Hypotheses Testing - Questionstar

5.2 Inferential Statistics: Can the Results Be Generalized to Population?

5.2.1 Hypotheses Testing

Hypothesis Testing

Hypothesis Testing is a five-step procedure using sample evidence and probability theory to determine whether the hypothesis is a reasonable statement.

In other words, it is a method to prove whether or not the results obtained on a randomly drawn sample are projectable to the whole population.

Procedure:

1.State null and alternative hypothesis

2.Select a level of significance

3.Identify the test statistic

4.Formulate a decision rule

5.Take a sample, arrive at a decision

People are ‘erroneously confident’ in their knowledge and underestimate the odds that their information or beliefs will be proved wrong. They tend to seek additional information in ways that confirm what they already believed.”

Max Bazerman

Sex and Internet Usage

	Sex
Internet usage	Male	Female	*Row total*
light	5	10	15
heavy	10	5	15
Column total	15	15	n=30

Based on this sample:

Q: Are there really more heavy internet users among males than among females in the general population?

Step 1: State null and alternative hypothesis

A null hypothesis (H0) is a statement of status quo, one of no difference or no effect.

An alternative hypothesis (H1) is one in which some difference or effect is expected.

H0: There is no difference between males and females w.r.t. internet usage.

IUm=IUf

H1: Males and females expose different internet usage behavior.

IUm≠IUf

Step 2: Select a level of significance

Significance (α) – probability of rejecting a true null hypothesis.

β – probability of accepting a false null hypothesis.

Null hypothesis (H0)
is true

Null hypothesis (H0)
is false

Reject null hypothesis

Type I error
False positive

Correct outcome
True positive

Fail to reject null hypothesis

Correct outcome
True negative

Type II error
False negative

Analogy: innocence in a criminal trial
H0: the defendant is innocent

Analogy: Rustle in the bush – is it a lion?
H0: there is no lion in the bush

Levels of significance in marketing research

α – level of significance

0.01 (1%)

0.05 (5%)

(1-α) – level of confidence

0.99 (99%)

0.95 (95%)

Step 3: Identify the test statistic

Sample	Application	Level of scaling	Test/Comments
One Sample	Distributions	Non-metric	Kolmogorow-Smirnow and χ2 test for goodness of fit; Runs test for randomness; Binomial test for goodness of fit of dichotomous variables
	Means	Metric	t test, if variance is unknown z test, if variance is known
	Proportions	Metric	z test
Two Independent Samples	Distributions	Non-metric	Kolmogorow-Smirnow two-sample test for equality of two distributions
	Means	Metric	Two-group t test F test for equality of variances
	Proportions	Metric Non-metric	z test χ2 test
	Ranking/Medians	Non-metric	Mann-Whitney U test is more powerful than the median test
Paired Samples	Means	Metric	paired t test
	Proportions	Non-metric	McNemar test for binary variables, χ2 test
	Ranking/Medians	Non-metric	Wilcoxon matched-pairs ranked-signs test is more powerful than the sign test

χ2 (chi-square) statistic for goodness of fit is used to test the statistical significance of the observed association in a cross-tabulation

H0: There is no association between the variables

χ2 (chi-square) tests the equality of frequency distributions.
Which distributions/frequencies should we test?

fe – cell frequencies that would be expected if no association were present between the variables

fo – actual observed cell frequencies

χ2 statistic should be estimated only on counts of data. When the data are in percentage form, they should be first converted to absolute values or numbers.

Step 4: Formulate a decision rule

TScal – observed value of the test statistic.

TScr – critical value of the test statistic for a given significance level.

If probability of TScal < significance level (α), then reject H0.

If TScal > TScr , then reject H0.

Table of critical values of χ2 for different levels of significance α

Step 5: Arrive at a decision

Is the evidence there?

What are the consequences?

H0 of no association cannot be rejected
Association is not statistically significant at the .05 level
The findings from the sample cannot be generalized to population