PL9fwy3NUQKwaIc7KWc8buW344941GStQL/GyiivmJglvM_raw.srt

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The last chapter we are going to talk in this

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semester is correlation and simple linearization.

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So we are going to explain two types in chapter

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12. One is called correlation. And the other type

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is simple linear regression. Maybe this chapter

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I'm going to spend about two lectures in order to

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cover these objectives. The first objective is to

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calculate the coefficient of correlation. The

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second objective, the meaning of the regression

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coefficients beta 0 and beta 1. And the last

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objective is how to use regression analysis to

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predict the value of dependent variable based on

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an independent variable. It looks like that we

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have discussed objective number one in chapter

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three. So calculation of the correlation

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coefficient is done in chapter three, but here

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we'll give some details about correlation also. A

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scatter plot can be used to show the relationship

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between two variables. For example, imagine that

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we have a random sample of 10 children.

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And we have data on their weights and ages. And we

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are interested to examine the relationship between

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weights and age. For example, suppose child number

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one, his

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or her age is two years with weight, for example,

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eight kilograms.

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His weight or her weight is four years, and his or

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her weight is, for example, 15 kilograms, and so

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on. And again, we are interested to examine the

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relationship between age and weight. Maybe they

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exist sometimes. positive relationship between the

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two variables that means if one variable increases

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the other one also increase if one variable

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increases the other will also decrease so they

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have the same direction either up or down so we

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have to know number one the form of the

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relationship this one could be linear here we

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focus just on linear relationship between X and Y.

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The second, we have to know the direction of the

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relationship. This direction might be positive or

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negative relationship.

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In addition to that, we have to know the strength

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of the relationship between the two variables of

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interest the strength can be classified into three

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categories either strong, moderate or there exists

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a weak relationship so it could be positive

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-strong, positive-moderate or positive-weak, the

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same for negative so by using scatter plot we can

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determine the form either linear or non-linear,

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but here we are focusing on just linear

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relationship. Also, we can determine the direction

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of the relationship. We can say there exists

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positive or negative based on the scatter plot.

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Also, we can know the strength of the

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relationship, either strong, moderate or weak. For

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example, suppose we have again weights and ages.

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And we know that there are two types of variables

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in this case. One is called dependent and the

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other is independent. So if we, as we explained

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before, is the dependent variable and A is

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independent variable.

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Always dependent

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variable

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is denoted by Y and always on the vertical axis so

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here we have weight and independent variable is

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denoted by X and X is in the X axis or horizontal

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axis now scatter plot for example here child with

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age 2 years his weight is 8 So two years, for

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example, this is eight. So this star represents

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the first pair of observation, age of two and

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weight of eight. The other child, his weight is

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four years, and the corresponding weight is 15.

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For example, this value is 15. The same for the

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other points. Here we can know the direction.

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In this case they exist. Positive. Form is linear.

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Strong or weak or moderate depends on how these

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values are close to the straight line. Closer

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means stronger. So if the points are closer to the

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straight line, it means there exists stronger

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relationship between the two variables. So closer

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means stronger, either positive or negative. In

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this case, there exists positive. Now for the

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negative association or relationship, we have the

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other direction, it could be this one. So in this

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case there exists linear but negative

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relationship, and this negative could be positive

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or negative, it depends on the points. So it's

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positive relationship. The other direction is

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negative. So the points, if the points are closed,

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then we can say there exists strong negative

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relationship. So by using scatter plot, we can

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determine all of these.

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and direction and strength now here the two

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variables we are talking about are numerical

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variables so the two variables here are numerical

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variables so we are talking about quantitative

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variables but remember in chapter 11 We talked

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about the relationship between two qualitative

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variables. So we use chi-square test. Here we are

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talking about something different. We are talking

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about numerical variables. So we can use scatter

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plot, number one. Next correlation analysis is

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used to measure the strength of the association

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between two variables. And here again, we are just

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talking about linear relationship. So this chapter

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just covers the linear relationship between the

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two variables. Because sometimes there exists non

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-linear relationship between the two variables. So

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correlation is only concerned with the strength of

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the relationship. No causal effect is implied with

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correlation. We just say that X affects Y, or X

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explains the variation in Y. Scatter plots were

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first presented in Chapter 2, and we skipped, if

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you remember, Chapter 2. And it's easy to make

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scatter plots for Y versus X. In Chapter 3, we

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talked about correlation, so correlation was first

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presented in Chapter 3. But here I will give just

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a review for computation about correlation

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coefficient or coefficient of correlation. First,

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coefficient of correlation measures the relative

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strength of the linear relationship between two

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numerical variables. So here, we are talking about

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numerical variables. Sample correlation

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coefficient is given by this equation. which is

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sum of the product of xi minus x bar, yi minus y

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bar, divided by n minus 1 times standard deviation

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of x times standard deviation of y. We know that x

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bar and y bar are the means of x and y

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respectively. And Sx, Sy are the standard

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deviations of x and y values. And we know this

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equation before. But there is another equation

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that one can be used For computation, which is

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called shortcut formula, which is just sum of xy

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minus n times x bar y bar divided by square root

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of this quantity. And we know this equation from

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chapter three. Now again, x bar and y bar are the

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means. Now the question is, Do outliers affect the

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correlation? For sure, yes. Because this formula

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actually based on the means and the standard

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deviations, and these two measures are affected by

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outliers. So since R is a function of these two

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statistics, the means and standard deviations,

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then outliers will affect the value of the

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correlation coefficient.

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Some features about the coefficient of

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correlation. Here rho is the population

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coefficient of correlation, and R is the sample

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coefficient of correlation. Either rho or R have

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the following features. Number one, unity free. It

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means R has no units. For example, here we are

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talking about whales. And weight in kilograms,

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ages in years. And for example, suppose the

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correlation between these two variables is 0.8.

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It's unity free, so it's just 0.8. So there is no

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unit. You cannot say 0.8 kilogram per year or

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whatever it is. So just 0.8. So the first feature

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of the correlation coefficient is unity-free.

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Number two ranges between negative one and plus

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one. So R is always, or rho, is always between

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minus one and plus one. So minus one smaller than

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or equal to R smaller than or equal to plus one.

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So R is always in this range. So R cannot be

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smaller than negative one or greater than plus

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one. The closer to minus one or negative one, the

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stronger negative relationship between or linear

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relationship between x and y. So, for example, if

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R is negative 0.85 or R is negative 0.8. Now, this

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value is closer to minus one than negative 0.8. So

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negative 0.85 is stronger than negative 0.8.

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Because we are looking for closer to minus 1.

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Minus 0.8, the value itself is greater than minus

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0.85. But this value is closer to minus 1 than

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minus 0.8. So we can say that this relationship is

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stronger than the other one.

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Also, the closer to plus 1, the stronger the

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positive linear relationship. Here, suppose R is 0

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.7 and another R is 0.8. 0.8 is closer to plus one

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than 0.7, so 0.8 is stronger. This one makes

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sense. The closer to zero, the weaker relationship

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between the two variables. For example, suppose R

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is plus or minus 0.05. This value is very close to

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zero. It means there exists weak. relationship.

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Sometimes we can say that there exists moderate

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relationship if R is close to 0.5. So it could be

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classified into these groups closer to minus 1,

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closer to 1, 0.5 or 0. So we can know the

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direction by the sign of R negative it means

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because here our ranges as we mentioned between

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minus one and plus one here zero so this these

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values it means there exists negative above zero

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all the way up to one it means there exists

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positive relationship between the two variables so

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the sign gives the direction of the relationship

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The absolute value gives the strength of the

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relationship between the two variables. So the

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same as we had discussed before. Now, some types

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of scatter plots for different types of

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relationship between the two variables is

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presented in this slide. For example, if you look

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carefully at figure one here, sharp one, this one,

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and the other one, In each one, all points are

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on the straight line, it means they exist perfect.

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So if all points fall exactly on the straight

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line, it means they exist perfect.

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Here there exists perfect negative. So this is

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perfect negative relationship. The other one

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perfect positive relationship. In reality you will

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never see something

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like perfect positive or perfect negative. Maybe

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in real situation. In real situation, most of the

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time, R is close to 0.9 or 0.85 or something like

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that, but it's not exactly equal one. Because

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equal one, it means if you know the value of a

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child's age, then you can predict the exact

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weight. And that never happened. If the data looks

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like this table, for example. Suppose here we have

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age and weight. H1 for example 3, 5, 7 weight for

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example 10, 12, 14, 16 in this case they exist

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perfect because x increases by 2 units also

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weights increases by 2 units or maybe weights for

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example 9, 12, 15, 18 and so on So X or A is

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increased by two units for each value for each

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individual and also weights are increased by three

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units for each person. In this case there exists

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perfect relationship but that never happened in

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real life. So perfect means all points are lie on

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the straight line otherwise if the points are

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close Then we can say there exists strong. Here if

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you look carefully at these points corresponding

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to this regression line, it looks like not strong

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because some of the points are not closed, so you

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can say there exists maybe moderate negative

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relationship. This one, most of the points are

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scattered away from the straight line, so there

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exists weak relationship. So by just looking at

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the scatter path, sometimes you can, sometimes

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it's hard to tell, but most of the time you can

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tell at least the direction, positive or negative,

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the form, linear or non-linear, or the strength of

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the relationship. The last one here, now x

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increases, y remains the same. For example,

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suppose x is 1, y is 10. x increases to 2, y still

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is 10. So as x increases, y stays the same

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position, it means there is no linear relationship

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between the two variables. So based on the scatter

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plot you can have an idea about the relationship

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between the two variables. Here I will give a

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simple example in order to determine the

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correlation coefficient. A real estate agent

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wishes to examine the relationship between selling

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the price of a home and its size measured in

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square feet. So in this case, there are two

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variables of interest. One is called selling price

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of a home. So here, selling price of a home and

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its size. Now, selling price in $1,000.

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And size in feet squared. Here we have to

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00:19:29,380 --> 00:19:35,640
distinguish between dependent and independent. So

258
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your dependent variable is house price, sometimes

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00:19:39,740 --> 00:19:41,620
called response variable.

260
00:19:45,750 --> 00:19:49,490
The independent variable is the size, which is in

261
00:19:49,490 --> 00:19:54,570
square feet, sometimes called sub-planetary

262
00:19:54,570 --> 00:19:54,850
variable.

263
00:19:59,570 --> 00:20:06,370
So my Y is ceiling rise, and size is square feet,

264
00:20:07,530 --> 00:20:12,910
or size of the house. In this case, there are 10.

265
00:20:14,290 --> 00:20:17,890
It's sample size is 10. So the first house with

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00:20:17,890 --> 00:20:26,850
size 1,400 square feet, it's selling price is 245

267
00:20:26,850 --> 00:20:31,670
multiplied by 1,000. Because these values are in

268
00:20:31,670 --> 00:20:37,950
$1,000. Now based on this data, you can first plot

269
00:20:37,950 --> 00:20:46,590
the scatterplot of house price In Y direction, the

270
00:20:46,590 --> 00:20:51,870
vertical direction. So here is house. And rise.

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00:20:54,230 --> 00:21:01,470
And size in the X axis. You will get this scatter

272
00:21:01,470 --> 00:21:07,370
plot. Now, the data here is just 10 points, so

273
00:21:07,370 --> 00:21:12,590
sometimes it's hard to tell. the relationship

274
00:21:12,590 --> 00:21:15,510
between the two variables if your data is small.

275
00:21:16,510 --> 00:21:21,170
But just this example for illustration. But at

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00:21:21,170 --> 00:21:25,370
least you can determine that there exists linear

277
00:21:25,370 --> 00:21:28,810
relationship between the two variables. It is

278
00:21:28,810 --> 00:21:35,490
positive. So the form is linear. Direction is

279
00:21:35,490 --> 00:21:41,880
positive. Weak or strong or moderate. Sometimes

280
00:21:41,880 --> 00:21:45,620
it's not easy to tell if it is strong or moderate.

281
00:21:47,720 --> 00:21:50,120
Now if you look at these points, some of them are

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00:21:50,120 --> 00:21:53,700
close to the straight line and others are away

283
00:21:53,700 --> 00:21:56,700
from the straight line. So maybe there exists

284
00:21:56,700 --> 00:22:02,720
moderate for example, but you cannot say strong.

285
00:22:03,930 --> 00:22:08,210
Here, strong it means the points are close to the

286
00:22:08,210 --> 00:22:11,890
straight line. Sometimes it's hard to tell the

287
00:22:11,890 --> 00:22:15,230
strength of the relationship, but you can know the

288
00:22:15,230 --> 00:22:20,990
form or the direction. But to measure the exact

289
00:22:20,990 --> 00:22:24,130
strength, you have to measure the correlation

290
00:22:24,130 --> 00:22:29,810
coefficient, R. Now, by looking at the data, you

291
00:22:29,810 --> 00:22:31,430
can compute

292
00:22:33,850 --> 00:22:42,470
The sum of x values, y values, sum of x squared,

293
00:22:43,290 --> 00:22:48,170
sum of y squared, also sum of xy. Now plug these

294
00:22:48,170 --> 00:22:50,610
values into the formula we have for the shortcut

295
00:22:50,610 --> 00:22:58,210
formula. You will get R to be 0.76 around 76.

296
00:23:04,050 --> 00:23:10,170
So there exists positive, moderate relationship

297
00:23:10,170 --> 00:23:13,770
between selling

298
00:23:13,770 --> 00:23:19,850
price of a home and its size. So that means if the

299
00:23:19,850 --> 00:23:24,670
size increases, the selling price also increases.

300
00:23:25,310 --> 00:23:29,550
So there exists positive relationship between the

301
00:23:29,550 --> 00:23:30,310
two variables.

302
00:23:35,800 --> 00:23:40,300
Strong it means close to 1, 0.8, 0.85, 0.9, you

303
00:23:40,300 --> 00:23:44,400
can say there exists strong. But fields is not

304
00:23:44,400 --> 00:23:47,960
strong relationship, you can say it's moderate

305
00:23:47,960 --> 00:23:53,440
relationship. Because it's close if now if you

306
00:23:53,440 --> 00:23:57,080
just compare this value and other data gives 9%.

307
00:23:58,830 --> 00:24:03,790
Other one gives 85%. So these values are much

308
00:24:03,790 --> 00:24:08,550
closer to 1 than 0.7, but still this value is

309
00:24:08,550 --> 00:24:09,570
considered to be high.

310
00:24:15,710 --> 00:24:16,810
Any question?

311
00:24:19,850 --> 00:24:22,810
Next, I will give some introduction to regression

312
00:24:22,810 --> 00:24:23,390
analysis.

313
00:24:26,970 --> 00:24:32,210
regression analysis used to number one, predict

314
00:24:32,210 --> 00:24:35,050
the value of a dependent variable based on the

315
00:24:35,050 --> 00:24:39,250
value of at least one independent variable. So by

316
00:24:39,250 --> 00:24:42,490
using the data we have for selling price of a home

317
00:24:42,490 --> 00:24:48,370
and size, you can predict the selling price by

318
00:24:48,370 --> 00:24:51,510
knowing the value of its size. So suppose for

319
00:24:51,510 --> 00:24:54,870
example, You know that the size of a house is

320
00:24:54,870 --> 00:25:03,510
1450, 1450 square feet. What do you predict its

321
00:25:03,510 --> 00:25:10,190
size, its sale or price? So by using this value,

322
00:25:10,310 --> 00:25:16,510
we can predict the selling price. Next, explain

323
00:25:16,510 --> 00:25:19,890
the impact of changes in independent variable on

324
00:25:19,890 --> 00:25:23,270
the dependent variable. You can say, for example,

325
00:25:23,510 --> 00:25:30,650
90% of the variability in the dependent variable

326
00:25:30,650 --> 00:25:36,790
in selling price is explained by its size. So we

327
00:25:36,790 --> 00:25:39,410
can predict the value of dependent variable based

328
00:25:39,410 --> 00:25:42,890
on a value of one independent variable at least.

329
00:25:43,870 --> 00:25:47,090
Or also explain the impact of changes in

330
00:25:47,090 --> 00:25:49,550
independent variable on the dependent variable.

331
00:25:51,420 --> 00:25:53,920
Sometimes there exists more than one independent

332
00:25:53,920 --> 00:25:59,680
variable. For example, maybe there are more than

333
00:25:59,680 --> 00:26:04,500
one variable that affects a price, a selling

334
00:26:04,500 --> 00:26:10,300
price. For example, beside selling

335
00:26:10,300 --> 00:26:16,280
price, beside size, maybe location.

336
00:26:19,480 --> 00:26:23,580
Maybe location is also another factor that affects

337
00:26:23,580 --> 00:26:27,360
the selling price. So in this case there are two

338
00:26:27,360 --> 00:26:32,240
variables. If there exists more than one variable,

339
00:26:32,640 --> 00:26:36,080
in this case we have something called multiple

340
00:26:36,080 --> 00:26:38,680
linear regression.

341
00:26:42,030 --> 00:26:46,710
Here, we just talk about one independent variable.

342
00:26:47,030 --> 00:26:51,610
There is only, in this chapter, there is only one

343
00:26:51,610 --> 00:26:58,330
x. So it's called simple linear

344
00:26:58,330 --> 00:26:59,330
regression.

345
00:27:02,190 --> 00:27:07,930
The calculations for multiple takes time. So we

346
00:27:07,930 --> 00:27:11,430
are going just to cover one independent variable.

347
00:27:11,930 --> 00:27:14,290
But if there exists more than one, in this case

348
00:27:14,290 --> 00:27:18,250
you have to use some statistical software as SPSS.

349
00:27:18,470 --> 00:27:23,390
Because in that case you can just select a

350
00:27:23,390 --> 00:27:25,970
regression analysis from SPSS, then you can run

351
00:27:25,970 --> 00:27:28,590
the multiple regression without doing any

352
00:27:28,590 --> 00:27:34,190
computations. But here we just covered one

353
00:27:34,190 --> 00:27:36,820
independent variable. In this case, it's called

354
00:27:36,820 --> 00:27:41,980
simple linear regression. Again, the dependent

355
00:27:41,980 --> 00:27:44,600
variable is the variable we wish to predict or

356
00:27:44,600 --> 00:27:50,020
explain, the same as weight. Independent variable,

357
00:27:50,180 --> 00:27:52,440
the variable used to predict or explain the

358
00:27:52,440 --> 00:27:54,000
dependent variable.

359
00:27:57,400 --> 00:28:00,540
For simple linear regression model, there is only

360
00:28:00,540 --> 00:28:01,800
one independent variable.

361
00:28:04,830 --> 00:28:08,450
Another example for simple linear regression.

362
00:28:08,770 --> 00:28:11,590
Suppose we are talking about your scores.

363
00:28:14,210 --> 00:28:17,770
Scores is the dependent variable can be affected

364
00:28:17,770 --> 00:28:21,050
by number of hours.

365
00:28:25,130 --> 00:28:31,030
Hour of study. Number of studying hours.

366
00:28:36,910 --> 00:28:39,810
Maybe as number of studying hour increases, your

367
00:28:39,810 --> 00:28:43,390
scores also increase. In this case, if there is

368
00:28:43,390 --> 00:28:46,330
only one X, one independent variable, it's called

369
00:28:46,330 --> 00:28:51,110
simple linear regression. Maybe another variable,

370
00:28:52,270 --> 00:28:59,730
number of missing classes or

371
00:28:59,730 --> 00:29:03,160
attendance. As number of missing classes

372
00:29:03,160 --> 00:29:06,380
increases, your score goes down. That means there

373
00:29:06,380 --> 00:29:09,400
exists negative relationship between missing

374
00:29:09,400 --> 00:29:13,540
classes and your score. So sometimes, maybe there

375
00:29:13,540 --> 00:29:16,580
exists positive or negative. It depends on the

376
00:29:16,580 --> 00:29:20,040
variable itself. In this case, if there are more

377
00:29:20,040 --> 00:29:23,180
than one variable, then we are talking about

378
00:29:23,180 --> 00:29:28,300
multiple linear regression model. But here, we

379
00:29:28,300 --> 00:29:33,630
have only one independent variable. In addition to

380
00:29:33,630 --> 00:29:37,230
that, a relationship between x and y is described

381
00:29:37,230 --> 00:29:40,850
by a linear function. So there exists a straight

382
00:29:40,850 --> 00:29:46,270
line between the two variables. The changes in y

383
00:29:46,270 --> 00:29:50,210
are assumed to be related to changes in x only. So

384
00:29:50,210 --> 00:29:54,270
any change in y is related only to changes in x.

385
00:29:54,730 --> 00:29:57,810
So that's the simple case we have for regression,

386
00:29:58,890 --> 00:30:01,170
that we have only one independent

387
00:30:03,890 --> 00:30:07,070
Variable. Types of relationships, as we mentioned,

388
00:30:07,210 --> 00:30:12,190
maybe there exist linear, it means there exist

389
00:30:12,190 --> 00:30:16,490
straight line between X and Y, either linear

390
00:30:16,490 --> 00:30:22,050
positive or negative, or sometimes there exist non

391
00:30:22,050 --> 00:30:25,830
-linear relationship, it's called curved linear

392
00:30:25,830 --> 00:30:29,290
relationship. The same as this one, it's parabola.

393
00:30:32,570 --> 00:30:35,150
Now in this case there is no linear relationship

394
00:30:35,150 --> 00:30:39,690
but there exists curved linear or something like

395
00:30:39,690 --> 00:30:45,910
this one. So these types of non-linear

396
00:30:45,910 --> 00:30:49,530
relationship between the two variables. Here we

397
00:30:49,530 --> 00:30:54,070
are covering just the linear relationship between

398
00:30:54,070 --> 00:30:56,570
the two variables. So based on the scatter plot

399
00:30:56,570 --> 00:31:00,620
you can determine the direction. The form, the

400
00:31:00,620 --> 00:31:03,860
strength. Here, the form we are talking about is

401
00:31:03,860 --> 00:31:04,720
just linear.

402
00:31:08,700 --> 00:31:13,260
Now, another type of relationship, the strength of

403
00:31:13,260 --> 00:31:16,940
the relationship. Here, the points, either for

404
00:31:16,940 --> 00:31:20,570
this graph or the other one, These points are

405
00:31:20,570 --> 00:31:24,570
close to the straight line, it means there exists

406
00:31:24,570 --> 00:31:28,210
strong positive relationship or strong negative

407
00:31:28,210 --> 00:31:31,230
relationship. So it depends on the direction. So

408
00:31:31,230 --> 00:31:35,710
strong either positive or strong negative. Here

409
00:31:35,710 --> 00:31:38,850
the points are scattered away from the regression

410
00:31:38,850 --> 00:31:41,790
line, so you can say there exists weak

411
00:31:41,790 --> 00:31:45,090
relationship, either weak positive or weak

412
00:31:45,090 --> 00:31:49,650
negative. It depends on the direction of the

413
00:31:49,650 --> 00:31:54,270
relationship between the two variables. Sometimes

414
00:31:54,270 --> 00:31:59,680
there is no relationship or actually there is no

415
00:31:59,680 --> 00:32:02,340
linear relationship between the two variables. If

416
00:32:02,340 --> 00:32:05,660
the points are scattered away from the regression

417
00:32:05,660 --> 00:32:09,800
line, I mean you cannot determine if it is

418
00:32:09,800 --> 00:32:13,160
positive or negative, then there is no

419
00:32:13,160 --> 00:32:16,220
relationship between the two variables, the same

420
00:32:16,220 --> 00:32:20,580
as this one. X increases, Y stays nearly in the

421
00:32:20,580 --> 00:32:24,540
same position, then there exists no relationship

422
00:32:24,540 --> 00:32:29,280
between the two variables. So, a relationship

423
00:32:29,280 --> 00:32:32,740
could be linear or curvilinear. It could be

424
00:32:32,740 --> 00:32:37,280
positive or negative, strong or weak, or sometimes

425
00:32:37,280 --> 00:32:41,680
there is no relationship between the two

426
00:32:41,680 --> 00:32:49,200
variables. Now the question is, how can we write

427
00:32:51,250 --> 00:32:55,290
Or how can we find the best regression line that

428
00:32:55,290 --> 00:32:59,570
fits the data you have? We know the regression is

429
00:32:59,570 --> 00:33:06,270
the straight line equation is given by this one. Y

430
00:33:06,270 --> 00:33:20,130
equals beta 0 plus beta 1x plus epsilon. This can

431
00:33:20,130 --> 00:33:21,670
be pronounced as epsilon.

432
00:33:24,790 --> 00:33:29,270
It's a great letter, the same as alpha, beta, mu,

433
00:33:29,570 --> 00:33:35,150
sigma, and so on. So it's epsilon. I, it means

434
00:33:35,150 --> 00:33:39,250
observation number I. I 1, 2, 3, up to 10, for

435
00:33:39,250 --> 00:33:42,710
example, is the same for selling price of a home.

436
00:33:43,030 --> 00:33:46,970
So I 1, 2, 3, all the way up to the sample size.

437
00:33:48,370 --> 00:33:54,830
Now, Y is your dependent variable. Beta 0 is

438
00:33:54,830 --> 00:33:59,810
population Y intercept. For example, if we have

439
00:33:59,810 --> 00:34:00,730
this scatter plot.

440
00:34:04,010 --> 00:34:10,190
Now, beta 0 is

441
00:34:10,190 --> 00:34:15,370
this one. So this is your beta 0. So this segment

442
00:34:15,370 --> 00:34:21,550
is beta 0. it could be above the x-axis I mean

443
00:34:21,550 --> 00:34:34,890
beta zero could be positive might be negative now

444
00:34:34,890 --> 00:34:40,270
this beta zero fall below the x-axis so beta zero

445
00:34:40,270 --> 00:34:43,850
could be negative or

446
00:34:46,490 --> 00:34:49,350
Maybe the straight line passes through the origin

447
00:34:49,350 --> 00:34:56,990
point. So in this case, beta zero equals zero. So

448
00:34:56,990 --> 00:34:59,890
it could be positive and negative or equal zero,

449
00:35:00,430 --> 00:35:05,510
but still we have positive relationship. That

450
00:35:05,510 --> 00:35:09,970
means The value of beta zero, the sign of beta

451
00:35:09,970 --> 00:35:13,310
zero does not affect the relationship between Y

452
00:35:13,310 --> 00:35:17,850
and X. Because here in the three cases, there

453
00:35:17,850 --> 00:35:22,390
exists positive relationship, but beta zero could

454
00:35:22,390 --> 00:35:25,370
be positive or negative or equal zero, but still

455
00:35:25,370 --> 00:35:31,720
we have positive relationship. I mean, you cannot

456
00:35:31,720 --> 00:35:35,060
determine by looking at beta 0, you cannot

457
00:35:35,060 --> 00:35:37,940
determine if there is a positive or negative

458
00:35:37,940 --> 00:35:41,720
relationship. The other term is beta 1. Beta 1 is

459
00:35:41,720 --> 00:35:46,900
the population slope coefficient. Now, the sign of

460
00:35:46,900 --> 00:35:50,010
the slope determines the direction of the

461
00:35:50,010 --> 00:35:54,090
relationship. That means if the slope has positive

462
00:35:54,090 --> 00:35:56,570
sign, it means there exists positive relationship.

463
00:35:57,330 --> 00:35:59,370
Otherwise if it is negative, then there is

464
00:35:59,370 --> 00:36:01,390
negative relationship between the two variables.

465
00:36:02,130 --> 00:36:05,310
So the sign of the slope determines the direction.

466
00:36:06,090 --> 00:36:11,290
But the sign of beta zero has no meaning about the

467
00:36:11,290 --> 00:36:15,470
relationship between Y and X. X is your

468
00:36:15,470 --> 00:36:19,630
independent variable, Y is your independent

469
00:36:19,630 --> 00:36:19,630
variable, Y is your independent variable, Y is

470
00:36:19,630 --> 00:36:19,650
your independent variable, Y is your independent

471
00:36:19,650 --> 00:36:21,250
variable, Y is your independent variable, Y is

472
00:36:21,250 --> 00:36:21,250
your independent variable, Y is your independent

473
00:36:21,250 --> 00:36:21,250
variable, Y is your independent variable, Y is

474
00:36:21,250 --> 00:36:21,250
your independent variable, Y is your independent

475
00:36:21,250 --> 00:36:24,370
variable, Y is your independent variable, Y is

476
00:36:24,370 --> 00:36:24,370
your independent variable, Y is your independent

477
00:36:24,370 --> 00:36:24,370
variable, Y is your independent variable, Y is

478
00:36:24,370 --> 00:36:24,370
your independent variable, Y is your independent

479
00:36:24,370 --> 00:36:24,370
variable, Y is your independent variable, Y is

480
00:36:24,370 --> 00:36:24,370
your independent variable, Y is your independent

481
00:36:24,370 --> 00:36:24,370
variable, Y is your independent variable, Y is

482
00:36:24,370 --> 00:36:24,370
your independent variable, Y is your independent

483
00:36:24,370 --> 00:36:24,370
variable, Y is your independent variable, Y is

484
00:36:24,370 --> 00:36:24,370
your independent variable, Y is your independent

485
00:36:24,370 --> 00:36:24,370
variable, Y is your independent variable, Y is

486
00:36:24,370 --> 00:36:24,370
your independent variable, Y is your independent

487
00:36:24,370 --> 00:36:24,430
variable, Y is your independent variable, Y is

488
00:36:24,430 --> 00:36:24,770
your independent variable, Y is your independent

489
00:36:24,770 --> 00:36:27,490
variable, Y is your independent variable, Y is

490
00:36:27,490 --> 00:36:30,110
your independent variable, Y is your It means

491
00:36:30,110 --> 00:36:32,450
there are some errors you don't know about it

492
00:36:32,450 --> 00:36:36,130
because you ignore some other variables that may

493
00:36:36,130 --> 00:36:39,410
affect the selling price. Maybe you select a

494
00:36:39,410 --> 00:36:42,490
random sample, that sample is small. Maybe there

495
00:36:42,490 --> 00:36:46,270
is a random, I'm sorry, there is sampling error.

496
00:36:47,070 --> 00:36:52,980
So all of these are called random error term. So

497
00:36:52,980 --> 00:36:57,420
all of them are in this term. So epsilon I means

498
00:36:57,420 --> 00:37:00,340
something you don't include in your regression

499
00:37:00,340 --> 00:37:03,280
modeling. For example, you don't include all the

500
00:37:03,280 --> 00:37:06,180
independent variables that affect Y, or your

501
00:37:06,180 --> 00:37:09,700
sample size is not large enough. So all of these

502
00:37:09,700 --> 00:37:14,260
measured in random error term. So epsilon I is

503
00:37:14,260 --> 00:37:18,840
random error component, beta 0 plus beta 1X is

504
00:37:18,840 --> 00:37:25,070
called linear component. So that's the simple

505
00:37:25,070 --> 00:37:31,430
linear regression model. Now, the data you have,

506
00:37:32,850 --> 00:37:38,210
the blue circles represent the observed value. So

507
00:37:38,210 --> 00:37:47,410
these blue circles are the observed values. So we

508
00:37:47,410 --> 00:37:49,370
have observed.

509
00:37:52,980 --> 00:37:57,940
Y observed value of Y for each value X. The

510
00:37:57,940 --> 00:38:03,360
regression line is the blue, the red one. It's

511
00:38:03,360 --> 00:38:07,560
called the predicted values. Predicted Y.

512
00:38:08,180 --> 00:38:14,760
Predicted Y is denoted always by Y hat. Now the

513
00:38:14,760 --> 00:38:19,740
difference between Y and Y hat. It's called the

514
00:38:19,740 --> 00:38:20,200
error term.

515
00:38:24,680 --> 00:38:28,000
It's actually the difference between the observed

516
00:38:28,000 --> 00:38:31,600
value and its predicted value. Now, the predicted

517
00:38:31,600 --> 00:38:34,720
value can be determined by using the regression

518
00:38:34,720 --> 00:38:39,180
line. So this line is the predicted value of Y for

519
00:38:39,180 --> 00:38:44,480
XR. Again, beta zero is the intercept. As we

520
00:38:44,480 --> 00:38:46,260
mentioned before, it could be positive or negative

521
00:38:46,260 --> 00:38:52,600
or even equal zero. The slope is changing Y.

522
00:38:55,140 --> 00:38:57,580
Divide by change of x.

523
00:39:01,840 --> 00:39:07,140
So these are the components for the simple linear

524
00:39:07,140 --> 00:39:10,840
regression model. Y again represents the

525
00:39:10,840 --> 00:39:14,960
independent variable. Beta 0 y intercept. Beta 1

526
00:39:14,960 --> 00:39:17,960
is your slope. And the slope determines the

527
00:39:17,960 --> 00:39:20,900
direction of the relationship. X independent

528
00:39:20,900 --> 00:39:25,270
variable epsilon i is the random error term. Any

529
00:39:25,270 --> 00:39:25,650
question?

530
00:39:31,750 --> 00:39:36,610
The relationship may be positive or negative. It

531
00:39:36,610 --> 00:39:37,190
could be negative.

532
00:39:40,950 --> 00:39:42,710
Now, for negative relationship,

533
00:39:57,000 --> 00:40:04,460
Or negative, where beta zero is negative.

534
00:40:04,520 --> 00:40:08,700
Or beta

535
00:40:08,700 --> 00:40:09,740
zero equals zero.

536
00:40:16,680 --> 00:40:20,620
So here there exists negative relationship, but

537
00:40:20,620 --> 00:40:22,060
beta zero may be positive.

538
00:40:25,870 --> 00:40:30,210
So again, the sign of beta 0 also does not affect

539
00:40:30,210 --> 00:40:31,990
the relationship between the two variables.

540
00:40:36,230 --> 00:40:40,590
Now, we don't actually know the values of beta 0

541
00:40:40,590 --> 00:40:44,510
and beta 1. We are going to estimate these values

542
00:40:44,510 --> 00:40:48,110
from the sample we have. So the simple linear

543
00:40:48,110 --> 00:40:50,970
regression equation provides an estimate of the

544
00:40:50,970 --> 00:40:55,270
population regression line. So here we have Yi hat

545
00:40:55,270 --> 00:41:00,010
is the estimated or predicted Y value for

546
00:41:00,010 --> 00:41:00,850
observation I.

547
00:41:03,530 --> 00:41:08,220
The estimate of the regression intercept P0. The

548
00:41:08,220 --> 00:41:11,360
estimate of the regression slope is b1, and this

549
00:41:11,360 --> 00:41:16,680
is your x, all independent variable. So here is

550
00:41:16,680 --> 00:41:20,340
the regression equation. Simple linear regression

551
00:41:20,340 --> 00:41:24,400
equation is given by y hat, the predicted value of

552
00:41:24,400 --> 00:41:29,380
y equals b0 plus b1 times x1.

553
00:41:31,240 --> 00:41:35,960
Now these coefficients, b0 and b1 can be computed

554
00:41:37,900 --> 00:41:43,040
by the following equations. So the regression

555
00:41:43,040 --> 00:41:52,920
equation is

556
00:41:52,920 --> 00:41:57,260
given by y hat equals b0 plus b1x.

557
00:41:59,940 --> 00:42:06,140
Now the slope, b1, is r times standard deviation

558
00:42:06,140 --> 00:42:10,540
of y Times standard deviation of x. This is the

559
00:42:10,540 --> 00:42:13,820
simplest equation to determine the value of the

560
00:42:13,820 --> 00:42:18,980
star. B1r, r is the correlation coefficient. Sy is

561
00:42:18,980 --> 00:42:25,080
xr, the standard deviations of y and x. Where b0,

562
00:42:25,520 --> 00:42:30,880
which is y intercept, is y bar minus b x bar, or

563
00:42:30,880 --> 00:42:38,100
b1 x bar. Sx, as we know, is the sum of x minus y

564
00:42:38,100 --> 00:42:40,460
squared divided by n minus 1 under square root,

565
00:42:40,900 --> 00:42:47,060
similarly for y values. So this, how can we, these

566
00:42:47,060 --> 00:42:52,380
formulas compute the values of b0 and b1. So we

567
00:42:52,380 --> 00:42:54,600
are going to use these equations in order to

568
00:42:54,600 --> 00:42:58,960
determine the values of b0 and b1.

569
00:43:04,670 --> 00:43:07,710
Now, what's your interpretation about the slope

570
00:43:07,710 --> 00:43:13,130
and the intercept? For example, suppose we are

571
00:43:13,130 --> 00:43:18,610
talking about your score Y and

572
00:43:18,610 --> 00:43:22,110
X number of missing classes.

573
00:43:29,210 --> 00:43:35,460
And suppose, for example, Y hat Equal 95 minus 5x.

574
00:43:37,780 --> 00:43:41,420
Now let's see what's the interpretation of B0.

575
00:43:42,300 --> 00:43:45,060
This is B0. So B0 is 95.

576
00:43:47,660 --> 00:43:51,960
And B1 is 5. Now what's your interpretation about

577
00:43:51,960 --> 00:43:57,740
B0 and B1? B0 is the estimated mean value of Y

578
00:43:57,740 --> 00:44:02,560
when the value of X is 0. that means if the

579
00:44:02,560 --> 00:44:08,500
student does not miss any class that means x

580
00:44:08,500 --> 00:44:13,260
equals zero in this case we predict or we estimate

581
00:44:13,260 --> 00:44:19,880
the mean value of his score or her score is 95 so

582
00:44:19,880 --> 00:44:27,500
95 it means when x is zero if x is zero then we

583
00:44:27,500 --> 00:44:35,350
expect his or Here, the score is 95. So that means

584
00:44:35,350 --> 00:44:39,830
B0 is the estimated mean value of Y when the value

585
00:44:39,830 --> 00:44:40,630
of X is 0.

586
00:44:43,370 --> 00:44:46,590
Now, what's the meaning of the slope? The slope in

587
00:44:46,590 --> 00:44:51,290
this case is negative Y. B1, which is the slope,

588
00:44:51,590 --> 00:44:57,610
is the estimated change in the mean of Y. as a

589
00:44:57,610 --> 00:45:03,050
result of a one unit change in x for example let's

590
00:45:03,050 --> 00:45:07,070
compute y for different values of x suppose x is

591
00:45:07,070 --> 00:45:15,510
one now we predict his score to be 95 minus 5

592
00:45:15,510 --> 00:45:25,470
times 1 which is 90 when x is 2 for example Y hat

593
00:45:25,470 --> 00:45:28,570
is 95 minus 5 times 2, so that's 85.

594
00:45:31,950 --> 00:45:39,970
So for each one unit, there is a drop by five

595
00:45:39,970 --> 00:45:43,750
units in his score. That means if number of

596
00:45:43,750 --> 00:45:47,550
missing classes increases by one unit, then his or

597
00:45:47,550 --> 00:45:51,790
her weight is expected to be reduced by five units

598
00:45:51,790 --> 00:45:56,150
because the sign is negative. another example

599
00:45:56,150 --> 00:46:05,910
suppose again we are interested in whales and

600
00:46:05,910 --> 00:46:16,170
angels and imagine that just

601
00:46:16,170 --> 00:46:21,670
for example y equal y hat equals three plus four x

602
00:46:21,670 --> 00:46:29,830
now y hat equals 3 if x equals zero. That has no

603
00:46:29,830 --> 00:46:34,510
meaning because you cannot say age of zero. So

604
00:46:34,510 --> 00:46:40,450
sometimes the meaning of y intercept does not make

605
00:46:40,450 --> 00:46:46,150
sense because you cannot say x equals zero. Now

606
00:46:46,150 --> 00:46:50,690
for the stock of four, that means as his or her

607
00:46:50,690 --> 00:46:55,550
weight increases by one year, Then we expect his

608
00:46:55,550 --> 00:47:00,470
weight to increase by four kilograms. So as one

609
00:47:00,470 --> 00:47:05,130
unit increase in x, y is our, his weight is

610
00:47:05,130 --> 00:47:10,150
expected to increase by four units. So again,

611
00:47:10,370 --> 00:47:16,950
sometimes we can interpret the y intercept, but in

612
00:47:16,950 --> 00:47:18,670
some cases it has no meaning.

613
00:47:24,970 --> 00:47:27,190
Now for the previous example, for the selling

614
00:47:27,190 --> 00:47:32,930
price of a home and its size, B1rSy divided by Sx,

615
00:47:33,790 --> 00:47:43,550
r is computed, r is found to be 76%, 76%Sy divided

616
00:47:43,550 --> 00:47:49,990
by Sx, that will give 0.109. B0y bar minus B1x

617
00:47:49,990 --> 00:47:50,670
bar,

618
00:47:53,610 --> 00:48:00,150
Y bar for this data is 286 minus D1. So we have to

619
00:48:00,150 --> 00:48:03,490
compute first D1 because we use it in order to

620
00:48:03,490 --> 00:48:08,590
determine D0. And calculation gives 98. So that

621
00:48:08,590 --> 00:48:16,450
means based on these equations, Y hat equals 0

622
00:48:16,450 --> 00:48:22,990
.10977 plus 98.248.

623
00:48:24,790 --> 00:48:29,370
times X. X is the size.

624
00:48:32,890 --> 00:48:39,830
0.1 B1

625
00:48:39,830 --> 00:48:45,310
is

626
00:48:45,310 --> 00:48:56,650
0.1, B0 is 98, so 98.248 plus B1. So this is your

627
00:48:56,650 --> 00:49:03,730
regression equation. So again, the intercept is

628
00:49:03,730 --> 00:49:09,750
98. So this amount, the segment is 98. Now the

629
00:49:09,750 --> 00:49:14,790
slope is 0.109. So house price, the expected value

630
00:49:14,790 --> 00:49:21,270
of house price equals B098 plus 0.109 square feet.

631
00:49:23,150 --> 00:49:27,630
So that's the prediction line for the house price.

632
00:49:28,510 --> 00:49:34,370
So again, house price equal B0 98 plus 0.10977

633
00:49:34,370 --> 00:49:36,930
times square root. Now, what's your interpretation

634
00:49:36,930 --> 00:49:41,950
about B0 and B1? B0 is the estimated mean value of

635
00:49:41,950 --> 00:49:46,430
Y when the value of X is 0. So if X is 0, this

636
00:49:46,430 --> 00:49:52,980
range of X observed X values and you have a home

637
00:49:52,980 --> 00:49:57,860
or a house of size zero. So that means this value

638
00:49:57,860 --> 00:50:02,680
has no meaning. Because a house cannot have a

639
00:50:02,680 --> 00:50:06,400
square footage of zero. So B0 has no practical

640
00:50:06,400 --> 00:50:10,040
application in this case. So sometimes it makes

641
00:50:10,040 --> 00:50:17,620
sense, in other cases it doesn't have that. So for

642
00:50:17,620 --> 00:50:21,790
this specific example, B0 has no practical

643
00:50:21,790 --> 00:50:28,210
application in this case. But B1 which is 0.1097,

644
00:50:28,930 --> 00:50:33,050
B1 estimates the change in the mean value of Y as

645
00:50:33,050 --> 00:50:36,730
a result of one unit increasing X. So for this

646
00:50:36,730 --> 00:50:41,640
value which is 0.109, it means This fellow tells

647
00:50:41,640 --> 00:50:46,420
us that the mean value of a house can increase by

648
00:50:46,420 --> 00:50:52,280
this amount, increase by 0.1097, but we have to

649
00:50:52,280 --> 00:50:55,700
multiply this value by a thousand because the data

650
00:50:55,700 --> 00:51:01,280
was in thousand dollars, so around 109, on average

651
00:51:01,280 --> 00:51:05,160
for each additional one square foot of a size. So

652
00:51:05,160 --> 00:51:09,990
that means if a house So if house size increased

653
00:51:09,990 --> 00:51:14,630
by one square foot, then the price increased by

654
00:51:14,630 --> 00:51:19,530
around 109 dollars. So for each one unit increased

655
00:51:19,530 --> 00:51:22,990
in the size, the selling price of a home increased

656
00:51:22,990 --> 00:51:29,590
by 109. So that means if the size increased by

657
00:51:29,590 --> 00:51:35,860
tenth, It means the selling price increased by

658
00:51:35,860 --> 00:51:39,400
1097

659
00:51:39,400 --> 00:51:46,600
.7. Make sense? So for each one unit increase in

660
00:51:46,600 --> 00:51:50,300
its size, the house selling price increased by

661
00:51:50,300 --> 00:51:55,540
109. So we have to multiply this value by the unit

662
00:51:55,540 --> 00:52:02,280
we have. Because Y was 8000 dollars. Here if you

663
00:52:02,280 --> 00:52:06,600
go back to the previous data we have, the data was

664
00:52:06,600 --> 00:52:11,120
house price wasn't thousand dollars, so we have to

665
00:52:11,120 --> 00:52:15,840
multiply the slope by a thousand.

666
00:52:19,480 --> 00:52:23,720
Now we

667
00:52:23,720 --> 00:52:30,380
can use also the regression equation line to make

668
00:52:30,380 --> 00:52:35,390
some prediction. For example, we can predict the

669
00:52:35,390 --> 00:52:42,290
price of a house with 2000 square feet. You just

670
00:52:42,290 --> 00:52:43,590
plug this value.

671
00:52:46,310 --> 00:52:52,210
So we have 98.25 plus 0.109 times 2000. That will

672
00:52:52,210 --> 00:53:01,600
give the house price. for 2,000 square feet. So

673
00:53:01,600 --> 00:53:05,920
that means the predicted price for a house with 2

674
00:53:05,920 --> 00:53:10,180
,000 square feet is this amount multiplied by 1

675
00:53:10,180 --> 00:53:18,260
,000. So that will give $317,850. So that's how

676
00:53:18,260 --> 00:53:24,240
can we make predictions for why I mean for house

677
00:53:24,240 --> 00:53:29,360
price at any given value of its size. So for this

678
00:53:29,360 --> 00:53:36,020
data, we have a house with 2000 square feet. So we

679
00:53:36,020 --> 00:53:43,180
predict its price to be around 317,850.

680
00:53:44,220 --> 00:53:50,920
I will stop at coefficient of correlation. I will

681
00:53:50,920 --> 00:53:54,190
stop at coefficient of determination for next time

682
00:53:54,190 --> 00:53:57,770
that's

683
00:53:57,770 --> 00:53:57,990
all