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Statistics Assignment

Task 1

Answer 1: Mean, Mode, Variance and Standard Deviation

The following data represent business startup costs (thousands of dollars) for shops.

X1 = startup costs for pizza

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X2 = startup costs for baker/donuts

X3 = startup costs for shoe stores

X4 = startup costs for gift shops

X5 = startup costs for pet stores

Statistics X1 X2 X3 X4 X5
Mean 83 92.0909091 72.3 87 51.63
Mode 35 #N/A #N/A 100 30
Median 80 87 70 97.5 49
Maxima 140 160 125 150 110
Minima 35 40 35 35 20
Range 105 120 90 115 90
Variance 1165.166667 1512.690909 983.7888889 1289.111111 733.05
Standard Deviation 34.13453774 38.89332731 31.36540911 35.9041935 27.0749

Answer 2: For business type construct


  1. a) frequency and relative frequency distributions

X1 Business (Frequency distribution)

Bin Frequency
25 0
50 3
75 2
100 4
125 3
150 1
More 0

X2 Business (Frequency distribution)

Bin Frequency
25 0
50 2
75 2
100 4
125 1
150 1
More 1

X3 Business (Frequency distribution)

Bin Frequency
25 0
50 4
75 2
100 2
125 2
150 0
More 0

X4 Business (Frequency distribution)

Bin Frequency
25 0
50 3
75 1
100 4
125 1
150 1
More 0

X5 Business (Frequency distribution)

Bin Frequency
25 3
50 6
75 4
100 2
125 1
150 0
More 0

  1. b) A relative frequency histogram

X1 business (relative frequency distribution)

Bin Relative frequency
25 0
50 0.230769231
75 0.153846154
100 0.307692308
125 0.230769231
150 0.076923077
More 0

X2 business (relative frequency distribution)

Bin Relative frequency
25 0
50 0.181818182
75 0.181818182
100 0.363636364
125 0.090909091
150 0.090909091
More 0.090909091

X3 business (relative frequency distribution)

Bin Relative frequency
25 0
50 0.4
75 0.2
100 0.2
125 0.2
150 0
More 0

X4 business (relative frequency distribution)

Bin Relative frequency
25 0
50 0.3
75 0.1
100 0.4
125 0.1
150 0.1
More 0

X5 business (relative frequency distribution)

Bin Relative frequency
25 0.1875
50 0.375
75 0.25
100 0.125
125 0.0625
150 0
More 0

Answer 3: Discussing result obtained from parts (answer) 1 and 2

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From the above calculation in Answer 1, it can be interpreted that X2 business is spending more in starting-up business as compared to the X5 business. From the above evaluation, it is also found that X2 has largest spread of values among the set of data of given businesses and this indicates that the mean value is not data representative.

However, large spread of data helps in interpreting that probably there may a large difference in individual scores which each business got. On the other hand, it is also determined that X3 and X5 business has low range of data representative that means that both business are less efficient to start a business. But in contrast to it, X2 business has low and high values comparatively as X2 business indicates that the mean value is widely spread.

At the same time, in Answer 2, it can be evaluated that frequency and relative frequency distributions in X2 business is also widely spread. From the above graphs, it can be easy to understand and identify that X2 distribution data is skewed to left as given data set value mostly are average i.e., some are small and large.

The change in the shape of data indicates that set of given data helped in analyzing the data in order to determine the average (mean) value. In similar manner, X5 business also has normally distributed data set.

Answer 4: Testing a significant difference in starting cost of business

Anova: Single Factor
SUMMARY
Groups Count Sum Average Variance
Column 1 13 1079 83 1165.167
Column 2 11 1013 92.09091 1512.691
Column 3 10 723 72.3 983.7889
Column 4 10 870 87 1289.111
Column 5 16 826 51.625 733.05
ANOVA
Source of Variation SS df MS F P-value F crit
Between Groups 14298.22 4 3574.556 3.246336 0.018391 2.539689
Within Groups 60560.76 55 1101.105
Total 74858.98 59

From the above ANOVA table, it can be easy to test and interpret the significant difference in starting cost of the business. With the help of this table, it can be determined that F value is more significant than the P value which is less than 0.05. However, P- value helped in determining that this data have major difference in the starting cost of business.

So, it can be stated that this types of business faces a significant difference in the starting cost of business.

Task 2

All Greens Franchise                       

The data (X1, X2, X3, X4, X5, X6) are for each franchise store.

X1 = annual net sales/$1000

X2 = number sq. ft./1000

X3 = inventory/$1000

X4 = amount spent on advertizing/$1000

X5 = size of sales district/1000 families

X6 = number of competing stores in district

Answer 1: Ms- Excel output and estimated regression equation

SUMMARY OUTPUT
Regression Statistics
Multiple R 0.996583914
R Square 0.993179497
Adjusted R Square 0.991555568
Standard Error 17.64924165
Observations 27
ANOVA
  df SS MS F Significance F
Regression 5 952538.9415 190507.7883 611.5903672 5.3973E-22
Residual 21 6541.410344 311.4957306
Total 26 959080.3519
  Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0%
Intercept -18.8594 30.1502 -0.6255 0.538 -81.5602 43.8414 -81.5602 43.8414
X Variable 1 16.2016 3.5444 4.5710 0.000 8.8305 23.5726 8.8305 23.5726
X Variable 2 0.1746 0.0576 3.0315 0.006 0.0548 0.2944 0.0548 0.2944
X Variable 3 11.5263 2.5321 4.5521 0.000 6.2605 16.7921 6.2605 16.7921
X Variable 4 13.5803 1.7705 7.6705 0.000 9.8984 17.2622 9.8984 17.2622
X Variable 5 -5.3110 1.7054 -3.1142 0.005 -8.8576 -1.7643 -8.8576 -1.7643

Regression Equation:

Y=m 1 X 1 + m 2 X 2 + m 3 X 3 + m 4 X 4+ m5 X 5 + C

Y= Annual sales

Y = 16.20*area+0.17* inventory + 11.53* advertising spending + 13.58*size of sales district + 5.31 * number of competing stores -18.86

Answer 2: Determining that how well model fit to the data

In order to determine that how well is model fit to the data, for that R-squared is used for measuring and this helps in determining the data are close to the fitted regression line.

In this measurement, if r-square is 0 % then it states that regression model demotes that there is none variability around the mean value. But, is it is found that R- square is around 100% then it indicates that there is variability near to mean value. In concern to it, the above calculated table shows that R-square value is 99.31% or 0.993179 which means that this model is well suited or fit with the data.

Answer 3: Testing the hypothesis (no significant relationship between the dependent and any independent variables)

The hypothesis test is conducted in order to determine the no significance difference between the dependent and independent variables. If P-value is less that 0.05 then it indicates that variables have a significant relationship and null hypothesis will get rejected. But is P-value is greater than 0.05 then null hypothesis accepted and both variables have no significance relationship between them.

Dependent and independent variables P-value Null Hypothesis

(Rejected or Accepted)
Annual sales and area 0.000 Rejected
Annual sales and inventory 0.006 Rejected
Annual sales and advertising spending 0.000 Rejected
Annual sales and size of sales district 0.000 Rejected
Annual sales and Competing Stores 0.005 Rejected

Answer 4: Interpret individual slope coefficients

Variable Slope Interpretation
Area 16.2016 The rate of change in the mean value of sales in respect to the area is 16.2016.
Inventory 0.1746 The rate of change of mean value of sales with respect to inventory is 0.1746.
Advertising spending 11.5363 The rate of change of mean value of sales with respect to advertising spending is 11.5363.
Size of sales district 13.5803 The rate of change of the conditional mean of sales with respect to size of sales district is about 13.5803.
Number of competing stores -5.3110 With respect to the rate of change of mean value of sales number, then competing stores is -5.3110

From the above stated table, it can be easily interpreted that the change in area of store is due to the change in the advertising spending and sales district size and also due to which there is maximum change in the sales observed and minimum impact on inventory of the firm.

Answer 5: Construct a 95% confidence interval for the slope coefficients of individual variables

Variable Lower 95.0% Upper 95.0%
Sales -81.5602 43.8414
Area 8.8305 23.5726
Inventory 0.0548 0.2944
Advertising spending 6.2605 16.7921
Size (sales district) 9.8984 17.2622
Competing stores -8.8576 -1.7643

Answer 6: For determining the significance of individual variables, then there is need to test the estimated slope coefficients

For identifying the significance of individual variable, the hypothesis is tested by estimating the slope coefficient. In this, if t-value is absolute value and greater than critical value then null hypothesis is rejected. But is t-value is less than critical value then null hypothesis is accepted.

t-Test: Paired Two Sample for Means (Area)
  Sales Area
Mean 286.5740741 3.32592593
Variance 36887.70584 4.044301983
Observations 27 27
Pearson Correlation 0.894092081
Hypothesized Mean Difference 0
df 26
t Stat 7.735497275
P(T<=t) one-tail 1.65162E-08
t Critical one-tail 1.705617901
P(T<=t) two-tail 3.30324E-08
t Critical two-tail 2.055529418
t-Test: Paired Two Sample for Means (Inventory )
  sales inventory
Mean 286.574074 387.4815
Variance 36887.7058 36545.11
Observations 27 27
Pearson Correlation 0.94550363
Hypothesized Mean Difference 0
df 26
t Stat -8.28771958
P(T<=t) one-tail 4.5308E-09
t Critical one-tail 1.7056179
P(T<=t) two-tail 9.0617E-09
t Critical two-tail 2.05552942
t-Test: Paired Two Sample for Means (advertising spending)
  sales advertising spending
Mean 286.5741 8.099999982
Variance 36887.71 14.24692313
Observations 27 27
Pearson Correlation 0.914024
Hypothesized Mean Difference 0
df 26
t Stat 7.671559
P(T<=t) one-tail 1.92E-08
t Critical one-tail 1.705618
P(T<=t) two-tail 3.85E-08
t Critical two-tail 2.055529
t-Test: Paired Two Sample for Means (size)
  sales size
Mean 286.5741 9.692593
Variance 36887.71 26.41994
Observations 27 27
Pearson Correlation 0.953683
Hypothesized Mean Difference 0
df 26
t Stat 7.686851
P(T<=t) one-tail 1.85E-08
t Critical one-tail 1.705618
P(T<=t) two-tail 3.71E-08
t Critical two-tail 2.055529
t-Test: Paired Two Sample for Means (competing stories)
  sales No. of competing stores
Mean 286.5741 7.740741
Variance 36887.71 23.96866
Observations 27 27
Pearson Correlation -0.91224
Hypothesized Mean Difference 0
df 26
t Stat 7.371908
P(T<=t) one-tail 3.96E-08
t Critical one-tail 1.705618
P(T<=t) two-tail 7.91E-08
t Critical two-tail 2.055529
Variables t-stat t-critical Accepted or Rejected Significance of individual variables
Area 7.735497275 2.055529 Rejected Statistically significant
Inventory -8.28771958 2.055529 Accepted Not significant
Advertising spending 7.671559 2.055529 Rejected Statistically significant
Size 7.686851 2.055529 Rejected Statistically significant
Competing stores 7.371908 2.055529 Rejected Statistically significant

However, this above stated table helped in determining that the value of individual variables i.e., inventory is not significant in this model.

Answer 7: Removing unnecessary significant variables and re-estimating the model by doing regression analysis.

SUMMARY OUTPUT
Regression Statistics
Multiple R 0.995085241
R Square 0.990194637
Adjusted R Square 0.988411844
Standard Error 20.67511795
Observations 27
ANOVA
  df SS MS F Significance F
Regression 4 949676.2208 237419.0552 555.4175271 9.5799E-22
Residual 22 9404.131054 427.4605024
Total 26 959080.3519
  Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0%
Intercept -39.46002205 34.41055873 -1.146741683 0.263807827 -110.823153 31.90311 -110.823 31.90311
X Variable 1 20.444 3.814801407 5.359095937 2.21824E-05 12.5324729 28.3553 12.53247 28.3553
X Variable 2 16.966 2.092787626 8.10695865 4.73185E-08 12.6259669 21.30632 12.62597 21.30632
X Variable 3 15.673 1.90985556 8.206359798 3.85791E-08 11.7121639 19.63376 11.71216 19.63376
X Variable 4 -4.043301284 1.936828415 -2.087588788 0.048629066 -8.06003755 -0.02657 -8.06004 -0.02657

Answer 8: Using the above model, for predicting annual sales for a franchise with floor area 1000sq ft:

Families in the area of operation = 5000

Competitors =2

Inventory= $150,000

Advertising expenses =$5,000

Equation: Annual Sales = Y=m 1 X 1 + m 2 X 2 + m 3 X 3 + m 4 X 4+ m5 X 5 + C

M1 = 20.44

M2 = 16.97

M3 = 15.67

M4 = 4.04

Intercept = 39.46

Annual Sales = 20.44*area + 16.97* advertising spending + 15.67*size of sales district + 4.04 * number of competing stores -39.46

= 20.44*1000+ 16.97*5000 + 15.67*5000+4.04*2 -39.46

= 20440+84850+ 78350+8.08-39.46

= 183609/$1000

= $ 183.609

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