This course provides an analytical framework to help you evaluate key problems in a structured fashion and will equip you with tools to better manage the uncertainties that pervade and complicate business processes. The course aim to cover statistical ideas that apply to managers. We will consider two basic themes: first, is recognizing and describing variations present in everything around us, and then modeling and making decisions in the presence of these variations. The fundamental concepts studied in this course will reappear in many other classes and business settings. Our focus will be on interpreting the meaning of the results in a business and managerial setting. While you will be introduced to some of the science of what is being taught, the focus will be on applying the methodologies. This will be accomplished through use of Excel and using data sets from many different disciplines, allowing you to see the use of statistics in very diverse settings. The course will focus not only on explaining these concepts but also understanding the meaning of the results obtained. Upon successful completion of this course, you will be able to: • Test for beliefs about a population.. • Compare differences between populations. • Use linear regression model for prediction. • Learn how to use Excel for statistical analysis. This course is part of the iMBA offered by the University of Illinois, a flexible, fully-accredited online MBA at an incredibly competitive price. For more information, please see the Resource page in this course and onlinemba.illinois.edu.
3-4.1. Using the Model
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来自 University of Illinois at Urbana-Champaign 的课程
Inferential and Predictive Statistics for Business
224 个评分
University of Illinois at Urbana-Champaign
224 个评分
课程 6（共 7 门，Specialization Managerial Economics and Business Analysis ）
从本节课中
Module 3: Simple Linear Regression
Does your job involve a lot of sitting? If so, you are at higher risk of coronary heart disease. How do I know this? We got to know the relationship between coronary heart disease and sitting when researchers studied a cohort of London bus drivers and bus conductors from 1947 to 1972. If you want to know more, then read on!
与讲师见面
Fataneh Taghaboni-DuttaClinical Professor of Business Administration
University of Illinois, Urbana-Champaign College of Business Department of Business Administration
[SOUND]
A national newspaper publisher wants to model the relationship between
the salaries of its field reporters with their years of experience.
They have collected for 34 randomly selected reporters.
Are they correct in thinking that a relationship does exist between one's
salary and years of experience?
So here the independent variable is the years of experience,
that would be x and the response variable is the salary and this would be y.
We will first begin by doing a scatter plot to see if we can detect a linear
relationship between the two variables.
From the scatter plot, we get a sense for
a strong positive correlation between years of experience and one's salary.
So we can go ahead and run the data through the regression analysis, but
before doing that,
let me alert you to a common mistake mixing up the two variables.
This scatter plot shows a positive correlation too,
but there's something major wrong with it.
Can you detect it?
If not, pay close attention to what variable is on the x-axis and
what is on the y-axis.
Salary.
The dependent variable is being displayed on the x-axis.
That's a no, no.
Just like I have said before, Excel is automating tasks for you.
It will not be checking your model.
No software will.
You will have to be sure that you model the problem correctly.
Otherwise, you will be looking at an invalid analysis.
Now, we can go ahead and do it correctly and run the analysis.
So, here is he output.
First, let's see what percent variations we see in someone's
salary can be explained by their number of years of experience.
So, here is he output.
First, let's see what percent variations we see in someone's salary can be
explained by their number of years of experience.
The answer to this is in R Square.
We can now see what the regression equation be and
that is found by looking here.
So then, the equation is
y=60.699+2.169x.
Now, lets understand this better.
The intercept of this equation is 60.699.
This means that if you put x equal to 0, this is the value of y.
Then based on our data, this should predict the predicted starting salary for
a reporter in this organization.
Then we have the coefficient of independent variable,
number of years of experience.
This means for every year, the salary has gone
up by 2.169, which is really $2,169.
Data was scaled by thousands.
This 2.169 represents the average raise that reporters get per year.
Now if I asked you what would you expect the average salary be if
I have a sample of reporters with an average of 15 years experience,
the answer will be 60.699+2.169 times 15, which is 93.24.
I will multiply this by 1,000.
Again, because the data was entered in thousands of dollars and
that would be $93,240.
Since we such a high value of r square, it is not surprising that we
get a very small p-value for years highlighted in yellow.
We have rejected the null hypothesis that one's years of experience has
nothing to do with the person's salary.
We also see in red font that salary and
years of experience are highly correlated at .91.
This is a positive correlation,
as we see the sign of the coefficient for years is positive.
This means that as the reporters length of stay has increased, so has their salary.
Again, on average about $2,169 per year.
Please be mindful that the coefficient we have used is a point estimate.
This is the midpoint between the confidence interval for
the coefficient, which means the true average value of salary increase
can be somewhere between 1,825 to $2,514.
To do better, we should calculate the confidence interval for the prediction.
This is not done every easily in Excel though.
To reduce production interruption for the robotic painting line due to break downs,
the plant manager would like to institute a preventative maintenance program for
the robots.
This initiative will require support from the upper management.
So, he has collected data on hours of total down time per month and
the line's productivity.
He believes he can use regression for this analysis.
In this study, the average number of the hours the line is down is the independent
variable x and the response variable is the productivity measure of the line.
Scatter plot for the data is shown here.
A strong negative correlation between the two variables
as the number of hours the line is down increases,
the productivity measure drops in a linear form.
So, we can apply simple linear regression to explore this relationship further.
We have used Excel to analyze the data.
The degree of correlation between average hours of downtime in a month and
productivity are negatively correlated.
And r, the correlation is -0.94.
Again, the sign comes from the sign for the downtime coefficient.
R square shows that 88.4% of variations in productivity
we seen are data can be explained by the regression variable down time.
The remaining 11.6% are due to other sources we call error.
So given that we have a strong relationship, we can go ahead and
develop the regression equation.
To develop the regression equation,
we find the coefficients in the third part of the output.
So, the equation is 98.0143-1.648x.
Looking at this equation, which is based on our data, the baseline for
productivity is about 98%.
And for every hour, the line is down,
the productivity goes down by an additional 1.648%.
The manager believes that on average, the line is down for
about eight hours per month.
If this is indeed the case, then what is the average productivity expected from
the production line for the months with this much down time?
To find the predicted productivity, we put value of 8 in variable x and
that will give us the average productivity to be 84.82%.
So, making the line more reliable by instituting a preventive maintenance
program will reduce breakdowns and that will improve productivity.
Now, the plant manager can use this insight and
do some economic analysis comparing the cost of maintenance program
versus the cost associated with lost productivity.
As you can see with the regression,
we not only have the ability to infer about the population, but now we can
predict the value of the response variable for a given independent variable.
Let's practice.
An advisor in the university would like to motivate her students to study hard and
graduate with high GPA.
To stress the importance of GPA, she's wondering if she can show data on
the relationship of a graduate's starting salary and the student's GPA.
She has collected data on 100 recent graduates.
What are the dependent and independent variables in this study?
The independent variable is the student's GPA,
which the advisor believes influences the starting salary, so
that would be the y variable, which is the response variable.
Now based on this scatter plot, how would you characterize the relationship between
GPA and starting salary?
There appears to be a strong and positive relationship between GPA and
starting salary.
What percent of variability seen in a student's starting
salary can be explained by the student's GPA?
Based on our analysis, r square is about 0.3365.
So, GPA explains 33.65% of the variations we see in the starting
salaries in our data, which means that remaining variation
you see is due to other sources, unexplained by our model.
Is this advisor wrong?
Does GPA appear to be not that important?
Actually, the answer to that is here in the p-value and
this value here is practically zero.
Recall that regression is running a hypothesis test on the independent
variable.
Null hypothesis is that no relationship exists.
You can think of regression, as a cynic.
You have to have data to change its mind.
In this case, GPA has such a small p-value that you would reject the null
hypothesis at any level of significance.
So then, why is r squared not higher?
What r square is saying is that while about
a third of the differences observed in a salary was explained by GPA,
two-thirds of the differences are due to other sources.
Based on the key value,
you have identified one very significant source and that's good.
But your regression equation is not very good,
because most of the variability is due to other factors.
For example, could one of those factors be students' major?
You would expect that to be a factor, wouldn't you?
So, the advisor needs to expand her data collection and
include more independent variables.
But then you will no longer be doing a simple linear regression, but
a multiple regression.
This is a topic of the last module in this class and
I will come back to this later on.
