HYPOTHESIS TESTINGNormal Distribution and Hypothesis Testing

Hypothesis Testing

• A hypothesis is a conjecture or assertion about a parameter

• Null v. Alternative hypothesis– Proof by contradiction – Null hypothesis is the hypothesis being tested– Alternative hypothesis is the operational

statement of the experiment that is believed to be true

Null Hypothesis

• The hypothesis stating that the manipulation has no effect and that there will be no difference between the two groups– H0: μ1 - μ2 = 0

– H0: μ1 = μ2

Alternative Hypothesis

• The hypothesis stating that the manipulation has an effect and that there will be difference between the two groups– HA: μ1 < μ2 (one-tailed)

– HA: μ1 > μ2 (one-tailed)

– HA: μ1 ≠ μ2 (two-tailed)

One-tailed test

• Alternative hypothesis specifies a one-directional difference for parameter– H0: μ = 10 v. Ha: μ < 10

– H0: μ = 10 v. Ha: μ > 10

– H0: μ1 - μ2 = 0 v. Ha: μ1 - μ2 > 0

– H0: μ1 - μ2 = 0 v. Ha: μ1 - μ2 < 0

Two-tailed test

• Alternative hypothesis does not specify a directional difference for the parameter of interest– H0: μ = 10 v. Ha: μ ≠ 10

– H0: μ1 - μ2 = 0 v. Ha: μ1 - μ2 ≠ 0

Example

Title: The NSAT Scores and Academic Achievement of the Students in Private School and Public Schools.

H0: There is no significant relationship between the NSAT performance and the academic achievement among the four learning areas of private schools, public schools and combination of private and public schools

Ha: There is a significant relationship between the NSAT performance and the academic achievement among the four learning areas of private schools, public schools and combination of private and public schools

Critical Region

• Also known as the “rejection region”• Critical region contains values of the test

statistic for which the null hypothesis will be rejected

• Acceptance and rejection regions are separated by the critical value, Z.

Type I error

• Error made by rejecting the null hypothesis when it is true.

• False positive• Denoted by the level of significance, α• Level of significance suggests the highest

probability of committing a type I error

Type II error

• Error made by not rejecting (accepting) the null hypothesis when it is false.

• False negative• Probability denoted by β

Decision H0 true H0 false

Reject H0Type I error

(α)Correct

decision (1-β)

Accept H0

Correct decision

(1-α)Type II error

(β)

Notes on errors

• Type I (α) and type II errors (β) are related. A decrease in the probability of one, increases the probability in the other.

• As α increases, the size of the critical region also increases

• Consequently, if H0 is rejected at a low α, H0 will also be rejected at a higher α.

critical value

test statistic

Reject H0

critical value

test statistic

Do not reject H0

Make a decision. Reject H0 if the value of the test statistic belongs to the critical region.

Collect the data and compute the value of the test statistic from the sample data

Select the appropriate test statistic and establish the critical region

Choose the level of significance, α

State the null hypothesis (H0) and the alternative hypothesis (Ha)

Independent-Groups and Correlated-Groups T Tests

Independent Group t Test Correlated Group t Test

What it is A parametric test for a two-group between-participants design

A parametric test for a two-group within-participants or matched participants design

What it does Compares performance of the two groups to determine whether they represent the same population or different populations

Analyzes whether each individual performed in a similar or different manner across conditions

Assumptions Interval-ratio dataBell-shaped distributionIndependent observationsHomogeneity of variance

Interval-ratio dataBell-shaped distributionDependent observationsHomogeneity of variance