What is ordinary least squares analysis and how is it used?

  

Please provide solution with an explaination
1) As an expert in marketing, you want predict the time that people spend on shopping
websites. You believe that time spent on shopping websites (on a per-week basis) is linearly
related to age, so you collect data from a random sample of 15 individuals. You can download
the data by clicking here: Homework5.xlsx In the data set, “time spent on shopping” is the
dependent
variable,
and
“age”
is
the
independent
variable.
a. Use Excel and conduct ordinary least squares analysis on the data. Based on the results,
what is your slope estimate (b1)? Round your answer to one decimal digit.
Answer
In order to find the OLS we should find some parameters. First we assume that age as independent
variable is called X and the minutes spent for shopping as dependent variable is called Y. Then we
compute average and deviation of all data points from average and we call them x, and y. Then we
compute sum of xy and x2 to find the slope based on the least square method. We know from least
square method that
xi yi
b1 =
xi 2
And considering the fact that the fitted line will pass through the averaged point we have
b2 =
Y b1
X
The calculations are provided in the excel file.
Y = 11.5X + 750
Thus the slop is -11.5.
b. Consider a teenager who was born in 2001. Based on your results from question 1, what is
your best estimate of the time spent on shopping websites by this teenager? Round your
answer to one decimal digit.
Hint: We are in year 2016
Answer:
Age=2016-2001=15
Thus
Y = 11.5 15 + 750=577.5 minutes
c. Use the “data analysis” tool of Excel and conduct simple linear regression analysis on the
data given in question 1. A video of how to conduct simple linear regression analysis on Excel
is also included on Blackboard. Please use a Windows PC since the data analysis option may
not be available on the Mac version of Excel.
Answer:
You want to guess if the true slope of the linear model differs from 0.
Based on the results from Excel, what is the p-value for this test? Round your answer to three
decimal digits.
We know that the smaller the P-value for the slope the more convinced we can be that the slope
something other than zero. Here the P-value of our test for the slope is 4.67e-14 which is really
small number
D. What is your conclusion (inference) about the true slope of the linear model? Use =0.05
Answer
Important facts:
We know that: alpha (): = 1 – (confidence level / 100) hence the confidence level is 95%. The
analysis is provided in the excel file. Besides, the Multiple R value is the correlation coefficient
which measures how well the data clusters around our regression line. The closer this value is to 1,
the more linear the data is. Another important factor is R Square which is the coefficient of
determination. This measures the percentage of variation in the dependent variable that can be
explained by the linear relationship between x and y.
Answer:
H0 : 1 = 0
H1 : 1 0
Considering the fact that the p-value for the slope is very small (P-value< ). Therefore we reject the null hypothesis Finally in order to evaluate the Fitness of the Model we should check the fitness interval (lower 95% and upper 95%) which is -12.75 and -10.24. Since it does not contain zero, there is a linear relationship. Thus the answer is 4. 1. We believe 1 0, and we infer a linear relationship between age vs time spent on shopping websites. 2.We do not reject b1 = 0, and we do not infer any relationship between age vs time spent on shopping websites. 3.We believe b1 0, and we infer a linear relationship between age vs time spent on shopping websites. 4.We believe 1 0, and we infer a negative linear relationship between age vs time spent on shopping websites. We do not reject 1 = 0, and we do not infer a linear relationship between age vs time spent on shopping websites. 5.We believe b1 0, and we infer a positive linear relationship between age vs time spent on shopping websites. 6. We do not reject b1 = 0, and we do not infer a linear relationship between age vs time spent on shopping websites. 7.We do not reject 1 = 0, and we do not infer any relationship between age vs time spent on shopping websites. 8.We believe 1 0, and we infer a positive linear relationship between age vs time spent on shopping websites. 9.We believe b1 0, and we infer a negative linear relationship between age vs time spent on shopping websites. E. What is your conclusion (inference) about the true slope of the linear model? Use =0.01 Solution: H0 : 1 = 0 H1 : 1 0 Considering the fact that the p-value for the slope is very small (P-value< ). Therefore we reject the null hypothesis Finally in order to evaluate the Fitness of the Model we should check the fitness interval (lower 95% and upper 95%) which is -12.75 and -10.24. Since it does not contain zero, there is a linear relationship. Thus the answer is 6. 1.We do not reject b1 = 0, and we do not infer a linear relationship between age vs time spent on shopping websites. 2.We believe 1 0, and we infer a linear relationship between age vs time spent on shopping websites. 3.We believe b1 0, and we infer a linear relationship between age vs time spent on shopping websites. 4.We do not reject b1 = 0, and we do not infer any relationship between age vs time spent on shopping websites. 5.We believe 1 0, and we infer a positive linear relationship between age vs time spent on shopping websites. 6.We believe 1 0, and we infer a negative linear relationship between age vs time spent on shopping websites. 7.We do not reject 1 = 0, and we do not infer a linear relationship between age vs time spent on shopping websites. 8.We believe b1 0, and we infer a positive linear relationship between age vs time spent on shopping websites. 9.We do not reject 1 = 0, and we do not infer any relationship between age vs time spent on shopping websites. 10.We believe b1 0, and we infer a negative linear relationship between age vs time spent on shopping websites. F. What percent of the total variation in "time spent on shopping websites" is explained by "age" ? Enter the percentage as a whole number (without the "%" sign). Based on the reference that I have provided for you, Considering the correlation the percentage that the age explains the time spent on shopping is r2=0.96 Thus it explains it with 96%. Q. NO 2 B. What is your conclusion (inference) about the true slope of the linear model? Use =0.05 1.We believe b1 0, and we infer a linear relationship between age vs time spent on shopping websites. 2.We believe 1 0, and we infer a negative linear relationship between age vs time spent on shopping websites. 3.We believe b1 0, and we infer a negative linear relationship between age vs time spent on shopping websites. 4.We believe 1 0, and we infer a positive linear relationship between age vs time spent on shopping websites. 5.We believe 1 0, and we infer a linear relationship between age vs time spent on shopping websites. 6.We do not reject b1 = 0, and we do not infer any relationship between age vs time spent on shopping websites. 7.We believe b1 0, and we infer a positive linear relationship between age vs time spent on shopping websites. 8.We do not reject 1 = 0, and we do not infer a linear relationship between age vs time spent on shopping websites. 9.We do not reject 1 = 0, and we do not infer any relationship between age vs time spent on shopping websites. 10.We do not reject b1 = 0, and we do not infer a linear relationship between age vs time spent on shopping websites. Solution: H0 : 1 = 0 H1 : 1 0 Considering the fact that the p-value for the slope is very small (P-value> ). Therefore we
cannot reject the null hypothesis. We do not have the limits of confidence level to check
whether it contains the zero or not. So we use the Multiple R we know closer to 1 represents a
linear relationship.
Thus the answer is 8.
C. What is your conclusion (inference) about the true slope of the linear
model? Use =0.10
1.We believe 1 0, and we infer a positive linear relationship between age vs time spent
on shopping websites.
2.We do not reject b1 = 0, and we do not infer any relationship between age vs time spent
on shopping websites.
3.We believe 1 0, and we infer a linear relationship between age vs time spent on
shopping websites.
4.We do not reject b1 = 0, and we do not infer a linear relationship between age vs time
spent on shopping websites.
5.We believe b1 0, and we infer a negative linear relationship between age vs time
spent on shopping websites.
6.We believe b1 0, and we infer a linear relationship between age vs time spent on
shopping websites.
7.We do not reject 1 = 0, and we do not infer a linear relationship between age vs time
spent on shopping websites.
8.We do not reject 1 = 0, and we do not infer any relationship between age vs time spent
on shopping websites.
9.We believe b1 0, and we infer a positive linear relationship between age vs time
spent on shopping websites.
10.We believe 1 0, and we infer a negative linear relationship between age vs time
spent on shopping websites.
Solution:
H0 : 1 = 0
H1 : 1 0
Considering the fact that the p-value for the slope is very small (P-value> ). Therefore we
cannot reject the null hypothesis. We do not have the limits of confidence level to check
whether it contains the zero or not. So we use the Multiple R we know closer to 1 represents a
linear relationship.
Thus the answer is 7.
D. Based on the Excel output, what is the total variation in battery life, which is the response
(dependent) variable?
Answer
Based on the reference that I have provided for you as well. The total variation is defined as R 2
which is 0.05.
E. Based on the Excel output, what is the variation in battery life that is explained
by maximum brightness?
Answer
Based on the reference 3. The SS Regression is the variation explained by the regression line; SS
Residual is the variation of the dependent variable that is not explained. Thus the variation that is
explained is 2.7.
F. What percent of the variation in battery life is explained by maximum brightness? Round
your answer to the nearest whole number. Do not enter the “%” sign in your answer.
Answer:
It will be determined from R2 hence it will be 5%. (check the reference again)
G. What is the correlation coefficient between battery life and maximum brightness?
Answer
Based on the reference (2) Multiple R This is the correlation coefficient which measures how
well the data clusters around our regression line. The closer this value is to 1, the more linear the
data is.
References:
1. https://people.richland.edu/james/lecture/m170/ch11-rsq.html
2. https://www.cgc.maricopa.edu/Academics/LearningCenter/Math/Documents/AnalyzingL
inearRegression.pdf
3. http://educ.jmu.edu/~drakepp/FIN360/readings/regression_excel.pdf

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In marketing, understanding consumer behavior is essential to design effective strategies that promote the products or services. One way to analyze this behavior is by predicting the time people spend on shopping websites. As an expert in the field, you believe that age is a relevant factor, and you collected data from a random sample of 15 individuals to test this hypothesis. Using Excel, you conducted ordinary least squares analysis, simple linear regression analysis, and calculated the p-value for the slope to draw conclusions about the relationship between age and time spent shopping.

Description

The data set provided consists of two variables, time spent on shopping website (dependent) and age (independent), collected from a sample of 15 individuals. The objective is to predict the time spent on shopping based on age and test whether this relationship is statistically significant. To achieve this, several statistical techniques were executed using Excel. Firstly, ordinary least squares (OLS) analysis was performed to estimate the slope of the regression line. The slope shows the rate at which time spent shopping increases or decrease with age. The slope estimate was -11.5 minutes spent shopping per week per year of age. Then, the analysis continued with simple linear regression analysis conducted using the “data analysis” tool of Excel. This analysis allowed you to test whether the true slope of the model is significantly different from zero, indicating the existence of a relationship between the variables. The p-value for the slope was 4.67e-14, which is much smaller than the significance level of 0.05, indicating that the slope is statistically significant. Finally, multiple R value was calculated, resulting in 0.957, indicating a strong correlation between age and time spent shopping. Thus, it can be concluded that there is a linear relationship between age and time spent on shopping websites, and the variation in age can explain 95.7% of the variation in time spent shopping.

Objectives:
– To understand how to perform ordinary least squares analysis in Excel
– To learn how to conduct simple linear regression analysis using the data analysis tool in Excel
– To understand how to interpret the results of the regression analysis, including identifying the slope estimate and p-value

Learning Outcomes:
– Students will be able to perform ordinary least squares analysis in Excel to determine the linear relationship between two variables.
– Students will be able to use Excel’s data analysis tool to conduct simple linear regression analysis.
– Students will be able to interpret the results of the regression analysis, including identifying the slope estimate and p-value and making inferences about the true slope of the linear model based on the p-value and confidence level.

Solution 1:

Using Ordinary Least Squares analysis provided in the excel file, the slope estimate (b1) is -11.5. This means that for every year increase in age, the time spent on shopping websites decreases by 11.5 minutes.

Solution 2:

Using Simple Linear Regression analysis in Excel, we can determine if there is a significant linear relationship between age and time spent on shopping websites. The p-value for this test is 4.67e-14 which is very small, indicating a strong evidence against the null hypothesis that the true slope is zero. Therefore, we can conclude that there is a significant linear relationship between age and time spent on shopping websites.

Suggested Resources/Books:
1. “Marketing Metrics: The Definitive Guide to Measuring Marketing Performance” by Paul W. Farris, Neil T. Bendle, Phillip E. Pfeifer, and David J. Reibstein
2. “Marketing Research: An Applied Orientation” by Naresh K. Malhotra
3. “Marketing Analytics: Data-Driven Techniques with Microsoft Excel” by Wayne L. Winston

Similar Asked Questions:
1. How can linear regression be used in marketing analysis?
2. What are some common tools and techniques used in marketing research?
3. How can data analysis be used to improve marketing strategies?
4. What is the significance of correlation coefficients in marketing research?
5. How can marketers predict consumer behavior using statistical techniques?

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