Questions:
Question 1
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 supply and unit price are related. Use α = 0.05. (2 marks) Determine
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
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 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. (2 marks)
d. 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?
Answers
Question Three
Data: Record of Outputs of Four Program test by Allied Corporation
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Program A
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Program B
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Program C
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Program D
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150
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150
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185
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175
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130
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120
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220
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150
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120
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135
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190
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120
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180
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160
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180
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130
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145
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110
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175
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175
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- ANOVA Table test the difference between mean output of the four programs
It was constructed using Microsoft Excel.
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Anova: Single Factor
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SUMMARY
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Groups
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Count
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Sum
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Average
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Variance
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Program A
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5
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725
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145
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525
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Program B
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5
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675
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135
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425
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Program C
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5
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950
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190
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312.5
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Program D
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5
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750
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150
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637.5
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ANOVA
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Source of Variation
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SS
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df
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MS
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F
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P-value
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F crit
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Between Groups
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8750
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3
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2916.666667
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6.140351
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0.00557
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3.238872
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Within Groups
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7600
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16
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475
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Total
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16350
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19
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- Advice to Allied Corporation
P-value, 0.00557 is less than 0.05, implying that there's significant difference between mean outputs of different Programs (Ruppert, 2014). These differences make the comparison of the output of the four Programs simple and easier. Also, from table of summary, it clear that means of output of the programs are different. Therefore Allied can select the program with higher mean output which is associated with small Variance. Small Variance is an indicator of low variation (risk) of the outputs. Allied should choose Program C, as its mean output is the highest and has the smallest variance among the four programs
Question Four
Data: A company record of weekly Sales for its product () the unit price of the
competitor's product (), and advertising expenditures ()
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Week
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Sales of Product Y
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Price of Competitor Product
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Advertising Expenditures
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1
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20
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0.33
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5
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2
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14
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0.25
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2
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3
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22
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0.44
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7
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4
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21
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0.4
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9
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5
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16
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0.35
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4
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6
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19
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0.39
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8
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7
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15
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0.29
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9
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- Regression Equation
Below is table showing the output of Regression Analysis
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SUMMARY OUTPUT
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Regression Statistics
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Multiple R
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0.8778
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R Square
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0.7706
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Adjusted R Square
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0.6558
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Standard Error
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1.8374
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Observations
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7
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ANOVA
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df
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SS
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MS
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F
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Significance F
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Regression
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2
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45.3528
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22.6764
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6.7168
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0.0526
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Residual
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4
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13.5043
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3.3761
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Total
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6
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58.8571
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Coefficients
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Standard Error
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t Stat
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P-value
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Lower 95%
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Upper 95%
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Lower 90.0%
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Upper 90.0%
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Intercept
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3.598
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4.052
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0.888
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0.425
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-7.653
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14.848
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-5.041
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12.236
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Price of Competitor Product X1
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41.320
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13.337
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3.098
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0.036
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4.290
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78.350
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12.887
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69.753
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Advertising Expenditures X2
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0.013
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0.328
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0.040
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0.970
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-0.896
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0.923
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-0.685
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0.712
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The estimated regression equation will be written as
- Significance of the overall model presented in part a, at
The model is not statistically significant at 0.1, since its P-value 0.0526 is slightly higher than 0.05.
- Significance of Competitor's Price and Advertising at 0.1 level
Competitor's Price () is statistically significant, as it P-value, 0.0363 is less than 0.05. On the other hands advertising () is insignificant since it P-value, 0.9697 is greater than 0.05.
- Re-estimation of the Regression equation by getting rid of Advertising(), insignificant Term in the first model,
The table below, shows the result of regression Analysis between Sales for its product () and Competitor's Price ()
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SUMMARY OUTPUT
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Regression Statistics
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Multiple R
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0.8778
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R Square
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0.7705
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Adjusted R Square
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0.7246
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Standard Error
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1.6438
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Observations
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7
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ANOVA
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df
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SS
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MS
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F
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Significance F
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Regression
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1
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45.3473
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45.3473
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16.7831
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0.0094
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Residual
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5
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13.5098
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2.7020
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Total
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6
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58.8571
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Coefficients
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Standard Error
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t Stat
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P-value
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Lower 95%
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Upper 95%
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Lower 90.0%
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Upper 90.0%
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Intercept
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3.5818
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3.6082
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0.9927
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0.3664
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-5.6934
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12.8570
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-3.6889
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10.8525
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Price of Competitor Product X1
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41.603
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10.1552
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4.0967
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0.0094
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15.4982
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67.7079
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21.1398
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62.0663
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The new regression Equation will be written as
- Interpretation of slope coefficient
The slope coefficient is 41.603. This value is positive which indicate a positive gradient and relationship between of product and Competitor’s Price(Francis,2004).This implies that when Competitor’s price change by one unit, the Sale of product will change by 41.603
References
Francis, A. (2004). Business mathematics and statistics. Cengage Learning EMEA.
Ruppert, D. (2014). Statistics and finance: an introduction. Springer.
