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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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Objectives for Hypothesis Testing

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Objectives for hypothesis testing l.jpg

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.


  • The model for hypothesis testing l.jpg

    The Model for Hypothesis Testing

    The Simple Linear Regression Model


    Hypothesis testing l.jpg

    Hypothesis Testing

    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 .


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    The t-distribution

    • 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.


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    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.


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    The t-distribution

    Figure 3.1 Critical Values from a t-distribution


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    Two-tail Tests with Alternative “Not Equal To” (≠)

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


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    Examples of Hypothesis Tests


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    Tests of Significance

    • 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.

    • 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


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    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 > |t|

      Intercept 1 486.18871 59.78625 8.13 <.0001

      distance 1 -2.57625 3.16619 0.4222


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    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 > -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.


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    The p-Value


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    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: 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?


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    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 > |t|

    Intercept 1 486.18871 59.78625 8.13 <.0001

    distance 1 -2.57625 3.16619 -0.81 0.4222


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    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 > |t|

    Intercept 1 65.53119 16.73502 3.92 0.0024

    age 1 -1.36918 0.58207 -2.35 0.0383


    Test of significance of age in coffee regression for sample of coffee drinkers l.jpg

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

    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.


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    Questions?


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