Objectives for Hypothesis Testing. Simple Regression B. Hypothesis Testing Calculate t-ratios and confidence intervals for b 1 and b 2 . Test the significance of b 1 and b 2 with: T-ratios Prob values Confidence intervals. Explain the meaning of Type I and Type II errors.
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B. Hypothesis Testing
The Simple Linear Regression Model
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 ≠ 0Alternative 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 .
t-distribution, i.e.,t = b2/[se(b2)] ~ t (N-2)
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
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.
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
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
t = -2.58/3.17 = - 0.81
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?