Predicting Annual Franchise Sales

This assignment involves analyzing a statistical model used to predict annual net sales for franchise businesses. The model considers factors such as floor area, advertising expenditures, population density, and the number of competing stores in the district. Students are tasked with interpreting the coefficients of the model, understanding their significance, and applying it to predict the annual net sales for a specific franchisee scenario.

Running head: STATISTICS
Statistics
Name:
Institution:

STATISTICS
Task 1
The start-up cost of business is important to understand so that a prospect venture
enterprise can understand how much it takes to start up a business. The report analysed five
businesses start-up costs. First, the descriptive measures were computed to determine the
distribution of the five start-up cost data. Second, it was evaluated whether the start-up cost
significantly differs.
The descriptive analysis was carried out, and the results are as summarized below.
Table 1: Descriptive statistics for start-up cost
Descriptive statistics
X1 X2 X3 X4 X5
count 13 11 10 10 16
mean 83.00 92.09 72.30 87.00 51.63
sample standard deviation 34.13 38.89 31.37 35.90 27.07
sample variance 1,165.17 1,512.69 983.79 1,289.11 733.05
minimum 35 40 35 35 20
maximum 140 160 125 150 110
range 105 120 90 115 90
standard error of the mean 9.47 11.73 9.92 11.35 6.77
1st quartile 58.00 67.50 45.75 56.25 29.50
median 80.00 87.00 70.00 97.50 49.00
3rd quartile 110.00 110.00 90.75 100.00 75.00
interquartile range 52.00 42.50 45.00 43.75 45.50
The summary shows that the average startup costs for pizza are $83.00 (SD = 34.13).
The mode ($35.00) is significantly lower than the mean, and the median close to the average
startup cost $80.00. The minimum pizza startup cost is $35.00 and a maximum of $140.00.
The range of the pizza business startup cost is $105.00. The difference between the 75th and
25th percent is $52.00. The average startup costs for baker/donuts is $92.09 (SD = $38.89).
The middle 50% of the startup cost is between $67.50 and $110.00. The median startup costs
for baker/donuts is $80.00. The average startup costs for shoe stores is $72.30 (SD = $31.37),
and a median of $70.00. The minimum startup cost for the shoe stores is $35.00 and the

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maximum of $90.00. This indicates that the range between the minimum and maximum is
$90.00.
The average startup costs for gift shops is $87.00 (SD = 35.90), and the median is
$97.50. The range between the minimum ($35.00) and the maximum ($150.00) is $115.00.
Lastly, the average startup costs for pet stores is $51.63 (SD = 27.07), and the median is
$49.00. The minimum amount requires to startup the pet store is $20.00 and the maximum
amount is $110.00.
The distribution of the data was carried out, and the frequency table and histogram are
as portrayed below.
Table 2: startup costs for pizza frequency table
Frequency Distribution - Quantitative
30
60
90
120
150
0
1
2
3
4
5
startup costs for pizza histogram
startup costs for pizza
Frequency
Figure 1: startup costs for pizza histogram

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The table and the histogram indicate that most of the businesses startup costs are on
the lower side of the plot. This means that the data are positively skewed with a long tail to
the right.
Table 3: startup costs for baker/donuts frequency table
Frequency Distribution - Quantitative
X2 cumulative
lower upper midpoint width frequency percent frequency percent
30 < 60 45 30 2 18.2 2 18.2
30
60
90
120
0
1
2
3
4
5
startup costs for pizza
Frequency
30
60
90
120
150
180
0
1
2
3
4
5
startup costs for baker/donuts histogram
X2
Frequency
Figure 2: startup costs for baker/donuts histogram
The chart indicates that most of the startup cost of the baker/donut business is
between $60.00 and $90.00. There is a long tail to the higher value side, which shows that
few of the business is starting with more cash.
Table 4: Startup costs for shoe stores frequency table
Frequency Distribution - Quantitative
30
60
90
120
150
0
1
2
3
4
5
startup costs for baker/donuts histogram
X2
Frequency

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30
60
90
120
150
0
1
2
3
4
5
startup costs for shoe stores histogram
X3
Frequency
Figure 3: startup costs for shoe stores histogram
The chart shows that there is a consequent decline in the number of businesses with
the increase of the startup cost. This indicates that the data are positively skewed (Blanca,
Arnau, López-Montiel, Bono, & Bendayan, 2013).
Table 5: startup costs for gift shops frequency table
30
60
90
120
0
1
2
3
4
5
startup costs for shoe stores histogram
X3
Frequency
30
60
90
120
150
180
0
1
2
3
4
5
startup costs for gift shops histogram
X4
Frequency
Figure 4: startup costs for gift shops histogram

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Most of the gift shops are started up with between $90.00 and $120.00. However,
there seems to be a skewness in the data, with a long tail to the right on the plot (Blanca,
Arnau, López-Montiel, Bono, & Bendayan, 2013).
Table 6: startup costs for pet stores frequency table
Frequency Distribution - Quantitative
X5 cumulative
lower upper midpoint width frequency percent frequency percent
30
60
90
120
150
0
1
2
3
4
5
X4
Frequency
0
30
60
90
120
0
1
2
3
4
5
6
7
startup costs for pet stores histogram
X5
Frequency
Figure 5: startup costs for pet stores histogram
The most frequent startup cost for the pet store is between $30.00 and $60.00, with a
slightly longer tail on the higher side (Johnson & Wichern., 2014).
One-factor ANOVA test was carried out test the hypothesis:
H0: All average startup cost for businesses are equal.
H1: At least one startup cost of businesses average is different.
The results are as summarized below.

STATISTICS
72.3 10 31.37 X3
87.0 10 35.90 X4
51.6 16 27.07 X5
75.2 60 35.62 Total
ANOVA table
Source SS df MS F
Treatment 14,298.22 4 3,574.556 3.25
Error 60,560.76 55 1,101.105
Total 74,858.98 59
72.3 10 31.37 X3
87.0 10 35.90 X4
51.6 16 27.07 X5
75.2 60 35.62 Total
The summary shows that there is enough evidence to support the claim that at least
one startup cost average is different (Keller, 2014). This implies that the at least one of the
averages is different from the other. In particular, at least one of the business require the
different amount to start. An investigation was carried out to determine which businesses
required different start-up amount.
X1 X2 X3 X4 X5
0.0
20.0
40.0
60.0
80.0
100.0
120.0
140.0
160.0
180.0
Comparison of Groups
Figure 6: Comparison of Groups
The comparison of the averages shows that businesses X1, X2, and X4, were not
statistically different and higher than those required to start-up businesses X3 and X5.
Businesses X3 and X5 average start-up cost were not statistically different.

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Task 2
Multiple regression models to predict the annual net sales/$1000 using the number sq.
Ft./1000, inventory/$1000, the amount spent on advertising/$1000, size of sales district/1000
families and number of competing stores in the district. The model is as illustrated below.
Regression Analysis
R² 0.993
Adjusted R² 0.992 n 27
R 0.997 k 5
Std. Error 17.649 Dep. Var. X1
ANOVA table
Source SS df MS F p-value
Regression 952,538.9415 5 190,507.7883 611.59 5.40E-22
Residual 6,541.4103 21 311.4957
Total 959,080.3519 26
Regression output confidence interval
variables coefficients std. error t (df=21) p-value 95% lower 95% upper VIF
Intercept -18.8594 30.1502 -0.626 .5384 -81.5602 43.8414
X2 16.2016 3.5444 4.571 .0002 8.8305 23.5726 4.241
X3 0.1746 0.0576 3.032 .0063 0.0548 0.2944 10.122
X4 11.5263 2.5321 4.552 .0002 6.2605 16.7921 7.624
X5 13.5803 1.7705 7.671 1.61E-07 9.8984 17.2622 6.912
X6 -5.3110 1.7054 -3.114 .0052 -8.8576 -1.7643 5.819
The p-value < .05 shows that the developed model is significant (Draper & Smith,
2014). However, in accordance with (Chatterjee & Hadi., 2015), the VIF value greater than
10 shows a serious multicollinearity. Therefore, the variable X3 (inventory/$1000) should be
deleted from the regression model. Another model was remodeled and the result was as
follows.

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R² 0.990
Adjusted R² 0.988 n 27
R 0.995 k 4
Std. Error 20.675 Dep. Var. X1
ANOVA table
Source SS df MS F p-value
Regression 949,676.2208 4 237,419.0552 555.42 9.58E-22
Residual 9,404.1311 22 427.4605
Total 959,080.3519 26
Regression output confidence interval
variables coefficients std. error t (df=22) p-value 95% lower 95% upper VIF
Intercept -39.4600 34.4106 -1.147 .2638 -110.8232 31.9031
X2 20.4439 3.8148 5.359 2.22E-05 12.5325 28.3553 3.580
X4 16.9661 2.0928 8.107 4.73E-08 12.6260 21.3063 3.795
X5 15.6730 1.9099 8.206 3.86E-08 11.7122 19.6338 5.862
X6 -4.0433 1.9368 -2.088 .0486 -8.0600 -0.0266 5.469
4.676
mean VIF
The VIF values are lower than 10, which means that the assumption of
multicollinearity is met (Chatterjee & Hadi., 2015). All the variables in the model are
significant (p-value < .05). The coefficient of determination (0.988), shows that the model
can take into account to 98.8% sources of variation. Therefore, it implies that 1.2% of the
variation could not be taken into account.
Model 2 developed is:
X1 (annual net sales/$1000) = -39.4600 + 20.4439X2 + 16.9661X4 + 15.6730X5 - 4.0433X6
The coefficient of the variables X1, X2, X4, and X5 are positive, which implies that
they positively influence the annual sales/$1000. First, an increase in one unit of X2 increases
the annual net sales by $20.44. An increase in one unit of X4 increases the net sale by $16.97.
Lastly, an increase in one unit of X5 increases the annual net sales/$1000 by $15.67. On the
other side, X6 has a negative, which means that an increase in the number of competing
stores in the district (X6) decreases the annual net sales/$1000. The model indicates that an

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increase in one unit of X6 causes a decline of $4.04 in the annual net sales/$1000. The 95%
confidence interval does not contain a zero, which implies that each coefficient is significant
(Draper & Smith, 2014).
A prediction is made on the annual sales for a franchisee with 1,000 sq ft floor area,
$5,000 spent on advertising, 5,000 families in the area of operation and two competitors.
X1 (annual net sales/$1000) = -39.4600 + 20.4439X2 + 16.9661X4 + 15.6730X5 - 4.0433X6
= -39.4600 + 20.4439(1000) + 16.9661(5000) + 15.6730(5000) - 4.0433(2)
= 183591.8534
It is predicted that the annual net sales will be $183,591,853.40, when the variables area as
given.

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References
Blanca, M. J., Arnau, J., López-Montiel, D., Bono, R., & Bendayan, R. (2013). Skewness and
kurtosis in real data samples. Methodology.
Chatterjee, S., & Hadi., A. S. (2015). Regression analysis by example. John Wiley & Sons.
Draper, N. R., & Smith, H. (2014). Applied regression analysis. John Wiley & Sons.
Johnson, R. A., & Wichern., D. W. (2014). Applied multivariate statistical analysis (Vol. 4).
New Jersey: Prentice-Hall.
Keller, G. (2014). Statistics for management and economics. Nelson Education.