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Hypothesis Testing

Subject: Mathematics

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29 March 2018

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Excellent resource - really clearly written. Thank you!

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Excellent and well written PPT to introduce the Hypothesis testing. Thank you ever so much for sharing

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Unit 12: Significance tests (hypothesis testing)

About this unit.

Significance tests give us a formal process for using sample data to evaluate the likelihood of some claim about a population value. Learn how to conduct significance tests and calculate p-values to see how likely a sample result is to occur by random chance. You'll also see how we use p-values to make conclusions about hypotheses.

The idea of significance tests

Simple hypothesis testing (Opens a modal)
Idea behind hypothesis testing (Opens a modal)
Examples of null and alternative hypotheses (Opens a modal)
P-values and significance tests (Opens a modal)
Comparing P-values to different significance levels (Opens a modal)
Estimating a P-value from a simulation (Opens a modal)
Using P-values to make conclusions (Opens a modal)
Simple hypothesis testing Get 3 of 4 questions to level up!
Writing null and alternative hypotheses Get 3 of 4 questions to level up!
Estimating P-values from simulations Get 3 of 4 questions to level up!

Error probabilities and power

Introduction to Type I and Type II errors (Opens a modal)
Type 1 errors (Opens a modal)
Examples identifying Type I and Type II errors (Opens a modal)
Introduction to power in significance tests (Opens a modal)
Examples thinking about power in significance tests (Opens a modal)
Consequences of errors and significance (Opens a modal)
Type I vs Type II error Get 3 of 4 questions to level up!
Error probabilities and power Get 3 of 4 questions to level up!

Tests about a population proportion

Constructing hypotheses for a significance test about a proportion (Opens a modal)
Conditions for a z test about a proportion (Opens a modal)
Reference: Conditions for inference on a proportion (Opens a modal)
Calculating a z statistic in a test about a proportion (Opens a modal)
Calculating a P-value given a z statistic (Opens a modal)
Making conclusions in a test about a proportion (Opens a modal)
Writing hypotheses for a test about a proportion Get 3 of 4 questions to level up!
Conditions for a z test about a proportion Get 3 of 4 questions to level up!
Calculating the test statistic in a z test for a proportion Get 3 of 4 questions to level up!
Calculating the P-value in a z test for a proportion Get 3 of 4 questions to level up!
Making conclusions in a z test for a proportion Get 3 of 4 questions to level up!

Tests about a population mean

Writing hypotheses for a significance test about a mean (Opens a modal)
Conditions for a t test about a mean (Opens a modal)
Reference: Conditions for inference on a mean (Opens a modal)
When to use z or t statistics in significance tests (Opens a modal)
Example calculating t statistic for a test about a mean (Opens a modal)
Using TI calculator for P-value from t statistic (Opens a modal)
Using a table to estimate P-value from t statistic (Opens a modal)
Comparing P-value from t statistic to significance level (Opens a modal)
Free response example: Significance test for a mean (Opens a modal)
Writing hypotheses for a test about a mean Get 3 of 4 questions to level up!
Conditions for a t test about a mean Get 3 of 4 questions to level up!
Calculating the test statistic in a t test for a mean Get 3 of 4 questions to level up!
Calculating the P-value in a t test for a mean Get 3 of 4 questions to level up!
Making conclusions in a t test for a mean Get 3 of 4 questions to level up!

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Introduction to Hypothesis Testing

Published by Emily Payne Modified over 10 years ago

Presentation on theme: "Introduction to Hypothesis Testing"— Presentation transcript:

Introductory Mathematics & Statistics for Business

Hypothesis Testing W&W, Chapter 9.

HYPOTHESIS TESTING. Purpose The purpose of hypothesis testing is to help the researcher or administrator in reaching a decision concerning a population.

Chapter 7 Hypothesis Testing

Our goal is to assess the evidence provided by the data in favor of some claim about the population. Section 6.2Tests of Significance.

Anthony Greene1 Simple Hypothesis Testing Detecting Statistical Differences In The Simplest Case:  and  are both known I The Logic of Hypothesis Testing:

Hypothesis Testing A hypothesis is a claim or statement about a property of a population (in our case, about the mean or a proportion of the population)

Statistical Issues in Research Planning and Evaluation

COURSE: JUST 3900 INTRODUCTORY STATISTICS FOR CRIMINAL JUSTICE Instructor: Dr. John J. Kerbs, Associate Professor Joint Ph.D. in Social Work and Sociology.

Hypothesis testing Week 10 Lecture 2.

HYPOTHESIS TESTING Four Steps Statistical Significance Outcomes Sampling Distributions.

Behavioural Science II Week 1, Semester 2, 2002

Evaluating Hypotheses Chapter 9. Descriptive vs. Inferential Statistics n Descriptive l quantitative descriptions of characteristics.

Cal State Northridge  320 Ainsworth Sampling Distributions and Hypothesis Testing.

Evaluating Hypotheses Chapter 9 Homework: 1-9. Descriptive vs. Inferential Statistics n Descriptive l quantitative descriptions of characteristics ~

Introduction to Hypothesis Testing CJ 526 Statistical Analysis in Criminal Justice.

PY 427 Statistics 1Fall 2006 Kin Ching Kong, Ph.D Lecture 6 Chicago School of Professional Psychology.

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Statistical Hypothesis Tests

Aug 01, 2014

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Statistical Hypothesis Tests. Notes of STAT6205 by Dr. Fan. Overview. Introduction of hypotheses tests ( Sections 7.1,7.2 ) General logic Two types of error Parametric tests for one mean and for proportions What is the best test for a given situation? Order Statistics (Section 8.3)

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Statistical Hypothesis Tests Notes of STAT6205 by Dr. Fan 6205

Overview • Introduction of hypotheses tests (Sections 7.1,7.2) • General logic • Two types of error • Parametric tests for one mean and for proportions • What is the best test for a given situation? • Order Statistics (Section 8.3) • Wilcoxon tests (Section 8.5) 6205

Statistical Hypotheses • A statistical hypothesis is an assumption or statement concerning one or more population parameters. • Simple vs. composite hypotheses E.g. A pharmaceutical company wants to be able to claim that for its newest medication the proportion of patients who experience side effects is less than 20%. Q. What are the two possible conclusions (hypotheses) here? 6205

Hypothesis Tests • A statistical test is to check a statistical hypothesis using data. It involves the five steps: • Set up the null (Ho) and alternative (H1) hypotheses • Find an appropriate test statistic (T.S.) • Find the rejection (critical) region (R.R.) • Reject Ho if the observed test statistic falls into R.R. and not reject Ho otherwise • Report the result in the context of the situation 6205

Determine Ho and H1 • The null hypothesis Ho is the no-change hypothesis • The alternative hypothesis H1 says that Ho is false The Logic of Hypothesis Tests: “Assume Ho is a possible truth until proven false” Analogical to “Presumed innocent until proven guilty” The logic of the US judicial system Q: What are the two possible conclusions? 6205

Determine Ho and H1 Golden Rule: Ho must be a simple hypothesis. Practical Rule: If possible, the hypothesis we hope to prove (called research hypothesis) goes to H1. Back to the drug example, setting Ho and H1. 6205

Types of Errors H0 true H0 false Type II Error, or “ Error” Good! (Correct!) we accept H0 Type I Error, or “ Error” Good! (Correct) we reject H0

More Terms • a= Significance level of a test = Type I error rate • Power of a test = 1-Type II error rate=1- b • We only control a not b,  so we don’t say “accept Ho”. 6205

Report the Conclusion • Reject Ho: the data shows strong evidence supporting Ha Eg. The data shows strong evidence that the proportion of users who will experience side effects is less than 20% at significant level of 10%. • Fail to reject Ho: the data does not provide sufficient evidence supporting Ha Eg. Based on the data, there is not sufficient evidence to support the proportion is less than 20% at significant level of 5%.

Tests for One Mean 6205

Z Test For normal populations or large samples (n > 30) And the computed value of Z is denoted by Z*. 6205

Types of Tests 6205

Example 1 (Conti.) Conduct a test for Ho: mu=2500 vs. H1: mu =3000 at 5% significant level. • What is the R.R.? • What is the power of the test? Z test is the most powerful test! 6205

P-Values • The p-value is the smallest level of significance to reject Ho at the observed value, also called the observed significance level. p-value > a fail to reject Ho p-value <a reject Ho (= accept Ha) • That is, p-value is the probability of seeing as extreme as (or more extreme) what we observe, given Ho is true.

P-Value • The level of significance (called a level) is usually 0.05 • p-value > a fail to reject Ho (??) • p-value <a reject Ho (= accept Ha)

Computing the p-Value for the Z-Test

Computing the p-Value for the Z-Test P-value = P(|Z| > |z*| )= 2 x P(Z > |z*|)

t Test • For normal populations with unknown s Eg. Revisit Example 1

One Population

Testing Hypotheses about a Proportion • Three possible Ho and Ha Write them all as p=po in the future

The z-test for a Proportion • When 1) the sample is a random sample 2) n(po) and n(1-po) are both at least 10, an appropriate test statistic for p is

Example: New Drug (Conti.) • Ho: p > 20% vs. Ha: p < 20% • Z-test statistic; a = 0.05 • Find rejection region or p-value • Decide if reject Ho or not • Report the conclusion in the context of the situation

Hypothesis Test for the Difference of Two Population Proportions • Step 1. Set up hypotheses Ho: p1 = p2 and three possible Ha’s: Ha: p1 = p2 (two-tailed) or Ha: p1 < p2 (lower-tailed) or Ha: p1 > p2 (upper-tailed)

Hypothesis Test for the Difference between Two Population Proportions • Step 2. calculate test statistic where

Hypothesis Test for the Difference between Two Population Proportions • Step 3: Find p value • Must be two independent random samples; both are large samples: And • When the above conditions are met, use Z-Table to find p-value. • Steps 4 and 5 are the same as before

Example: Bike to School For 80 randomly selected men, 30 regularly bicycled to campus; while for 100 randomly selected women, 20 regularly bicycled to campus. • Find the p-value for testing: Ho: p1 = p2 vs. Ha: p1 > p2 Answer: z=2.60, p=0.0047 1: men; 2: women

Order Statistics • Min & Max • Joint and other orders 6205

Order Statistics Problem 1: Suppose X1, X2, …, X5 are a random sample from U[0,1]. Find the pdf of X(2). Problem 2: Suppose X1, X2, …, Xnare a random sample from U[0,1]. Show that X(k)~ beta(k,n-k+1). 6205

Order Statistics The CDF of X(k)and example 6205

Wilcoxon Tests Ho: median of X = median of Y vs. H1: Ho is false Wilcoxon tests (p. 448 - 450) • assume the two distributions are of similar shapes but do not need to be normal • See the supplementary material 6205

Exercise 8.5-9 X = the life time of light bulb of brand A Y = the life time of light bulb of brand B Data: (in 100 hours) X: 5.6 4.6 6.8 4.9 6.1 5.3 4.5 5.8 5.4 4.7 Y: 7.2 8.1 5.1 7.3 6.9 7.8 5.9 6.7 6.5 7.1 • Conduct the Wilcoxon test at 5 % level to test if brand B has longer life time in general. A: W(Y)=145 > 128 or Z= 3.024 > 1.645; reject Ho • Construct and interpret a Q-Q plot of these data. 6205

R Code for Q-Q Plot > x<-c(5.6, 4.6, 6.8, 4.9, 6.1, 5.3, 4.5, 5.8, 5.4, 4.7) > y<-c(7.2, 8.1, 5.1, 7.3, 6.9, 7.8, 5.9, 6.7, 6.5, 7.1) > qqplot(x,y,xlab="life time of brand A", ylab="life time of brand B", main="qqplot of Life time of Brand A vs. Brand B") 6205

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