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### Statistics for Business and Economics

Dr. TANG Yu

Department of Mathematics

Soochow University

May 28, 2007

Types of Correlation

Positivecorrelation

Slope is positive

Negativecorrelation

Slope is negtive

No correlation

Slope is zero

Hypothesis Test

- For the simple linear regression model

- If x and y are linearly related, we must have

- We will use the sample data to test the following hypotheses about the parameter

Sampling Distribution

- Just as the sampling distribution of the sample mean, X-bar, depends on the the mean, standard deviation and shape of the X population, the sampling distributions of the β0-hat and β1-hat least squares estimators depend on the properties of the {Yj } sub-populations (j=1,…, n).

- Given xj, the properties of the {Yj } sub-population are determined by the εj error/random variable.

Model Assumption

- As regards the probability distributions of εj (j =1,…, n),it is assumed that:

- Each εj is normally distributed,

Yj is also normal;

- Each εj has zero mean,

E(Yj) = β0 + β1 xj

- Each εj has the same variance, σε2,

Var(Yj) =σε2 is also constant;

- The errors are independent of each other,

{Yi} and {Yj}, i j, are also independent;

- The error does not depend on the independent variable(s).

The effects of X and ε on Y can be separated from each other.

Graph Show

Yi: N (β0+β1xi ;σ )

Yj: N (β0+β1xj ;σ )

xi

xj

The y distributions have the same shape at each x value

Sum of Squares

Sum of squares due to error (SSE)

Sum of squares due to regression (SSR)

Total sum of squares (SST)

Example

Total

ANOVA Table

- As F=35.93 > 6.61, where 6.61 is the critical value for F-distribution with degrees of freedom 1 and 5 (significant level takes .05), we reject H0, and conclude that the relationship between x and y is significant

Hypothesis Test

- For the simple linear regression model

- If x and y are linearly related, we must have

- We will use the sample data to test the following hypotheses about the parameter

Standard Errors

Standard error of estimate: the sample standard deviation ofε.

Replacing σε with its estimate, sε, the estimated standard errorofβ1-hat is

Example

Total

Calculation

where 2.571 is the critical value for t-distribution with degree of freedom 5 (significant level takes .025), so we reject H0, and conclude that the relationship between x and y is significant

Confidence Interval

β1-hat is an estimator of β1

follows a t-distribution with n-2 degrees of freedom

The estimated standard errorofβ1-hat is

So the C% confidence interval estimatorsof β1 is

Example

The 95% confidence interval estimatorsof β1 in the previous example is

i.e., from –12.87 to -5.15, which does not contain 0

Regression Equation

- It is believed that the longer one studied, the better one’s grade is. The final mark (Y) on study time (X) is supposed to follow the regression equation:

- If the fit of the sample regression equation is satisfactory, it can be used to estimate its mean value or to predict the dependent variable.

Estimate and Predict

Estimate

Predict

For the expected value of a Y sub-population.

For a particular element of a Y sub-population.

E.g.: What is the mean final mark of all those students who spent 30 hours on studying?

I.e., given x= 30, how large is E(y)?

E.g.: What is the final mark of Tom who spent 30 hours on studying?

I.e., given x= 30, how large is y?

What Is the Same?

For a given X value, the point forecast (predict) of Y and the point estimator of the mean of the {Y} sub-population are the same:

Ex.1 Estimate the mean final mark of students who spent 30 hours on study.

Ex.2 Predict the final mark of Tom, when his study time is 30 hours.

What Is the Difference?

The interval prediction of Y and the interval estimation of the mean of the {Y} sub-population are different:

- The prediction

- The estimation

The prediction interval is wider than the confidence interval

Example

Total

Estimation and Prediction

- The point forecast (predict) of Y and the point estimator of the mean of the {Y} are the same:

For

The confidence interval

when xg =

The confidence interval

when xg =

The confidence interval

when xg =

Moving Rule- As xg moves away from x the interval becomes longer. That is, the shortest interval is found at x.

The confidence interval

when xg =

The confidence interval

when xg =

The confidence interval

when xg =

Moving Rule- As xg moves away from x the interval becomes longer. That is, the shortest interval is found at x.

Residual Analysis

- Regression Residual– the difference between an observed y value and its corresponding predicted value

- Properties of Regression Residual
- The mean of the residuals equals zero
- The standard deviation of the residuals is equal to the standard deviation of the fitted regression model

Standardized Residual

- Standard deviation of the ith residual

where

- Standardized residual for observation i

Standardized Residual

- The standardized residual plot can provide insight about the assumption that the error term has a normal distribution

- If the assumption is satisfied, the distribution of the standardized residuals should appear to come from a standard normal probability distribution

- It is expected to see approximately 95% of the standardized residuals between –2 and +2

Detecting Outlier

Outlier

Influential Observation

Outlier

Influential Observation

Influential observation

Contact Information

- Tang Yu (唐煜)
- [email protected]
- http://math.suda.edu.cn/homepage/tangy

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