Regression Analysis: Federated Islands Sales, Simple & Multiple Models

This assignment provides a comprehensive analysis of sales data using both simple and multiple regression models. The student begins by constructing a simple regression model to analyze the relationship between the sales of Federated Islands and time. The least square regression line equation is derived and the coefficients are interpreted, including the slope and intercept. The model's performance is evaluated using the coefficient of determination (R²), correlation coefficient (R), and standard error (S.E.). The 95% confidence interval for the slope coefficient is also calculated and interpreted. The assignment then progresses to a multiple regression model, incorporating multiple independent variables like year, GDP, price index, population, survey score, advertisement, and number of stores. The least square regression line equation is presented, and the coefficients are interpreted. The assignment includes the interpretation of R², adjusted R square, and the correlation coefficient. An ANOVA table and hypothesis testing are conducted to assess the significance of the model and individual variables. Finally, 95% confidence intervals for the regression coefficients are provided for each independent variable. The analysis demonstrates how to interpret the significance of each variable in the model, providing a thorough understanding of the statistical techniques used in regression analysis.

STUDENT NAME/ID
SIMPLE & MULTIPLE
REGRESSION MODEL

Simple Regression Model
Federated Islands
Sales (USD) = Dependent variable
Year (time) = Independent variable
Least Square Regression Line Equation
y = mx+ c
Ln(Sales (USD)) = 13.0191 + {0.0328 (Time)}

Ln(Sales $) = 13.0191 + {0.0328 (Time)}
• Interpretation of Coefficients
Slope coefficient:
m = 13.0191
The above value indicates the rate of change
Intercept
c = 0.0328
At t =0, the above value would be observed
The value of sales (USD) for Federated Islands would be determined based on the
given time.
For example, the sales for Federated Islands for a period of 8 years would be computed
below:
Ln(Sales $) = 13.0191 + {0.0328 (Time)}
Time = 8
Ln(Sales $) = 13.0191 + {0.0328*8}
Sales = $587,428
Hence, the mean sales is $587,428
Simple Regression Model

• Interpretation of coefficient of determination
R² = 0.8067
80.67% change in dependent variable explained
• Interpretation of correlation coefficient
R = 0.8982
High positive correlation between the variables
• Interpretation of standard error
S.E.= 0.1206
Average error stands at 0.1206
• Interpretation of 95% confidence interval for slope coefficient
Lower limit of 95% confidence interval = 0.0259
Upper limit of 95% confidence interval = 0.0397
There is 95% likelihood that the ln(sales) would increase by 0.0259 to 0.0397,
when the time is increased by 1 unit
Simple Regression Model

Hypothesis Testing
To check whether the slope coefficient is significant or not.
• Hypotheses
Hₒ : = 0
H : ≠ 0₁
• Level of significance = 5%
• Test statistics
t = slope coefficient / Standard error
t = 0.0328/0.0033 = 9.798
• The p value corresponding to slope coefficient.
P = 0.000
• It can be seen that the p value is lower than the level of significance and
therefore, sufficient evidence present to reject the null hypothesis and to
accept the alternative hypothesis.
Thus, the slope coefficient (time) is significant.
Simple Regression Model

Sales (USD) = Dependent variable
Year, GDP, Price Index, Population, Survey Score, Advertisement, Store =
Independent variables
Multiple Regression Model

Least Square Regression Line Equation
Ln(y) = mx+ c
Ln (Sales $) = 12.0094-(0.0125 *year) +(0.00*GDP) –(0.0281 *Price Index) + (0.00*
Population)+ (0.0160* Survey Score) + (0.0605 * Advertisement) + (0.0399* Stores)
Intercept
• Interpretation of coefficient of determination
R² = 0.9947
99.47% variation in dependent variable jointly explained
• Interpretation of correlation coefficient
R = 0.9974
High positive correlation between variables
• Interpretation of standard error
S.E.= 0.0232
Average standard error stands at 0.0232
Multiple Regression Model

• Interpretation of adjusted R square
Adjusted R square = 0.9925
Comparable to R square implying that majority of the independent
variables are significant
F test
Hₒ : All slopes can be assumed as insignificant and hence zero
H :Atleast one slope is significant and hence non-zero₁
ANOVA Table
Significance F or p value is 0 indicating rejection of H0 and hence linear
regression model is significant
Multiple Regression Model

Hypothesis testing
Hₒ : Slope coefficient is insignificant and hence zero
H :Slope coefficient is significant and hence non-zero₁
Level of significance = 5%
From the above table, p values for all slope coefficients except
population are less than 0.05 indicating that all independent variables
except population are significant for estimating of sales.
Multiple Regression Model

• 95% confidence interval
The mean regression coefficients for
The given independent variables lie
between.:
Years (-0.0187 to -0.0063)
GDP US$ (0.0000 to 0.0000)
Price Index (-0.0367 to -0.0195)
Population (0.0000 to 0.0000)
Survey Score (0.0053 to 0.0268)
Advertisement (0.0461 to 0.0747)
Stores (0.0212 to 0.0585)
Multiple Regression Model