# Hypothesis Testing Judicial Analogy Hypothesis Testing Hypothesis testing  Null hypothesis Purpose  Test the viability Null hypothesis  Population

• View
256

3

Tags:

Embed Size (px)

### Text of Hypothesis Testing Judicial Analogy Hypothesis Testing Hypothesis testing  Null hypothesis...

• Hypothesis Testing

• Judicial Analogy

• Hypothesis TestingHypothesis testing Null hypothesisPurpose Test the viabilityNull hypothesis Population parameter Reverse of what the experimenter believes

• Hypothesis Testing1. State the null hypothesis, H02. State the alternative hypothesis, HA3. Choose , our significance level4. Select a statistical test, and find the observed test statistic5. Find the critical value of the test statistic ( and p value) 6. Compare the observed test statistic with the critical value, (or compare the p value with ), and decide to accept or reject H0

• Coin Example

• Coin Analogy

• Types of ErrorsYou used a decision rule to make a decision, but was the decision correct?

ACTUAL

DECISION

Fair Coin

Not Fair Coin

Fair Coin

correct

Type II error

Not Fair Coin

Type I error

correct

• Modified Coin ExperimentWhich coins are fair?

• Cases in Hypothesis Testing Means - variance known - variance unknown

Comparison of means - unpaired, variance known - comparison of variances - unpaired, variances unknown but equal - unpaired, variances unknown and unequal - paired

Proportion

Comparison of Proportion

• One sample t-test

• Statistical Hypothesis Test

• Two-Sided Test of HypothesisThe test of hypothesis is two-sided if the null is rejected when the actual value of interest is either less than or greater than the hypothesized value.H0: 15.00HA: 15.00

• Two-Sided Test of Hypothesis

• One-Sided Test of HypothesisIn many situations, you are only interested in one direction. Perhaps you only want evidence that the mean is significantly lower than fifteen.For example, instead of testingH0: = 15 versus H1: 15you testH0: 15 versus H1: < 15

• One-Sided Test of Hypothesis

• The Critical Values of Z to memorizeTwo tailed hypothesisReject the null (H0) if z z/2, or z - z/2One tailed hypothesisIf HA is > Xbar, then reject H0 if z zIf HA is < Xbar, then reject H0 if z - z

• The Z-test an exampleSuppose that you took a sample of 25 people off the street in Morgantown and found that their personal income is \$24,379And you have information that the national average for personal income per capita is \$31,632 in 2003.Is the Morgantown significantly different from the National Average

Sources: (1) Economagic(2) US Bureau of Economic Analysis

• What to conclude?Should you conclude that West Virginia is lower than the national average? Is it significantly lower?Could it simple be a randomly bad sampleAssume that it is not a poor sampling techniqueHow do you decide?

• Example (cont.)We will hypothesize that WV income is lower than the national average.H0: WVInc = USInc (Null Hypothesis)HA: WVInc < USInc (Alternate Hypothesis)

Statistician can write by :H0: \$31,632HA: < \$31,632

Since we know the national average (\$31,632) and standard deviation (15000), we can use the z-test to make decide if WV is indeed significantly lower than the nation

• Example (cont.)Using the z-test, we get

For = 5% -z = -z0.05 = -1.645

THE DECISION IS REJECT H0 SO West Virginia is lower than the national average

• The t testWhen we cannot use the population standard deviation, we must employ a different statistical testThink of it this way:The sample standard deviation is biased a little low, but we know that as the sample size gets larger, it becomes closer to the true value.As a result, we need a sampling distribution that makes small sample estimates conservative, but gets closer to the normal distribution as the sample size gets larger, and the sample standard deviation more closely resembles the population standard deviation.

• The t-test (cont.)The t-test is a very similar formula.

Note the two differencesusing s instead of The resultant is a value that has a t-distribution instead of a standard normal one.

• The Critical Values of tTwo tailed hypothesisReject the null (H0) if t t/2(n-1), or t - t/2(n-1) Reject H0 if |t| t/2(n-1)One tailed hypothesisIf HA is > Xbar, then reject H0 if t t(n-1)If HA is < Xbar, then reject H0 if t - t(n-1)

Reject H0 if |t| t(n-1)

• T-test exampleSuppose we decided to look at Oregon, but do not know the population standard deviationAnd we have a small sample anyway (N=25).

without an a priori reason to hypothesize higher or lower, use the 2-tailed testAssume Oregon has a mean of 29,340, and that we collected a sample of 169.Using the t-test, we get

Critical value = t.025(168) = 1.96 Since |-1.9684| > 1.96REJECT H0

• Two sample t-test Two-Sample t-Tests

• Cereal Example

• Other ExamplesIs the income of blacks lower than whites?Are teachers salaries in West Virginia and Mississippi alike?Is there any difference between the background well and the monitoring well of a landfill?

• The Difference of means Test Frequently we wish to ask questions that compare two groups.Is the mean of A larger (smaller) than B?Are As different (or treated differently) than Bs?Are A and B from the same population?To answer these common types of questions we use the standard two-sample t-test

• Assumptions

independent observationsnormally distributed data for each groupequal variances for each group.

• The Difference of means Test The standard two-sample t-test is:

• The equal Variance testIf the variances from the two samples are the same we may use a more powerful variation

Where

• If the variances from the two samples are the same we may use a more powerful variation

With degree of freedom:

The unequal Variance test

• Which test to Use?In order to choose the appropriate two-sample t-test, we must decide if we think the variances are the same.Hence we perform a preliminary statistical test the equal variance F-test.

• The Equal Variance F-testOne of the fortunate properties on statistics is that the ratio of two variances will have an F distribution.Thus with this knowledge, we can perform a simple test.

• F Test for Equality of Variances

• Interpretation of F-testIf we find that F > F (n1-1,n2-1) , (P(F) > .05), we conclude that the variances are unequal.

If we find that F F (n1-1,n2-1) , (P(F) .05), we conclude that the variances are unequal.

We then select the equal or unequal-variance t-test accordingly.

• Test Statistics and p-ValuesF Test for equal variances:H0: 12 = 22Variance Test:F = 1.51DF = (3,3)t-Tests for equal means:H0: 1 = 2Unequal Variance t-test:T = 7.4017 DF = 5.8Equal Variance t-test:T = 7.4017 DF = 6.0

What would we conclude?

• PAIRED T-test

• Paired Samples

• Proportion

• Large SampleH0 : p = p0HA : p p0 or HA: p < p0 or HA: p > p0

Test statistic :

• ExampleDo you think it shoul or should not be government implementation the law of pornography and pornoaction?Let p denote the population proportion of Indonesia adults who believe it should bep < .5 minorityp > .5 majority

• Continued exampleThis data is not real, just for ilustrationSuppose from 1534 adults, 812 believe it should beH0 : p = .5HA : p .5

The critical value for = 5% z.025 = 1.96The conclusion - Reject H0 majority adults agree if government implementation the law of pornography and pornoaction

• Comparison two proportion

• Large sampleH0 : p1 = p2HA : p1 p2

Test statistic :

= x1/n1, = x2/n2, = (x1 + x2)/(n1 + n2)

• ReferenceAgresti, A. & Finlay, B. 1997. Statistical Methods for the Social Sciences 3rd Edition. Prentice Hall.Mac Gregor. 2006. Lecture 3: Review of Basic Statistics. McMaster UniversityPS 601 Notes Part II Statistical Tests SAS IncTang, A. 2004. Lecture 9 Common Statistical Test. Tufts University

*INTERNAL USE FIG. 01s04f01

***INTERNAL USE FIG. 01s04f02

*INTERNAL USE FIG. 01s04f03

*INTERNAL USE FIG. 01s04f04

*INTERNAL USE FIG. 01s04f05

*INTERNAL USE FIG. 01S04F06

*INTERNAL USE FIG. 01S04F07

*INTERNAL USE FIG. 01s05f01

*INTERNAL USE FIG. 01s05f02

*INTERNAL USE FIG. 01s05f03

*INTERNAL USE FIG. 01s04f08 ##### Pitfalls of Hypothesis Testing. Hypothesis Testing The Steps: 1. Define your hypotheses (null, alternative) 2. Specify your null distribution 3. Do an
Documents ##### REVIEW Hypothesis testing starts with a null ... REVIEW Hypothesis testing starts with a null hypothesis and a null distribution. We compare what we have to the null distribution,
Documents ##### 06. Hypothesis testing - UMass biep540w/pdf/Whitlock and... · PDF file Hypothesis testing Hypothesis testing asks how unusual it is to get data that differ from the null hypothesis
Documents ##### Hypothesis Testing. Outline The Null Hypothesis The Null Hypothesis Type I and Type II Error Type I and Type II Error Using Statistics to test the Null
Documents ##### COMP 3704 Computer Security - · PDF file Christian Grotho Null-hypothesis Testing A null hypothesis is a hypothesis set up to be refuted in order to support an alternative hypothesis
Documents ##### Hypothesis Testing - University of Manchester · PDF file 2020-01-03 · Hypothesis Testing Form the Null Hypothesis Calculate probability of observing data if null hypothesis is true
Documents ##### Section 7.1 Hypothesis Testing: Hypothesis: Null Hypothesis (H 0 ): Alternative Hypothesis (H 1 ):
Documents ##### Statistical Hypotheses & Hypothesis Testing. Statistical Hypotheses There are two types of statistical hypotheses. Null Hypothesis The null hypothesis,
Documents ##### HYPOTHESIS TESTING Null Hypothesis and Research Hypothesis ?
Documents ##### Testing a Nonparametric Null Hypothesis Against a ... meg/MEG2004/Beresteanu-Arie.pdfTesting a Nonparametric Null Hypothesis Against a Nonparametric Alternative Preliminary - comments
Documents ##### Introduction to hypothesis testing 2015 · PDF file Introduction to hypothesis testing Review: Logic of Hypothesis Tests • Usually, we test (attempt to falsify) a null hypothesis
Documents ##### Introduction to statistics for hypothesis testing ... Detection and Hypothesis testing Rejecting a hypothesis aka detection H 0: The \null" hypothesis i.e., the hypothesis that the
Documents ##### Null Hypothesis Significance Testing: A Review of an Old and
Documents ##### Null Hypothesis Significance Testing ... null hypothesis is that each number (0, 00, and 1 through 36) comes up 1 million times. 3 A failure to reject the null hypothesis, given sample
Documents ##### Testing of hypothesis: Null hypothesis, Alternate hypothesis, type l, & type Il errors ¢â‚¬â€‌ critical
Documents ##### Objective Bayesian Hypothesis Testing · PDF file Pathology of Null Hypothesis Statistical Testing Null and Alternative is asymmetric. Test only try to reject null, and gather evidence
Documents ##### 14. hypothesis testing · PDF file hypothesis testing 8 By convention, the null hypothesis is usually the “simpler” hypothesis, or “prevailing wisdom.” E.g., Occam’s Razor
Documents ##### NULL HYPOTHESIS TESTING: PROBLEMS, PREVALENCE ??NULL HYPOTHESIS TESTING: PROBLEMS, PREVALENCE, ... This paper presents a review and critique of statistical ... We find that null hypothesis
Documents ##### Testing Hypothesis About Proportions Chapter 20. Objectives Hypothesis Null hypothesis Alternative hypothesis Two-sided alternative One-sided alternative
Documents ##### PSY 1950 Null Hypothesis Significance Testing September 29, 2008
Documents ##### Doing Bayesian Data Analysis Chapter 11. Null Hypothesis Significance Testing
Education ##### Testing the null hypothesis of stationarity against the ... of Econometrics 54 (1992) 159-178. North-Holland Testing the null hypothesis of stationarity against the alternative of
Documents ##### Testing the null hypothesis of stationarity against the alternative of a
Documents ##### Null Hypothesis Signficance Testing Consider the general approach and associated problems
Documents Documents