you are given the examination scores of 20 students (data set also provided in accompanying MS Excel file).
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 width of 10.

b. Construct a histogram showing the percent frequency distribution of the examination scores. Comment on the shape of the distribution.

 

Question 2 
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). 
a. What has been the sample size for this problem
b. Determine whether or not demand and unit price are related. Use α = 0.05. (2 marks) Determine whether or not demand 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 (data set also provided in accompanying MS Excel file).
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

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 questions (data set also provided in accompanying MS Excel file).
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. 
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). 

1. The frequency distribution tables were constructed in MS Excel and have been provided below.

Table 1: Frequency Distribution of Examination Scores

CLASS INTERVAL	LOWER BOUND	UPPPER BOUND	CLASS MID-POINT	FREQUENCY
50-60	50	59	54.5	3
60-70	60	69	64.5	2
70-80	70	79	74.5	5
80-90	80	89	84.5	4
90-100	90	99	94.5	6

Table 2: Cumulative Frequency Distribution of Examination Scores

CLASS INTERVAL	LOWER BOUND	UPPPER BOUND	CLASS MID-POINT	FREQUENCY	CUMULATIVE FREQUENCY
50-60	50	59	54.5	3	3
60-70	60	69	64.5	2	5
70-80	70	79	74.5	5	10
80-90	80	89	84.5	4	14
90-100	90	99	94.5	6	20

Table 3: Relative Frequency Distribution of Examination Scores

CLASS INTERVAL	LOWER BOUND	UPPPER BOUND	CLASS MID-POINT	FREQUENCY	RELATIVE FREQUENCY
50-60	50	59	54.5	3	0.15
60-70	60	69	64.5	2	0.10
70-80	70	79	74.5	5	0.25
80-90	80	89	84.5	4	0.20
90-100	90	99	94.5	6	0.30

Table 4: Cumulative Relative Frequency Distribution of Examination Scores

CLASS INTERVAL	LOWER BOUND	UPPPER BOUND	CLASS MID-POINT	FREQUENCY	RELATIVE FREQUENCY	CUMULATIVE RELATIVE FREQUENCY
50-60	50	59	54.5	3	0.15	0.15
60-70	60	69	64.5	2	0.10	0.25
70-80	70	79	74.5	5	0.25	0.50
80-90	80	89	84.5	4	0.20	0.70
90-100	90	99	94.5	6	0.30	1.00

Table 5: Percent Frequency Distribution of Examination Scores

CLASS INTERVAL	LOWER BOUND	UPPPER BOUND	CLASS MID-POINT	FREQUENCY	PERCENT FREQUENCY
50-60	50	59	54.5	3	15.00%
60-70	60	69	64.5	2	10.00%
70-80	70	79	74.5	5	25.00%
80-90	80	89	84.5	4	20.00%
90-100	90	99	94.5	6	30.00%

Answer 2

41. It is known that the “degrees of freedom” of residual in a regression model is taken as (n-p-1), where p is the degrees of freedom of the regression model. Here , so the value of n (sample size) is calculated as 41.
42. The margin of error at 5% level of significance is and the confidence interval of the slope is [0.029 – 0.041, 0.029 + 0.041] = [-0.012, 0.07]. The standard error of the slope indicates that the values of supply are very closely accumulated around the regression line of X, unit price (in thousands of dollars).
43. The given regression table is incomplete, where the values of SS for regression model and residue are provided. Now coefficient of determinant is known as , where SST = SSE + SSM = 7389.951. Hence, coefficient of determinant is calculated to be . This implies that 95.2% variation of supply (Y) is explained by the unit price (X).
44. Coefficient of correlation implies that the supply (Y) and unit price (X) are positively correlated. The value of the correlation indicated a steep increase in supply for change in unit price.
45. The regression equation is

Now for Unit Price = $ 50,000, the dependent variable is calculated as (in thousands of units). Hence predicted supply is approximately 55,530 units (Black, 2011). 

Answer 3

1. The ANOVA table is given in table 6 (constructed in MS Excel).

Table 6: ANOVA for programs in Allied Corporation

ANOVA: Single Factor
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	6.1403	0.0055	3.2388
Within Groups	7600	16	475
Total	16350	19

Answer 4

1. The table 7 shows the regression output at 5% level of significance, calculated in Excel. The regression equation from the model is formed as,

Table 7: Regression Model for Weekly Sales and Price of Competitor’s Product at 5% Level of significance

SUMMARY OUTPUT	Regression 95%
Regression Statistics
Multiple R	0.8778
R Square	0.7706
Adjusted R Square	0.6558
Standard Error	1.8374
Observations	7
ANOVA
	df	SS	MS	F	Significance F
Regression	2.0000	45.3528	22.6764	6.7168	0.0526
Residual	4.0000	13.5043	3.3761
Total	6.0000	58.8571
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	3.5976	4.0522	0.8878	0.4248	-7.6532	14.8484
Price	41.3200	13.3374	3.0981	0.0363	4.2896	78.3505
Advertising	0.0132	0.3276	0.0404	0.9697	-0.8963	0.9228

Table 8: Regression Model for Weekly Sales and Price of Competitor’s Product at 10% Level of significance

SUMMARY OUTPUT	Regression 90%
Regression Statistics
Multiple R	0.8778
R Square	0.7706
Adjusted R Square	0.6558
Standard Error	1.8374
Observations	7
ANOVA
	df	SS	MS	F	Significance F
Regression	2.0000	45.3528	22.6764	6.7168	0.0526
Residual	4.0000	13.5043	3.3761
Total	6.0000	58.8571
	Coefficients	Standard Error	t Stat	P-value	Lower 90.0%	Upper 90.0%
Intercept	3.5976	4.0522	0.8878	0.4248	-5.0411	12.2364
Price	41.3200	13.3374	3.0981	0.0363	12.8868	69.7532
Advertising	0.0132	0.3276	0.0404	0.9697	-0.6851	0.7116

1. The table 9 shows the regression output at 10% level of significance with unit price of competitors’ (X2) as the only variable (Vittinghoff et al., 2011). The new regression equation is calculated as

Table 9: Regression Model for Weekly Sales and Price of Competitor’s Product at 10% Level of significance with Single Factor

SUMMARY OUTPUT	Regression 90%
Regression Statistics
Multiple R	0.8778
R Square	0.7705
Adjusted R Square	0.7246
Standard Error	1.6438
Observations	7
ANOVA
	df	SS	MS	F	Significance F
Regression	1.0000	45.3473	45.3473	16.7831	0.0094
Residual	5.0000	13.5098	2.7020
Total	6.0000	58.8571
	Coefficients	Standard Error	t Stat	P-value	Lower 90.0%	Upper 90.0%
Intercept	3.5818	3.6082	0.9927	0.3664	-3.6889	10.8525
Price	41.6031	10.1552	4.0967	0.0094	21.1398	62.0663

References

Black, K., 2009. Business statistics: Contemporary decision making. John Wiley & Sons.

Black, K., 2011. Business statistics: for contemporary decision making. John Wiley & Sons.

Burns, A.C., Bush, R.F. and Sinha, N., 2014. Marketing research (Vol. 7). Harlow: Pearson.

Cox, D.R. and Wermuth, N., 2014. Multivariate dependencies: Models, analysis and interpretation. Chapman and Hall/CRC.

Montgomery, D.C., Peck, E.A. and Vining, G.G., 2012. Introduction to linear regression analysis (Vol. 821). John Wiley & Sons.

Triola, M.F., 2013. Elementary statistics using Excel. Pearson.

Vittinghoff, E., Glidden, D.V., Shiboski, S.C. and McCulloch, C.E., 2011. Regression methods in biostatistics: linear, logistic, survival, and repeated measures models. Springer Science & Business Media.