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Introduction to Hypothesis Testing
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Hypothesis Testing
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Inferential Statistics: Hypothesis Testing
Jul 17, 2014
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Inferential Statistics: Hypothesis Testing. Testing Population Variances Analysis of Variance – ANOVA. Content. Estimation Estimate population means Estimate population proportion Estimate population variance Hypothesis testing Testing population means
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Inferential Statistics:Hypothesis Testing Testing Population Variances Analysis of Variance – ANOVA
Content • Estimation • Estimate population means • Estimate population proportion • Estimate population variance • Hypothesis testing • Testing population means • Testing categorical data / proportion • Testing population variances • Hypothesis about many population means
Testing Population Variances • Single population variance • Chi-square test • Two populations variance • F-test • Analysis of Variance – test many population means • One-way ANOVA • Two-way ANOVA
Single Population Variance • Assumption • Normal distribution of population • Test the population variance σ2 from a sample against a specified population variance σ02. • Hypothesis
Single Population Variance Test statistic – chi-square Critical Region
Example 1 A fluorescent lamp factor knows that the lifespan of the lamps is normally distributed with variance of 10,000 hr2. In an inspection, 20 sample lamps are tested and it is found that the variance is 12,000 hr2. Can a conclusion be drawn that the variance of lamp’s lifespan has changed at significant level 0.05? Hypothesis H0: σ2 = 10,000 H1: σ2 ≠ 10,000 α = 0.05
Example 1 Calculate test statistic Degree of freedom = 20 – 1 = 19 χ2(1-0.025),19 = 8.91 , χ2(0.025),19 = 32.85 The calculated chi-square: 8.91 < 22.8 < 32.85, not falling in two-tailed critical region. Accept H0 and reject H1 The variance of lamp’s lifespan has not changed at significant level 0.05
Example 2 A company claims that the standard deviation of its thermometer does not exceed 0.5 oC. To verify this, 16 thermometers are sampled. It is found that the standard deviation is 0.7 oC. Is the claim true at significant level 0.01? Hypothesis H0: σ2≤ 0.25 H1: σ2 > 0.25 α = 0.01
Example 2 Calculate test statistic Degree of freedom = 16 – 1 = 15 χ2(0.01),15 = 30.58 The calculated chi-square: 29.4 < 30.58, not falling in two-tailed critical region. Accept H0 and reject H1 The claim is true at significant level 0.01
Two Populations Variances • Campare between two variances • Sir Ronald Fisher found that • Given s12 and s22 are variances of the first and second sample groups of size n1 and n2 respectively, • Both sample groups are randomly selected from normally distributed populations, • The skewness of graph changes corresponding to the degree of freedoms of samples, which are n1-1 and n2-1. • This distribution has become known as Fisher Distribution or F-Distribution
http://en.wikipedia.org/wiki/F-distribution
Application and Limitation F-test is used in model fit such as ANOVA and Linear Regression Analysis in order to determine error (i.e. variance) from the model. Applicable to 2 populations only If populations are not normally distributed, the test will be inaccurate.
Two Populations Variances df1 = n1-1 และ df2 = n2-1
Two Populations Variances Alternate form
Example 1 The sale count of bicycles in one week of two retailers are as follows Retailer A: 65 46 57 43 58 Retailer B: 52 41 43 47 32 49 57 If the sale count in one week is normally distributed, test if the variances of the sale count the two retailers are different at significant 0.02. Hypothesis α = 0.02
Example 1 • Calculate variances • SA2 = 82.7, SB2 = 66.143 • Calculate test statistic • From H0, σ12 =σ22 • Degree of freedom • df1 = 5-1 = 4, df2 = 7-1 = 6
Example 1 Critical region Accept H0 and reject H1 The variances of the sale count the two retailers are not different at significant 0.02
Example 2 From a previous study, the variance of time required for female workers to assemble a product is less than that of male workers. To re-verify the study, 11 male workers and 14 female workers are sampled. The observed standard deviations of the assembling time are 6.1 for male and 5.3 for female. Assuming normal distribution of assembling time, test if the result of the previous study is accurate at significant level 0.01. Hypothesis H0: σm2 ≤ σf2 H1: σm2 > σf2 α = 0.01
Example 2 • Calculate test statistic • Degree of freedom • df1 = 11-1 = 10, df2 = 14-1 = 13 • Critical region F0.01(10,13) = 4.10 • Calculated F is 1.325 < 4.10 • Accept H0 and reject H1 • The variance of time required for female workers to assemble a product is not less than that of male workers at significant level 0.01
Analysis of Variance (ANOVA) • Test if any of multiple means are different from each other • One-way ANOVA: 1 variables – 3 or more groups • Dependent variable is assumed is of interval or ratio scale • Also used with ordinal scale data • Can describe the effect of independent variable on dependent variable • Two-way ANOVA: two independent, one dependent variables • MANOVA: Two or more dependent variables • Can describe interaction between two independent variables
One-way ANOVA • Test the means (of dependent variable) between groups as specified by an independent variable that are organized in 3 or more groups (dichotomous) • Occupation: Student, Lecturer, Doctor (1 var - 3 groups) • Salary: dependent variable • Assumptions • Dependent variable is either an interval or ratio (continuous) • Dependent variable is approximately normally distributed for each category of the independent variable • There is equality of variances between the independent groups (homogeneity of variances). • Independence of cases.
One-way ANOVA Concept • Total Variance = Between-Group Variance + Within-Group Variance • Between-Group Variance • Describe the difference of means between groups, which is the effect on variable of interest • Within-Group Variance • Describe the difference of means within each group, which is the effect caused by other factors, called Error H0 : μ1 = μ2 = μ3 = … = μn H1 : μ1 ≠ μ2 ≠ μ3 ≠ … ≠ μn (at least one different pair)
One-way ANOVA Table • k: number of groups n: number of samples SST = SSB + SSW
One-way ANOVA Table Tj: sum of frequencies in each group T: sum of all frequencies nj: frequency in each group k: number of group xij: the ith data (row) of jth group (column) : the mean of group j : overall mean
Example 1 The survey result of the attitude of the executives in small, medium, and large companies toward management administration is shown in the table. Test if the attitudes of the executives from different company sizes are different at significant level 0.05.
Example 1 Hypothesis Ho : 1 = 2 = 3 H1 : 1 ≠ 2 ≠ 3 α = 0.05 Calculate test statistic = (7-6)2 + (7-6)2 + (5-6)2 + (4-6)2 + (7-6)2 + (4-6)2 + (4-6)2 + (2-6)2 + (2-6)2 + (3-6)2 + (10-6)2 + (10-6)2 + (9-6)2 + (6-6)2 + (10-6)2 = 114
Example 1 = 24 / (15-3) = 2 = 45 /2 = 22.5 = 5(6-6)2 + 5(3-6)2 + 5(9-6)2 = 0+45+45 = 90 SSW = SST – SSB = 114 - 90 = 24 = 90 / (3-1) = 45
Example 1 Degree of freedom dfB=3-1=2, dfW=15-3=12 F0.05(2,12)= 3.89 The calculated F is 22.5 > 3.89 Reject H0 and accept H1 The attitudes of the executives from different company sizes are different at significant level 0.05
Post-hoc Test One-way ANOVA does not tell which pairs have different means Post-hoc test (or Post-hoc Analysis or Multiple Comparison) is used to identify the different pairs. *No need if ANOVA accepts H0 (means are not different)
Post-hoc Test Methods requiring equality of variances 1. Least-Significant Different(LSD) 2. Waller – Duncan 3. S-N-K(Student-Newman-Keuls) 4.Dunnett’s C 5. Bonferroni6. Sidak7. Scheffe8. R-E-G-WF 9.Tukey’s HSD 10.R-E-G-WQ11. Tukey’s–b 12.Duncan 13.Hochberg’s GT2 14.Gabriel Methods not requiring equality of variances 1. Tamhane’s T22.Dunnett’s T3 3.Games-Howell4.Dunnett’s C
Least-Significant Different(LSD) • Fisher’s Least-Significant Difference proposed by R.A. Fisher to compare multiple pairs at the same time • Calculate LSD • If n1 = n2 then • Compare to LSD value • If > LSD then the means of the pair are different i ≠ j • Otherwise, the means are not different i = j
From Example 1 n are equal in the 3 groups =0.05, n – k = 15 – 3 = 12, t(0.025, 12) = 2.18 Comparison At significant level 0.05, the attitude of the executives from small companies is higher than that of the medium ones. And the attitude of the executives from large companies is higher than that of the medium and small ones.
Tukey’s Honesty Significant Difference (HSD) • Require the sample sizes to be the same • k = number of groups, dfw = n-k, q is obtained from q-table • Compare to HSD value • If > HSD then the means of the pair are different i ≠ j • Otherwise, the means are not different i = j
From Example 1 n are equal in the 3 groups so HSD is applicable =0.05, k = 3, n – k = 15 – 3 = 12, q0.05, 3, 12 = 3.77 Comparison At significant level 0.05, the attitude of the executives from small companies is higher than that of the medium ones. And the attitude of the executives from large companies is higher than that of the medium and small ones.
Scheffe • Scheffe or S-Method is applicable to different sample sizes • k = number of groups, dfB= k-1, dfw = n-k, MSW from ANOVA • Compare to S value • If > S then the means of the pair are different i ≠ j • Otherwise, the means are not different i = j
From Example 1 =0.05, k=3, dfb=2, dfW=12, F0.05(2,12) = 3.88
From Example 1 Comparison At significant level 0.05, the attitude of the executives from small companies is higher than that of the medium ones. And the attitude of the executives from large companies is higher than that of the medium and small ones.
Example 2 Four teaching methods are applied to 4 groups of students. Based on the exam scores in the table, test if the four methods give different results at significant level 0.01
Example 2 Hypothesis H0 : 1 = 2 = 3 = 4 H1 : 1 ≠ 2 ≠ 3 ≠ 4 α = 0.01 SSB
Example 2 SSW of each group
Example 2 F0.01(3,22) = 4.82 The calculated F is 7.01 > 4.82 Reject H0 and accept H1 The four teaching methods give different results at significant level 0.01
Two-way ANOVA • Use to determine the effect of 2 factors /treatments (independent variables) on one dependent variable • Occupation: Student, Lecturer, Doctor • Age: less than 20, 20-30, 31-40, 41 or older • Salary: dependent variable • Assumptions • Dependent variable is either interval or ratio (continuous) • The dependent variable is approximately normally distributed for each combination of levels of the two independent variables • Homogeneity of variances of the groups formed by the different combinations of levels of the two independent variables. • Independence of cases
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5. Statistical Hypothesis • Definition: A statistical hypothesis is an assertion or conjecture concerning one or more populations. An assumption or statement, which may or may not be true concerning one or more population. • Two types of Statistical Hypothesis: a) The NULL HYPOTHESIS, Ho b) The ALTERNATIVE HYPOTHESIS, H1 a) Nondirectional Hypothesis- Asserts that one value is different ...
This document provides an overview of hypothesis testing including: - Defining null and alternative hypotheses - Types of errors like Type I and Type II - Test statistics and significance levels for comparing means, proportions, and standard deviations of one and two populations - Examples are given for hypothesis tests on population means, proportions, and comparing two population means.
Enter statistics. Hypothesis testing formalizes our intuition on this question. It quantifies: in what % of parallel worlds would the results have come out this way? This is what we call a p-value. p<.05 intuitively means "a result like this is likely to have come up in at least 95% of parallel worlds" (parallel world = sample)
14. Step 5: Critical value Demystifying statistics! - Lecture 4 SBCM, Joint Program - RiyadhSBCM, Joint Program - Riyadh • Critical value - a line on a graph that splits the graph into sections. One or two of the sections is the "rejection region"; if your test value falls into that region, then you reject the null hypothesis • These values are obtained from statistical tables ...
23.1 How Hypothesis Tests Are Reported in the News 1. Determine the null hypothesis and the alternative hypothesis. 2. Collect and summarize the data into a test statistic. 3. Use the test statistic to determine the p-value. 4. The result is statistically significant if the p-value is less than or equal to the level of significance.
7-1 Basics of Hypothesis Testing. Hypothesis in statistics, is a statement regarding a characteristic of one or more populations Definition. Statement is made about the population Evidence in collected to test the statement Data is analyzed to assess the plausibility of the statement Steps in Hypothesis Testing.
Review: steps in hypothesis testing about the mean 1.Hypothesis a value ( 0) and set up H 0 and H 1 2.Take a random sample of size n and calculate summary statistics (e.g., sample mean and sample variance) 3.Determine whether it is likely or unlikely that the sample, or one even more extreme, came from a population with mean
Step 2: Collect data. For a statistical test to be valid, it is important to perform sampling and collect data in a way that is designed to test your hypothesis. If your data are not representative, then you cannot make statistical inferences about the population you are interested in. Hypothesis testing example.
This document provides an overview of statistical inference and hypothesis testing. It discusses key concepts like point estimation, confidence intervals, sample means, and hypothesis testing errors. Specific hypothesis tests covered include tests for a mean (using z-tests and t-tests) and tests for a proportion. Examples are provided to illustrate hypothesis testing procedures, type I and ...
1 Introduction to Hypothesis Testing. 2 What is a Hypothesis Test? A hypothesis test is a statistical method that uses sample data to evaluate a hypothesis about a population. 3 Falsifiability A good hypothesis is one that is falsifiable. You cannot prove something that cannot be disproved Better yet, you cannot support a hypothesis if you ...
Hypothesis Testing. •The intent of hypothesis testing is formally examine two opposing conjectures (hypotheses), H. 0. and H. A. •These two hypotheses are mutually exclusive and exhaustive so that one is true to the exclusion of the other •We accumulate evidence - collect and analyze sample information - for the purpose of determining ...
Unit test. 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.
3. Testing of HypothesisTesting of Hypothesis A hypothesis is an assumption about the population parameter (say population mean) which is to be tested. For that we collect sample data , then we calculate sample statistics (say sample mean) and then use this information to judge/decide whether hypothesized value of population parameter is correct or not.
Chapter 8 Introduction to Hypothesis Testing. PowerPoint Lecture Slides Essentials of Statistics for the Behavioral Sciences Seventh Edition by Frederick J. Gravetter and Larry B. Wallnau. Chapter 8 Learning Outcomes. Concepts to review. z- Scores (Chapter 5)
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.
AP Statistics PowerPoints. Unit 1: Chapter 1 Notes Edition 5. Chapter 2 Notes Edition 5. Unit 2: Chapter 3 PowerPoint 2013-2014, ... Hypothesis Testing: Section 10.2, Section 12.1, Section 11.1, Section 10.4. Hypothesis Testing pdf: ...
Review: statistics • The language of statistics -Describes a universe where we sample datasets from a population • Interesting properties are proved for sampling distributions of parameter estimates • Statistical hypothesis testing -Helps us decide if a sample belongs to a population • A priori calculation of important statistical
15. Chap 9-15 6 Steps in Hypothesis Testing 1. State the null hypothesis, H0 and the alternative hypothesis, H1 2. Choose the level of significance, , and the sample size, n 3. Determine the appropriate test statistic (two-tail, one-tail, and Z or t distribution) and sampling distribution 4.
Hypothesis Testing. Subject: Mathematics. Age range: 16+. Resource type: Other. File previews. pptx, 616.63 KB. A powerpoint giving an introduction to hypothesis testing using the normal distribution. It also covers type I and type II errors, and has some examples of hypothesis testing using the Poisson and Binomial distributions, plus various ...
Presentation Transcript. Hypothesis Testing: Inferential statistics These will help us to decide if we should: 1) believe that the relationship we found in our sample data is the same as the relationship we would find if we tested the entire population OR 2) believe that the relationship we found in our sample data is a coincidence produced by ...
Hypothesis testing ppt final. Mar 4, 2013 • Download as PPTX, PDF •. 192 likes • 144,438 views. P. piyushdhaker. Business. 1 of 16. Download now. Hypothesis testing ppt final - Download as a PDF or view online for free.
•The objective of Hypothesis Testing to verify the Null Hypotheses, not prove it. •Chi-square is one statistic use to find the P value (Hypothesis Testing). •If P Value is less than level of significance, then reject the null hypothesis. •The other most popular test statistics are z2 test and t test. 15
Hypothesis Testing in the Real World. Hypothesis Testing in the Real World. Thus far, Hypothesis testing has been presented for situations in which the population variances are known. T-tests allow you to compare means for which the population variances are unknown T-test for a single sample. Hypothesis Testing in the Real World. 2k views ...