Business Statistics - QBM117

Interval estimation for the slope and y-intercept

Hypothesis tests for regression

Objectives

To determine confidence interval estimators of the slope and the y intercept.

To test hypotheses about the slope of the regression line.

Estimating the slope and the y-intercept

The point estimators for the slope and the y - intercept can easily be determined from the Excel output generated when fitting the regression.

As we are aware from our study on confidence interval estimators previously, there are two types of estimators when estimating a population parameter:

point estimators and interval estimators.

The interval estimators can be just as easily determined from the Excel output generated.

SUMMARY OUTPUT

Regression StatisticsMultiple R 0.86673932R Square 0.75123705Adjusted R Square 0.68904631Standard Error 6.51658723Observations 6

ANOVAdf SS MS F Significance F

Regression 1 512.9697 512.9697 12.07956 0.025454365Residual 4 169.8636 42.46591Total 5 682.8333

Coefficients Std Error t Stat P-value Lower 95% Upper 95%Intercept 15.3181818 6.213322 2.465377 0.06929 -1.93280173 32.5691654Experience 1.67272727 0.481282 3.475567 0.025454 0.336471833 3.00898271

Experience Residual Plot

-20

-10

0

10

0 5 10 15 20 25

Experience

Resi

dual

s

Therefore the 95% confidence interval estimate of the slope is from 0.336 to 3.009 ie from $336 to $3009.

Excel also generates a confidence interval estimate for the y-intercept. This will only be considered if the y-intercept has a sensible interpretation in the situation described.

For our salary and experience example, the y- intercept does has a sensible interpretation ie it is the salary for a person with no experience. As such, we would also be interested in determining a confidence interval estimate of the intercept.

Therefore the 95% confidence interval estimate of the intercept is from -1.933 to 32.569 ie from -$1933 to $32 569.

We can easily summarise the relationship between two variables, whether it exists or not.

Hypothesis testing will tell us whether the relationship that appears to be there, is pure coincidence or, there is in fact a significant relationship between the two variables.

The null hypothesis states that there is no relationship between x and y.

Therefore the hypotheses for testing a significant relationship are

Testing whether the relationship is real or coincidence

0:

0:

1

10

AH

H

Why Statistical Inference?

Because there can seem to be a relationship• when, in fact, the population is just random

Below are plots of the data from samples of size n = 10 • from a population with no relationship (correlation 0)

• Notice that the sample correlations are not zero!

• This is due to the randomness of samplingr = – 0.471 r = 0.089 r = 0.395

0:

0:

1

10

AH

HStep 1

Step 2

1ˆ

11ˆ

s

t

Step 3

776.205.0 4,025.02,2/ tt n

For our example, we would be testing: is there a significant relationship between salary and experience?

Step 5

output) Excel (from 476.3samplet

Step 4

776.2or 776.2if Reject 0 samplesample ttH

Coefficients Std Error t Stat P-value Lower 95% Upper 95%Intercept 15.3181818 6.213322 2.465377 0.06929 -1.93280173 32.5691654Experience 1.67272727 0.481282 3.475567 0.025454 0.336471833 3.00898271

48.348.0

067.1

ˆ

1ˆ

11

s

t

Since 3.48 > 2.776 we reject H0.

Step 5

48.3samplet

Step 6

There is sufficient evidence at = 0.05 to conclude that there is a significant linear relationship between salary and experience.

Step 4

776.2or 776.2if Reject 0 samplesample ttH

0:

0:

1

10

AH

H

05.0

Using the p-value to test: is there a significant relationship between salary and experience?

Level of significance:

Decision rule: 05.0if Reject 0 valuepH

Coefficients Std Error t Stat P-value Lower 95% Upper 95%Intercept 15.3181818 6.213322 2.465377 0.06929 -1.93280173 32.5691654Experience 1.67272727 0.481282 3.475567 0.025454 0.336471833 3.00898271

0reject we05.0025.0 Since Hvaluep There is sufficient evidence at = 0.05 to conclude that there is a significant linear relationship between salary and experience.

An important point to remember about using the p-value to test a hypothesis is that the p-value can give us a good indication of how much evidence exists to support the alternative hypothesis.

The smaller the p-value, the more overwhelming is the evidence to support the alternative hypothesis.

In our example here, the p-value was only 0.025. This allows us to conclude that a linear relationship exists when testing at = 0.05 and 0.1, but our conclusion would be different at = 0.01

In situations where we are interested in how the independent variable affects the dependent variable, we estimate and test hypotheses about the linear regression model.

In many situation however, one variable does not influence the other and therefore we are not interested in estimating how the independent variable affects the dependent variable.

We simply want to test whether there is a linear correlation between the two variables.

Testing for a significant correlation

For these situations the null hypothesis states that there is no linear correlation between x and y.

Therefore the hypotheses for testing a significant linear correlation are

Testing for a significant correlation

0:

0:0

AH

H

When we test for a significant correlation, you will find that the value of the test statistic and the conclusion are exactly the same as when we test for a significant relationship between two variables.

This is because we are in fact testing the same thing. Are the two variables linearly related (correlated)?

Therefore we perform one test or the other - not both!

0:

0:0

AH

HStep 1

Step 2

2

1 where

2

n

rs

s

rt r

r

Step 3

776.205.0 4,025.02,2/ tt n

For our previous example, we would be testing: is there a significant linear correlation between salary and experience?

Step 5

output) Excel (from 476.3samplet

Step 4

776.2or 776.2if Reject 0 samplesample ttH

Coefficients Std Error t Stat P-value Lower 95% Upper 95%Intercept 15.3181818 6.213322 2.465377 0.06929 -1.93280173 32.5691654Experience 1.67272727 0.481282 3.475567 0.025454 0.336471833 3.00898271

48.326

751.01 where

249.0

0751.0

2

1 where

2

r

rr

s

n

rs

s

rt

Since 3.48 > 2.776 we reject H0.

Step 5

48.3samplet

Step 6

There is sufficient evidence at = 0.05 to conclude that there is a significant linear correlation between salary and experience.

Step 4

776.2or 776.2if Reject 0 samplesample ttH

0:

0:0

AH

H

05.0

Using the p-value to test: is there a significant correlation between salary and experience?

Level of significance:

Decision rule: 05.0if Reject 0 valuepH

Coefficients Std Error t Stat P-value Lower 95% Upper 95%Intercept 15.3181818 6.213322 2.465377 0.06929 -1.93280173 32.5691654Experience 1.67272727 0.481282 3.475567 0.025454 0.336471833 3.00898271

0reject we05.0025.0 Since Hvaluep There is sufficient evidence at = 0.05 to conclude that there is a significant linear correlation between salary and experience.

Reading for next lecture

Read Chapter 18 Sections 18.6

(Chapter 11 Sections 11.6 abridged)

Exercises to be completed before next lecture

S&S 18.27 18.29

(11.27 11.29 abridged)