parametric tests, hypothesis testing, correlation and anova & z-test and chi-square test

PARAMETRIC TESTS

the tests applied to data that are normally distributed, the levels, of measurements which are expressed in the interval and ratio.

the distribution is normal

We use a parametric test when

The level of measurement to be analyzed is expressed in interval and ratio data.

We use a parametric test when

t-test for independent sample

t-test is a test of the difference between two independent groups where the means are compared (-x1 & -x2)

When we compare the means of two independent groups

When do we use the t-test for independent samples?

When the samples are less than 30

When do we use the t-test for independent samples?

formula for t-test for independent samples

T-tests for Correlated Samples (paired t-test)

another parametric test applied to one group of samples

When the samples are less than 30.

when do we use paired t-test / t-test for correlated samples?

T-tests for Correlated Samples (paired t-test)

can be used in the evaluation of certain program or treatment

When the mean before (pretest) and the mean after (post test) are being compared

When do we use the t-test for a correlated sample?

to find out if a difference exists between the before and

after means.

Why do we use the t-test for correlated samples?

an appropriate test for the evaluation of government programs. This is used in an experimental design to test the effectiveness of a certain technique or method or program that had been developed

Why do we use the t-test for correlated samples?

formula for t-test for correlated samples / paired t-test

Correlation

is a measure of relationship between two variables.

Coefficient of correlation

determines validity, reliability and objectivity of an examination prepared.

Coefficient of correlation

It also indicates the amount of agreement or disagreement between

groups of scores, measurements, or individuals.

from -1.00 through 0.00 up to +1.00

Correlation ranges in value

Positive Correlation

it indicates that as the value of x increases the value of y also increases. Likewise, if the value of x decreases, the value of y also decreases.

negative correlation

it indicates that for every increase of x, there’s a corresponding decrease of y. likewise, for every decrease of x, there’s a corresponding of y.

the relationship is inverse.

Pearson Product Moment Coefficient of Correlation

Pearson r

It is used to find the correlation between interval and ratio data.

What is the Pearson r?

used to determine the index of relationship between two variables, the independent and the dependent variables.

What is the Pearson r?

x

in pearson r, the independent variable can be represented by

y

in pearson r, the dependent variable can be represented by

-1, zero to +1

the value of r is

there exist a perfect correlation between x and y.

If the value r is +1 or -1,

then x and y are independent of each other

if r equals zero

We want to analyze if a relationship exists between two variables.

Why do we use Pearson r?

Why do we use Pearson r?

If there is a relationship that exist between x and y, then we can

determine the extent that of x influences y by means of the

coefficient of determination (COD) which is equal to the square

of r and multiplied by 100.

When do we use Pearson r?

used to determine the index of relationship between two variables, the independent and the dependent variables.

perfect positive correlation

if r = +1

negative perfect correlation

if the value of r = -1

then there is no correlation between two variables x and y.

if r = 0

formula for pearson r

verbal interpretation:

No Correlation

correlation interpretation:

range = 0

verbal interpretation:

Indifferent, almost negligible relationship

correlation interpretation:

range = ±0. 01 - ± 0.20

verbal interpretation:

Low correlation, slight relationship

correlation interpretation:

range = ± 0.21 - ± 0.40

verbal interpretation:

Moderate correlation, substantial or marked relationship

correlation interpretation:

range = ± 0.41 - ± 0.70

verbal interpretation:

High to very high correlation/relationship

correlation interpretation:

range = ± 0.71 - ± 0.99

Perfect Correlation

correlation interpretation:

range = ± 1

ANOVA or F-TEST

ANALYSIS OF VARIANCE

Another parametric test used to compare the mean of three or more groups of independent samples.

What is F-Test?

Sir Ronald Aylmer Fisher

analysis of variance (ANOVA) / f-test was developed by __________________

He was a British statistician, geneticist, and academic.

One-way analysis of variance, Two-way analysis of variance and Three-way analysis of variance

3 Kinds of ANOVA

When there is a normal distribution and when the level of measurements is expressed in interval and ratio data just like t-test and the z-test.

When do we use F-Test?

When do we use F-Test?

We want to find out if there is a significant difference between and among the means of the three or more independent groups.

used to determine if there are significant differences between the means of three or more independent (unrelated) groups

what is one-way anova?

formula for anova

anova summary table

between groups

df of anova is k - 1

within groups

df of anova is N - k

hypothesis testing

It first determines the probability that the pattern could have been produced by chance alone.

the pattern can reasonably be explained by chance

what happens if probability is large enough?

we can rule out chance as a plausible explanation to conclude that some meaningful, systematic force created the pattern

what happens if the probability is extremely small?

hypothesis testing

It deals with the problem of testing specific assertions about the population.

hypothesis testing

is a statistical procedure that allows researchers to use sample data to draw inferences about the population of interest.

tentative solution

it needs further testing and series of experimentation before it can be accepted as a solution to the problems a researcher is trying to arrive

1. Null Hypothesis (Ho)

2. Alternative Hypothesis (Ha)

what are the kinds of hypothesis

Null Hypothesis (Ho)

no difference relationship hypothesis.

Null Hypothesis (Ho)

implies neutrality and objectively which must be present in any research undertaking.

Null Hypothesis (Ho)

a statement that the value of a population parameter is equal to some claimed value.

Alternative Hypothesis (Ha)

stated opposite the very way how the null hypothesis

is stated

Alternative Hypothesis (Ha)

an alternative solution just in case one failed to find that the null hypothesis is the true to the problem a researcher is working.

TYPE I ERROR OR ALPHA ERROR

When the null hypothesis is rejected and the alternative

hypothesis is accepted.

TYPE II ERROR OR BETA ERROR

When the null hypothesis accepted and the alternative

hypothesis is rejected.

Level of Significance

The probability of rejecting the null hypothesis when it is true, (also known as a type 1 error).

Level of Significance

This is decided by the individual but is normally set at 5% (0.05) which means that there is a 1 in 20 chance of rejecting the null hypothesis when it is true.

there’s a 1 in 20 chance of rejecting the null hypothesis when it’s true

what does a level of significance at 5% or 0.05 mean?

1. state the hypothesis about a population

2. use hypothesis to predict characteristics of sample

3. obtain a random sample

4. compare obtained sample data with prediction that’s made from the hypothesis

what is the hypothesis-testing procedure?

then we conclude that the hypothesis is reasonable.

if the sample mean is consistent with the prediction

we decide that the hypothesis is wrong.

if there is a big discrepancy between the data and the prediction

Independent t-test

Compare two independent groups for parametric test

Mann–Whitney U Test

Compare two independent groups for nonparametric equivalent

Paired t-test

Compare two related groups for parametric test

Wilcoxon Signed-Rank Test

Compare two related groups for non-parametric equivalent

One-way ANOVA

Compare 3+ independent groups (1 factor) for parametric test

Kruskal–Wallis Test

Compare 3+ independent groups (1 factor) for non-parametric test

Repeated Measures ANOVA

Compare 3+ related groups for parametric test

Friedman Test

Compare 3+ related groups for non-parametric equivalent

Two-Way ANOVA

Compare 3+ groups with 2 factors for parametric test

No exact direct equivalent; sometimes use Scheirer–Ray–Hare test

Compare 3+ groups with 2 factors for non-parametric equivalent

Pearson r

Measure relationship (correlation) for parametric test

Spearman rho

Measure relationship (correlation) for non-parametric test

Linear Regression

Prediction / effect of variables for parametric test

Rank-based / nonparametric regression

Prediction / effect of variables of nonparametric equivalent

— (Not parametric)

Test association between categorical variables for parametric test

Chi-Square Test of Independence

Test association between categorical variables for non-parametric test

parametric test

a kind of hypothesis test which gives generalizations for generating records regarding the mean of the primary/original population.

non-parametric test

does not require any population distribution, which is meant by distinct parameters.

Compare exam scores of male vs female students

(EXAMPLE)

Compare two independent groups - Independent t-test - Mann–Whitney U Test

Scores before and after a tutorial

(EXAMPLE)

Compare two related groups - Paired t-test - Wilcoxon Signed-Rank

Test

Scores of Sections A, B, C

(EXAMPLE)

Compare 3+ independent groups (1 factor) - One-way ANOVA - Kruskal–Wallis Test

Same students’ scores across 3 exams

(EXAMPLE)

Compare 3+ related groups - Repeated Measures ANOVA - Friedman Test

Effect of teaching method (online vs face-to-face) and gender on

scores

(EXAMPLE)

Compare 3+ groups with 2 factors - Two-Way ANOVA - (No exact direct

equivalent; sometimes use Scheirer–Ray–Hare test)

Study time vs exam scores

(EXAMPLE)

Measure relationship (correlation) - Pearson r - Spearman rho

Predict score based on study hours

(EXAMPLE)

Prediction / effect of variables - Linear Regression - Rank-based /nonparametric regression

Relationship between gender and strand chosen

(EXAMPLE)

Test association between categorical variables - — (Not

parametric) - Chi-Square Test of

Independence

It is used to compare perceived population mean against the sample mean.

what is z-test one sample group?

the samples are equal or more than 30

you use z-test when