II. Simple Regression

B. Hypothesis Testing

• Calculate t-ratios and confidence intervals for b1 and b2 .

• Test the significance of b1 and b2 with:

• T-ratios

• Prob values

• Confidence intervals.

• Explain the meaning of Type I and Type II errors.

y x e 1 2

( ) 0E e 1 2( )E y x 2var( ) var( )e y

cov( , ) cov( , )e e y yi j i j 0

e N~ ( , )0 2

Goal: Does X significantly affect Y?

Is β2 = 0 ?

Conduct a hypothesis test of the form:

H0: β2 = 0 Null hypothesis (X does not affect Y)

H1: β2 ≠ 0 Alternative hypothesis (X affects Y)

Use a test statistic, the t-ratio, given by:

t = b2/[se(b2)]

where se(b2) = the standard error of b2 .

• Under the null hypothesis, β2 = 0, the t-ratio is distributed according to the

t-distribution, i.e., t = b2/[se(b2)] ~ t (N-2)

• If the null hypothesis is not true, the t-ratio does not have a t-distribution.

• The t-distribution is a bell-shaped curve.

• It looks like the normal distribution, except it is more spread out, with a larger variance and thicker tails.

• The t-distribution converges to a normal distribution as the sample size gets large (as N ).

• The shape of the t-distribution is controlled by a single parameter called the degrees of freedom, often abbreviated as df, where df = N – k , N = number of observations and k = number of parameters.

The t-distribution was developed by William Sealy Gosset, a brewing

chemist at the Guinness brewery in Ireland. He developed the t-test to

ensure consistent quality from each batch of Guinness beer. Guinness

allowed Gosset to publish his results, but only under the condition that

the data remain confidential and that he publish under a different

name. Gosset published under the pseudonym “Student” and the

distribution became known as the Student t-distribution.

Figure 3.1 Critical Values from a t-distribution

Figure 3.4 The rejection region for a two-tail test of H0: βk = c against H1: βk ≠ c

PROCEDURE FOR TESTING HYPOTHESES

1.Specify the model for estimation.

2.Determine the null and alternative hypotheses.

3.Specify the test statistic and its distribution if the null

hypothesis is true.

4.Select α and determine the rejection region.

5.Calculate the sample value of the test statistic.

6.State your conclusion.

1. Model: Rent = β1 + β2 Distance + e

Assume all other assumptions of the simple regression model are met.

2. The null hypothesis is : H0: β2 = 0

The alternative hypothesis is: H1: β2 ≠ 0 .

3. The test statistic is: t = b2/[se(b2)] ~ t (N-2)

if the null hypothesis is true.

4. Let us select α = .05. Note N = 32.

What are the degrees of freedom and critical values?

df = 30; t = 2.042

What is the rejection region? t - 2.042; t 2.042;

if - 2.042 < t < 2.042, we do not reject the null hypothesis

1. Calculate the t-ratio on the distance parameter.2. Test for the significance of the distance parameter on rent using a t-test.

The REG Procedure Model: MODEL1 Dependent Variable: rent Number of Observations Read 32

Parameter Estimates Parameter Standard Variable DF Estimate Error t Value Pr > |t| Intercept 1 486.18871 59.78625 8.13 <.0001 distance 1 -2.57625 3.16619 0.4222

5. What is the value of the t-ratio on b 2?

t = -2.58/3.17 = - 0.81

6. What do you conclude?

• Since t > -2.042 and t < 2.042, we do not reject the null hypothesis.

• The parameter estimate on distance, b2, is not significantly different from zero.

• Distance is not a significant determinant of rent.

P-value (or prob-value) – For a particular sample value of

t, the p-value is the probability of rejecting the null

hypothesis when the null is true.

p-value rule: Reject the null hypothesis when the p-value is

less than, or equal to, the level of significance α. That is, if

p ≤ α then reject H0. If p > α then do not reject H0.

Graphically, the P-value is the area in the tails of the distribution beyond |t|.

That is, if t is the calculated value of the t-statistic, and if H1: βK ≠ 0, then:

p = sum of probabilities to the right of |t| and to the left of – |t|

According to the p-value on the parameter on distance in the rent equation, do you reject the null hypothesis?

The REG Procedure Model: MODEL1 Dependent Variable: rent Number of Observations Read 32 Number of Observations Used 32 Parameter Estimates Parameter Standard Variable DF Estimate Error t Value Pr > |t| Intercept 1 486.18871 59.78625 8.13 <.0001 distance 1 -2.57625 3.16619 -0.81 0.4222

For the following regression estimates, test the hypothesis that the parameter estimate on age is significantly different from zero using a t-test and a p-value test.

The REG Procedure Model: MODEL1 Dependent Variable: coffee Number of Observations Read 13

Parameter Estimates Parameter Standard Variable DF Estimate Error t Value Pr > |t| Intercept 1 65.53119 16.73502 3.92 0.0024 age 1 -1.36918 0.58207 -2.35 0.0383

α = .05. N = 13 df = 13-2 = 11.

t = 2.201

Since t = -2.35 < -2.201 we reject the null hypothesis and

conclude that age significantly affects coffee

consumption for coffee drinkers.

The p-value = 0.0383 < .05 also indicates we should

reject H0.

QUESTIONS?