Objectives for Hypothesis Testing

1
Objectives for Hypothesis Testing

• 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.

2
The Model for Hypothesis Testing

• The Simple Linear Regression Model

3
Hypothesis Testing

• Goal Does X significantly affect Y?
• Is ß2 0 ?
• Conduct a hypothesis test of the form
• H0 ß2 0 Null hypothesis (X does not
  affect Y)?
• H1 ß2 ? 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 .

4
The t-distribution

• Under the null hypothesis, ß2 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.

5
Statistics Trivia

• 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.

6
The t-distribution

• Figure 3.1 Critical Values from a t-distribution

7
Two-tail Tests with Alternative Not Equal To
(?)

• Figure 3.4 The rejection region for a two-tail
  test of H0 ßk c against H1 ßk ? c

8
Examples of Hypothesis Tests
9
Tests of Significance

• Model Rent ß1 ß2 Distance e
• Assume all other assumptions of the simple
  regression model are met.
• 2. The null hypothesis is H0 ß2 0
• The alternative hypothesis is H1 ß2 ? 0
  .
• 3. The test statistic is t b2/se(b2) t
  (N-2) if the null hypothesis is true.
• Let us select a .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 lt t lt 2.042, we do not reject the
  null hypothesis

10
Regression of Rent on Distance

• Calculate the t-ratio on the distance parameter.
• 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 gt t
• 
  
• Intercept 1 486.18871 59.78625
  8.13 lt.0001
• distance 1 -2.57625 3.16619
  0.4222

11
Significance of the Distance Parameter in the
Rent Equation

• What is the value of the t-ratio on b 2?
• t -2.58/3.17 - 0.81
• What do you conclude?
• Since t gt -2.042 and t lt 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.

12
The p-Value
13
The p-Value

• 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 ßK ? 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?

14
Regression of Rent on Distance

• 
  
• 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 gt t
• 
  
• Intercept 1 486.18871 59.78625
  8.13 lt.0001
• distance 1 -2.57625 3.16619
  -0.81 0.4222

15
Regression of Coffee on Age for Coffee Drinkers

• 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 gt t
• 
  
• Intercept 1 65.53119 16.73502
  3.92 0.0024
• age 1 -1.36918 0.58207
  -2.35 0.0383

16
Test of Significance of Age in Coffee Regression
for Sample of Coffee Drinkers

• a .05. N 13 ? df 13-2 11.
• t ?2.201
• Since t -2.35 lt -2.201 we reject the null
  hypothesis and conclude that age significantly
  affects coffee consumption for coffee drinkers.
• The p-value 0.0383 lt .05 also indicates we
  should reject H0.

17

• Questions?