Hypothesis Testing and T-Tests

Hypothesis Tests Related to Differences

Copyright © 2009 Pearson Education, Inc.

Chapter 17 - 2

Tests of Differences

One Sample

Means

Proportions

Two IndependentSamples

PairedSamples

More ThanTwo Samples

Means Means Means

Proportions Proportions Proportions

The t Distribution• The t statistic assumes that the variable is normally distributed and

the mean is known (or assumed to be known) and the population variance is estimated from the sample.

• Assume that the random variable X is normally distributed, with mean and unknown population variance s2, which is estimated by the sample variance s 2.

• Then, is t distributed with n - 1 degrees of freedom.

• The t distribution is similar to the normal distribution in appearance. Both distributions are bell-shaped and symmetric. As the number of degrees of freedom increases, the t distribution approaches the normal distribution.

Copyright © 2009 Pearson Education, Inc. Chapter 17 - 3

Hypothesis Testing Using the t Statistic1. Formulate the null (H0) and the alternative (H1) hypotheses.

2. Select the appropriate formula for the t statistic.

3. Select a significance level, a, for testing H0. Typically, the 0.05 level is selected.

4. Take one or two samples and compute the mean and standard deviation for each sample.

5. Calculate the t statistic assuming H0 is true.

6. Calculate the degrees of freedom and estimate the probability of getting a more extreme value of the statistic from Table 4 (Alternatively, calculate the critical value of the t statistic).

Copyright © 2009 Pearson Education, Inc. Chapter 17 - 4

Hypothesis Testing Using the t Statistic (Cont.)

7. If the probability computed in step 5 is smaller than the significance level selected in step 2, reject H0. If the probability is larger, do not reject H0. (Alternatively, if the value of the calculated t statistic in step 4 is larger than the critical value determined in step 5, reject H0. If the calculated value is smaller than the critical value, do not reject H0). Failure to reject H0 does not necessarily imply that H0 is true. It only means that the true state is not significantly different than that assumed by H0.

8. Express the conclusion reached by the t test in terms of the marketing research problem.

Copyright © 2009 Pearson Education, Inc. Chapter 17 - 5

Conducting t-TestsFormulate H0 and H1

Select Appropriate t-Test

Choose Level of Significance,

Collect Data and Calculate Test Statistic

a) Determine Probability Associated with Test Statistic (TSCAL)

b) Determine Critical Value of Test Statistic TSCR

a) Compare with Level of Significance,

b) Determine if TSCAL falls into (Non) Rejection Region

Reject or Do Not Reject H0

Draw Marketing Research ConclusionChapter 17 - 6Copyright © 2012 Pearson Education, Inc.

Sample Test/Commets One Sample Means t test, if variance is unknown z test, if variance is known Proportions z test Two Independent Samples Means Two-group t test F test for equality of variances Proportions z test Chi-square test Paired Samples Means Paired t test Proportions Chi-square test More Than Two Samples Means One-way analysis of variance Proportions Chi-square test

A Summary of Hypothesis Testing

Chapter 17 - 7Copyright © 2012 Pearson Education, Inc.