TESTING OF HYPOTHESES

What is a Hypothesis?

A hypothesis is an assumption or a statement that may or may not be true.

The hypothesis is tested on the basis of information obtained from a sample.

Test hypotheses

Hypothesis = predictionInvolves association between variables

Children who watch violent television are more violent than children who do not watch violent television

The more spinach people eat, the longer they live

H0: No change in interest rate is warranted.

H1: Raise interest rate to control inflation.

Null and Alternative HypothesesConvert the research question to null and alternative

hypotheses The null hypothesis (H0) is a claim of “no difference

in the population” The alternative hypothesis (Ha) claims “H0 is false”

Collect data and seek evidence against H0 as a way of bolstering Ha (deduction)

Steps in Testing of Hypothesis Exercise

Make a prior expectation or judgment

Define an alternative supposition

Apply appropriate approach to test whether the prior expectation is statistically different from the alternative supposition or not.

13:0 XH

13:1 XH 13:1 XH 13:1 XH

Types of Errors…A Type I error occurs when we reject a true null hypothesis (i.e. Reject H0 when it is TRUE)

A Type II error occurs when we don’t reject a false null hypothesis (i.e. Do NOT reject H0 when it is FALSE)

11.7

H0 T F

Reject I

Reject II

Hypothesis Testing Procedures

HypothesisTesting

Procedures

NonparametricParametric

Z Test

Kruskal-WallisH-Test

WilcoxonRank Sum

Test

t Test One-WayANOVA

Many More Tests Exist!

Parametric Test Procedures

Enables us to determine if there is a significant difference between sample means with underlying assumptions of normality, homogeneity of variances

1. Involve Population Parameters (Mean)

2. Have Stringent Assumptions

(Normality)

Examples: Z Test, t Test, 2 Test,

F test

Nonparametric Test Procedures (also called “distribution free statistics”)

1. Do Not Involve Population ParametersExample: Probability Distributions,

Independence

2. Data Measured on Any Scale (Ratio or Interval, Ordinal or Nominal)

3. Example: Wilcoxon Rank Sum Mann-Whitney Test , Kruskal-Wallis Test

Parametric Test Nonparametric Test

One Sample One Sample t-test One sample sign test

Two independent Two-sample independent Wilcoxson rank-sum test Samples test Mann-Whitney U test

Two dependent Two-paired t-test Wilcoxson signed-rank test Samples Sign test

Correlation Pearson r Spearman (rho) rank-order correlation

Multiple Groups One-way ANOVA Kruskal-Wallis one-way One Factor ANOVA

Statistical Methods to test Hypothesis

Statistical procedures for continuous data:- t-test- Analysis of variance- Correlation- RegressionStatistical procedures for discrete data:- Chi-Square test- Mann-Whitney U test

T –Test: two groups

Paired t-test: This is a special t-test to use for data from paired samples e.g. if we collect data from the same individual before and after a particular treatment

Paired t-test: CalculationSubject Week 1 Week 2 Diff (D) Diff2 (D2)

1 10 122 50 523 20 254 8 105 115 1206 75 807 45 508 170 175

∑D = ∑D2 =Steps 1 & 2: Complete this table

Paired t-test: Calculation

Step 3:

Calculate the t statistic

t = n x ∑D2 – (∑D)2 = √ (n - 1)

∑D

GT = the sum of all observations Ti = the sum of the observations in the i-th ni = the number of observations in the i-th sampleN = total number of observationsΣ ------ = the sum of the square of each sample’s total divided by its sample sizeΣi Σj (x ij )2 = the sum of the squares of all observationsk = the number of groups

Analysis of Variance Anova- To compare three or more groups using F distribution

Correlation This method is used to examine a relationship between two (or more) quantitative variables, e.g. height and weight.

REGRESSION

To determine the strength of the relationship between one dependent variable (usually denoted by Y) and a series of other changing variables (known as independent variables). 

the formula for a straight line: y = a + bx

Y= the variable that we are trying to predictX= the variable that we are using to predict Y a= the intercept b= the slope

e) Chi-Square testChi square: determines relationship between variables. Chi-square test can handle any number of groups and any number of possible outcomes. The formula for calculating Chisquare a 2 x 2 situation (two groups and two possible outcomes) is as follows:

Mann-Whitney U TestActual measurements not used – ranks of the

measurements usedData can be ranked from highest to lowest or

lowest to highest valuesCalculate Mann-Whitney U statistic

U = n1n2 + n1(n1+1) – R1

2

Mann-Whitney U: CalculationStep 1:

Rank all the data from both groups in one series, then total each

Student

School A School B

StudentGrade GradeRank Rank

J. S.

L. D.

H. L. M. J.

T. J.

M. M

K. S.

P. S

B

c

A+

D

D

C+

C+

A

9

14

1

45

Median = ; Median = ;∑RA = ∑RB =

43

11

13

3