Chapter 14: 1-6, 9, 12; Chapter 15: 8 Solutions When is it appropriate to use the normal approximation to the binomial distribution?

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Chapter 14: 1-6, 9, 12; Chapter 15: 8 Solutions When is it appropriate to use the normal approximation to the binomial distribution?"

Transcription

1 Chapter 14: 1-6, 9, 1; Chapter 15: 8 Solutions 14-1 When is it appropriate to use the normal approximation to the binomial distribution? The usual recommendation is that the approximation is good if np > 5 and ( p) n 1 > Describe the function of a continuity correction When is it most useful? The continuity correction is used to make the probabilities evaluated under a normal approximation to the binomial closer to the actual binomial probabilities It is used when the binomial parameter, p, is known This means that the continuity correction is good to use if you are trying to evaluate a probability when p is known When testing hypotheses, we often need to evaluate the p-value A closer estimate of the p-value can be obtained from the normal approximation to the binomial distribution using the continuity correction When evaluating confidence intervals for the parameter p, simulation study results suggest that the continuity correction should not be used As an example, the P( X 4 p 5, n 10) using Table A1 is given by The normal approximation to the binomial is that X N μ np, σ np 1 p As a result, standardizing X, the z-statistic is given by ( ( )) X μ X np Z σ np( 1 p) Without the continuity correction, we would approximate this probability by 4 5 P( X 4) P σ 5( 075) 4 5 P Z 5( 075) P( Z 1095) With the continuity correction, be540-binomial-ch14-15v1-solutionsdoc 1

2 45 5 P( X 45) P σ 5( 075) 45 5 P Z 5( 075) P( Z 14606) The normal approximation with the continuity correction comes closer to the actual probability from the Binomial distribution If we evaluate a probability greater than or equal to a given value, we subtract 05 for the continuity correction Thus, the P( X 4 p 5, n 10) using Table A1 is given by 041 Without the continuity correction, we would approximate this probability by 4 5 P( X 4) P σ 5( 075) 4 5 P Z 5( 075) P( Z 1095) 0138 With the continuity correction, 35 5 P( X 35) P σ 5( 075) 35 5 P Z 5( 075) P( Z 07303) 033 Once again, with the continuity correction, the approximation is closer 14-3) What factors affect the length of a confidence interval for a proportion? Explain The factors are the sample size, the degree of confidence (α level), and the estimate of the proportion This is true since the length is proportional to z np( 1 p) ˆ ˆ α ) When your are working with a difference in proportions, why does the estimated standard error used in the construction of a confidence interval differ from that used in a hypothesis test? be540-binomial-ch14-15v1-solutionsdoc

3 A confidence interval for a difference in proportions (means) is constructed assuming that there may be a true non-zero difference, meaning that the parameters for each group may actually differ If the parameters (p) differ, they have different variances and the different estimates should be used A hypothesis test of equal proportions is constructed under the assumption that the parameters in the two groups are equal If this is true, a common estimate of the parameter using all the data is better than two separate estimates of the parameter 145) Suppose that you select a random sample of 40 children from the population of newborn infants in Mexico The probability that a child in this population weights at most 500 grams is 015 a) For the sample of size 40, what is the probability that four or fewer of the infants weigh at most 500 grams? Compute the exact binomial probability We interpret the statement weigh at most 500 grams to mean weigh less than 500 grams In order to compute this probability exactly, we need to evaluate the probability that there are 0 infants, 1 infant, infants, 3 infants, or 4 infants that weigh less than 500 grams These probabilities are given by n x n x P( X x) p ( 1 p) x 40 x ( 015 ) ( 085 ) 40 x x We use a spreadsheet to culate these probabilities Notice that 1, 40, ( 39) ( 39)( 38) , ( 39)( 38)( 37) 9880, and ( ) 4 3 ( )( 4) n p x n-x n choose x p to x (1-p) to (n-x) Prob(Xx) Source: be540c14xls Sum 0633 b) Using the normal approximation to the binomial distribution, estimate the probability that four or fewer of the children weight at most 500 grams be540-binomial-ch14-15v1-solutionsdoc 3

4 We use the continuity correction to evaluate this, where X N μ np, σ np 1 p σ np 1 p 51 so that ( ( )) and μ ( ) while ( ) σ P( X 45) P σ 58 P( Z 664) 055 c) Do these two methods provide consistent results? The results are pretty close 14-6 A study was conducted to investigate the relationship between maternal smoking during pregnancy and the presence of congenital malformations in the child Among children who suffer from an abnormality other than Down s syndrome or an oral cleft, 38% have mothers who smoked during pregnancy This proportion is homogeneous for children with various types of defects a) If you were to select repeated samples of size 5 from this population, what could you say about the distribution of sample proportions? List three properties I would expect the distribution of sample proportions of children with smoking mothers to be binomially distributed with mean 038 The properties would be the usual binomial properties ( possible outcomes (ie smoke, don t smoke); n trials independent; the probability of smoking is constant for each trial) b) Among the samples of size 5, what fraction has a sample proportion of 045 or higher? With a sample proportion of 45, there are x115 smokers In order to have a sample proportion greater than 45, we would need to have 1 or more smokers The question can be stated as P( X 1 p 38, n 5 )? We will use the continuity correction in approximating this probability Note that np 8 and σ 5( 038)( 1 038) Using a continuity correction, P( X 115) P σ 38 P( Z 1385) 0084 c) What fraction has a sample proportion of 00 or lower? With a sample proportion of 0, there are x5 smokers In order to have a sample proportion less than or equal to 0, we would need to have 5 or fewer smokers The question P X 5 p 38, n 5? We will use the continuity correction in can be stated as ( ) be540-binomial-ch14-15v1-solutionsdoc 4

5 approximating this probability Note that np 8 and σ 5( 038)( 1 038) Using a continuity correction, 55 8 P( X 55) P σ 38 P( Z 1133) 019 d) What value of p cuts off the lower 10% of the distribution? This question can be interpreted as asking what sample proportions, ˆp would fall in the lower 10% of the distribution of all possible values In order to determine this, we need to work backwards The lower 10% corresponds to a 18 P Z z, so that ( ) x 8 Using this value of z, we need to determine a value of x such that 18 or x Rel that with the continuity correction, this value of x corresponds to a count of 465 This means that sample proportions p ˆ 4/5 16 would be in the lower 10%, while sample proportions p ˆ 5/5 would be above 10% of the sampling distribution 14-9 In NY City, a study was conducted to evaluate whether any information that is available at the time of birth can be used to identify children with special educational needs In a random sample of 45 third-graders enrolled in the special education program of the public school system, 4 have mothers who have had more than 1 years of schooling a) Construct a 90% confidence interval for the population proportion of children with special educational needs whose mothers have had more than 1 years of schooling We use a normal approximation to the confidence interval: pˆ( 1 pˆ) pˆ ± z95 n ( 091) ± ± 054 (0034,014) b) In 1980, % of all third-graders enrolled in the NY city public school system had mothers who had had more than 1 years of schooling Suppose you wish to know whether this proportion is the same for children n the special education program What are the null and alternative hypotheses? H : 0 0 p H : p 0 a be540-binomial-ch14-15v1-solutionsdoc 5

6 c, d Conduct the test at the 005 level of significance What do you conclude? 4 p ˆ z 137 0( 078) We reject the null hypothesis if z > 196 Since this is true, we reject the null hypothesis e) If the true population proportion of children with special educational needs whose mother have had more than 1 years of schooling is as low as 010, you want to risk only a 5% chance of failing to reject the null hypothesis If you are conducting a two-sided test at the 005 level of significance, how large a sample would be required? We use the sample size formula given on page 331 to solve this problem The formula is given by z ( ) ( ) α/ p0 1 p0 zβ p1 1 p 1 n p1 p0 In this problem, p 0 0, p 1 01, α β 005, z α / 196, z β 1645, so that ( ) ( ) n so select a sample of 119 subjects Suppose you are interested in investigating the factors that affect the prevalence of tuberculosis among intravenous drug users In a group of 97 individuals who admit to sharing needles, 47% had a positive tuberculin skin test result; among 161 drug users who deny sharing needles, 174% had a positive test result a) Assuming that the population proportions of positive skin test results are in fact equal, estimate their common value, p npˆ pˆ n npˆ n ( ) ( ) b, c) Test the null hypothesis that the proportions of intravenous drug users who have a positive tuberculin skin test results are identi for those who share needles and those who do not State your conclusion be540-binomial-ch14-15v1-solutionsdoc 6

7 H : p p 0 1 H : p p a 1 We reject the null hypothesis if 196 z ˆ ( ) ( 1 ) ( 1 ) pˆ1 p p1 p pˆ pˆ pˆ pˆ n1 n (8) (8) z > Since this is not true, we fail to reject the null hypothesis, and conclude that the current data is insufficient to conclude that there is a difference in proportions d) Construct a 95% confidence interval for the true difference in proportions ( 1 ) ( 1 ) pˆ pˆ pˆ pˆ pˆ ˆ 1 p ± zα / n n ( ) ( ) ± ± ± 0104 ( 0031,0177) 158 The following data come from a study designed to investigate drinking problems among college students In 1983, a group of students were asked whether they had ever driven an automobile while drinking In 1987, after the legal drinking age was raised, a different group of college students were asked the same question Year Year Drove while Drinking Total yes no Total be540-binomial-ch14-15v1-solutionsdoc 7

8 a,b) Use a the chi-square test to evaluate the null hypothesis that the population proportions of students who drove while drinking are the same in the two endar years What are your conclusions? Ho: Population proportions driving while drinking are homogenous between the years Ha: Population proportions are not homogeneous between years Year Year Drove while Drinking Total yes no Total Prop(Yes Expected Year Year Drove while Drinking Total yes no Total check sum (Obs-Exp)/Exp Year Year Drove while Drinking Total yes no Total Chisq check sum We compare the chi-square statistic, χ 5535 with the criti value from a chi-square distribution (equal to 384 when α 005 with 1 degree of freedom) see Table A8 We conclude that the population proportions are not homogeneous between years c) Again test the null hypothesis that the proportions of students who drove while drinking are identi for the two endar years using the test in Section 146 H : p p 0 1 H : p p a 1 be540-binomial-ch14-15v1-solutionsdoc 8

9 We reject the null hypothesis if 196 In fact, ( ) z χ z ˆ ( ) ( 1 ) ( 1 ) pˆ1 p p1 p pˆ pˆ pˆ pˆ n1 n (58) 4(58) z > Since this is true, we reject the null hypothesis d) Construct a 95% confidence interval for the true difference in population proportions pˆ pˆ pˆ pˆ pˆ1 pˆ ± zα / n1 n 047( 053) 037( 063) 01± ± ± 006 (0074,016) ( 1 ) ( 1 ) 1 1 be540-binomial-ch14-15v1-solutionsdoc 9

Comparing Proportions Between Two Independent Populations. John McGready Johns Hopkins University

Comparing Proportions Between Two Independent Populations. John McGready Johns Hopkins University This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike License. Your use of this material constitutes acceptance of that license and the conditions of use of materials on this

More information

Section 7.2 Confidence Intervals for Population Proportions

Section 7.2 Confidence Intervals for Population Proportions Section 7.2 Confidence Intervals for Population Proportions 2012 Pearson Education, Inc. All rights reserved. 1 of 83 Section 7.2 Objectives Find a point estimate for the population proportion Construct

More information

Hypothesis Testing with One Sample. Introduction to Hypothesis Testing 7.1. Hypothesis Tests. Chapter 7

Hypothesis Testing with One Sample. Introduction to Hypothesis Testing 7.1. Hypothesis Tests. Chapter 7 Chapter 7 Hypothesis Testing with One Sample 71 Introduction to Hypothesis Testing Hypothesis Tests A hypothesis test is a process that uses sample statistics to test a claim about the value of a population

More information

STA 2023H Solutions for Practice Test 4

STA 2023H Solutions for Practice Test 4 1. Which statement is not true about confidence intervals? A. A confidence interval is an interval of values computed from sample data that is likely to include the true population value. B. An approximate

More information

Lecture Topic 6: Chapter 9 Hypothesis Testing

Lecture Topic 6: Chapter 9 Hypothesis Testing Lecture Topic 6: Chapter 9 Hypothesis Testing 9.1 Developing Null and Alternative Hypotheses Hypothesis testing can be used to determine whether a statement about the value of a population parameter should

More information

6.1 The Elements of a Test of Hypothesis

6.1 The Elements of a Test of Hypothesis University of California, Davis Department of Statistics Summer Session II Statistics 13 August 22, 2012 Date of latest update: August 20 Lecture 6: Tests of Hypothesis Suppose you wanted to determine

More information

A) to D) to B) to E) to C) to Rate of hay fever per 1000 population for people under 25

A) to D) to B) to E) to C) to Rate of hay fever per 1000 population for people under 25 Unit 0 Review #3 Name: Date:. Suppose a random sample of 380 married couples found that 54 had two or more personality preferences in common. In another random sample of 573 married couples, it was found

More information

PROBLEM SET 1. For the first three answer true or false and explain your answer. A picture is often helpful.

PROBLEM SET 1. For the first three answer true or false and explain your answer. A picture is often helpful. PROBLEM SET 1 For the first three answer true or false and explain your answer. A picture is often helpful. 1. Suppose the significance level of a hypothesis test is α=0.05. If the p-value of the test

More information

MAT X Hypothesis Testing - Part I

MAT X Hypothesis Testing - Part I MAT 2379 3X Hypothesis Testing - Part I Definition : A hypothesis is a conjecture concerning a value of a population parameter (or the shape of the population). The hypothesis will be tested by evaluating

More information

Math 62 Statistics Sample Exam Questions

Math 62 Statistics Sample Exam Questions Math 62 Statistics Sample Exam Questions 1. (10) Explain the difference between the distribution of a population and the sampling distribution of a statistic, such as the mean, of a sample randomly selected

More information

p-values / Tests on Proportions Solutions STAT-UB.0103 Statistics for Business Control and Regression Models

p-values / Tests on Proportions Solutions STAT-UB.0103 Statistics for Business Control and Regression Models p-values / Tests on Proportions Solutions STAT-UB.0103 Statistics for Business Control and Regression Models p-values 1. In the Quarter Pounder example, we we tested the null hypothesis that the weight

More information

Hypothesis Testing. Bluman Chapter 8

Hypothesis Testing. Bluman Chapter 8 CHAPTER 8 Learning Objectives C H A P T E R E I G H T Hypothesis Testing 1 Outline 8-1 Steps in Traditional Method 8-2 z Test for a Mean 8-3 t Test for a Mean 8-4 z Test for a Proportion 8-5 2 Test for

More information

Statistics for Management II-STAT 362-Final Review

Statistics for Management II-STAT 362-Final Review Statistics for Management II-STAT 362-Final Review Multiple Choice Identify the letter of the choice that best completes the statement or answers the question. 1. The ability of an interval estimate to

More information

MCQ TESTING OF HYPOTHESIS

MCQ TESTING OF HYPOTHESIS MCQ TESTING OF HYPOTHESIS MCQ 13.1 A statement about a population developed for the purpose of testing is called: (a) Hypothesis (b) Hypothesis testing (c) Level of significance (d) Test-statistic MCQ

More information

E205 Final: Version B

E205 Final: Version B Name: Class: Date: E205 Final: Version B Multiple Choice Identify the choice that best completes the statement or answers the question. 1. The owner of a local nightclub has recently surveyed a random

More information

Example 11-3, pg. 495 The Chi-Square Goodness-of-Fit Test

Example 11-3, pg. 495 The Chi-Square Goodness-of-Fit Test 132 Chapter 11 Chi-Square Tests Chi-Square Tests Chapter 11 Section 11.2 Example 11-3, pg. 495 The Chi-Square Goodness-of-Fit Test A bank manager wanted to investigate if the percentage of people who use

More information

Chapter 8. Professor Tim Busken. April 20, Chapter 8. Tim Busken. 8.2 Basics of. Hypothesis Testing. Works Cited

Chapter 8. Professor Tim Busken. April 20, Chapter 8. Tim Busken. 8.2 Basics of. Hypothesis Testing. Works Cited Chapter 8 Professor April 20, 2014 In Chapter 8, we continue our study of inferential statistics. Concept: Inferential Statistics The two major activities of inferential statistics are 1 to use sample

More information

UCLA STAT 13 Statistical Methods - Final Exam Review Solutions Chapter 7 Sampling Distributions of Estimates

UCLA STAT 13 Statistical Methods - Final Exam Review Solutions Chapter 7 Sampling Distributions of Estimates UCLA STAT 13 Statistical Methods - Final Exam Review Solutions Chapter 7 Sampling Distributions of Estimates 1. (a) (i) µ µ (ii) σ σ n is exactly Normally distributed. (c) (i) is approximately Normally

More information

STA218 Introduction to Hypothesis Testing

STA218 Introduction to Hypothesis Testing STA218 Introduction to Hypothesis Testing Al Nosedal. University of Toronto. Fall 2015 October 29, 2015 Who wants to be a millionaire? Let s say that one of you is invited to this popular show. As you

More information

AP Statistics 2013 Scoring Guidelines

AP Statistics 2013 Scoring Guidelines AP Statistics 2013 Scoring Guidelines The College Board The College Board is a mission-driven not-for-profit organization that connects students to college success and opportunity. Founded in 1900, the

More information

Hypothesis testing. Null hypothesis H 0 and alternative hypothesis H 1.

Hypothesis testing. Null hypothesis H 0 and alternative hypothesis H 1. Hypothesis testing Null hypothesis H 0 and alternative hypothesis H 1. Hypothesis testing Null hypothesis H 0 and alternative hypothesis H 1. Simple and compound hypotheses. Simple : the probabilistic

More information

Hypothesis testing: Examples. AMS7, Spring 2012

Hypothesis testing: Examples. AMS7, Spring 2012 Hypothesis testing: Examples AMS7, Spring 2012 Example 1: Testing a Claim about a Proportion Sect. 7.3, # 2: Survey of Drinking: In a Gallup survey, 1087 randomly selected adults were asked whether they

More information

I. Basics of Hypothesis Testing

I. Basics of Hypothesis Testing Introduction to Hypothesis Testing This deals with an issue highly similar to what we did in the previous chapter. In that chapter we used sample information to make inferences about the range of possibilities

More information

In the general population of 0 to 4-year-olds, the annual incidence of asthma is 1.4%

In the general population of 0 to 4-year-olds, the annual incidence of asthma is 1.4% Hypothesis Testing for a Proportion Example: We are interested in the probability of developing asthma over a given one-year period for children 0 to 4 years of age whose mothers smoke in the home In the

More information

Mind on Statistics. Chapter 12

Mind on Statistics. Chapter 12 Mind on Statistics Chapter 12 Sections 12.1 Questions 1 to 6: For each statement, determine if the statement is a typical null hypothesis (H 0 ) or alternative hypothesis (H a ). 1. There is no difference

More information

AP Statistics 2004 Scoring Guidelines

AP Statistics 2004 Scoring Guidelines AP Statistics 2004 Scoring Guidelines The materials included in these files are intended for noncommercial use by AP teachers for course and exam preparation; permission for any other use must be sought

More information

Chapter 9: Hypothesis Tests of a Single Population

Chapter 9: Hypothesis Tests of a Single Population Chapter 9: Hypothesis Tests of a Single Population Department of Mathematics Izmir University of Economics Week 12 2014-2015 Introduction In this chapter we will focus on Example developing hypothesis

More information

1. Rejecting a true null hypothesis is classified as a error. 2. Failing to reject a false null hypothesis is classified as a error.

1. Rejecting a true null hypothesis is classified as a error. 2. Failing to reject a false null hypothesis is classified as a error. 1. Rejecting a true null hypothesis is classified as a error. 2. Failing to reject a false null hypothesis is classified as a error. 8.5 Goodness of Fit Test Suppose we want to make an inference about

More information

Chapter 8 Hypothesis Testing

Chapter 8 Hypothesis Testing Chapter 8 Hypothesis Testing Chapter problem: Does the MicroSort method of gender selection increase the likelihood that a baby will be girl? MicroSort: a gender-selection method developed by Genetics

More information

Power and Sample Size Determination

Power and Sample Size Determination Power and Sample Size Determination Bret Hanlon and Bret Larget Department of Statistics University of Wisconsin Madison November 3 8, 2011 Power 1 / 31 Experimental Design To this point in the semester,

More information

Wording of Final Conclusion. Slide 1

Wording of Final Conclusion. Slide 1 Wording of Final Conclusion Slide 1 8.3: Assumptions for Testing Slide 2 Claims About Population Means 1) The sample is a simple random sample. 2) The value of the population standard deviation σ is known

More information

Testing: is my coin fair?

Testing: is my coin fair? Testing: is my coin fair? Formally: we want to make some inference about P(head) Try it: toss coin several times (say 7 times) Assume that it is fair ( P(head)= ), and see if this assumption is compatible

More information

6. Duality between confidence intervals and statistical tests

6. Duality between confidence intervals and statistical tests 6. Duality between confidence intervals and statistical tests Suppose we carry out the following test at a significance level of 100α%. H 0 :µ = µ 0 H A :µ µ 0 Then we reject H 0 if and only if µ 0 does

More information

The Purpose of Hypothesis Testing

The Purpose of Hypothesis Testing Section 8 1A: An Introduction to Hypothesis Testing The Purpose of Hypothesis Testing Seeʼs Candy states that a box of itʼs candy weighs 16 oz. They do not mean that every single box weights exactly 16

More information

13.2 The Chi Square Test for Homogeneity of Populations The setting: Used to compare distribution of proportions in two or more populations.

13.2 The Chi Square Test for Homogeneity of Populations The setting: Used to compare distribution of proportions in two or more populations. 13.2 The Chi Square Test for Homogeneity of Populations The setting: Used to compare distribution of proportions in two or more populations. Data is organized in a two way table Explanatory variable (Treatments)

More information

Math 10 MPS Homework 6 Answers to additional problems

Math 10 MPS Homework 6 Answers to additional problems Math 1 MPS Homework 6 Answers to additional problems 1. What are the two types of hypotheses used in a hypothesis test? How are they related? Ho: Null Hypotheses A statement about a population parameter

More information

N Mean Std. Deviation Std. Error of Mean

N Mean Std. Deviation Std. Error of Mean DEPARTMENT OF POLITICAL SCIENCE AND INTERNATIONAL RELATIONS Posc/Uapp 815 SAMPLE QUESTIONS FOR THE FINAL EXAMINATION I. READING: A. Read Agresti and Finalay, Chapters 6, 7, and 8 carefully. 1. Ignore the

More information

Objectives. 9.1, 9.2 Inference for two-way tables. The hypothesis: no association. Expected cell counts. The chi-square test.

Objectives. 9.1, 9.2 Inference for two-way tables. The hypothesis: no association. Expected cell counts. The chi-square test. Objectives 9.1, 9.2 Inference for two-way tables The hypothesis: no association Expected cell counts The chi-square test Using software Further reading: http://onlinestatbook.com/2/chi_square/contingency.html

More information

Hypothesis Testing - II

Hypothesis Testing - II -3σ -2σ +σ +2σ +3σ Hypothesis Testing - II Lecture 9 0909.400.01 / 0909.400.02 Dr. P. s Clinic Consultant Module in Probability & Statistics in Engineering Today in P&S -3σ -2σ +σ +2σ +3σ Review: Hypothesis

More information

A null hypothesis must always have μ or p, an equal sign, and the claimed value for the parameter in it!

A null hypothesis must always have μ or p, an equal sign, and the claimed value for the parameter in it! HOSP 1207 (Business Stats) Learning Centre Formal Hypothesis: Making Decisions with a Single Sample This worksheet continues to build on the previous concepts of inferential statistics, only now, we re

More information

Example for testing one population mean:

Example for testing one population mean: Today: Sections 13.1 to 13.3 ANNOUNCEMENTS: We will finish hypothesis testing for the 5 situations today. See pages 586-587 (end of Chapter 13) for a summary table. Quiz for week 8 starts Wed, ends Monday

More information

Lecture 13: Kolmogorov Smirnov Test & Power of Tests

Lecture 13: Kolmogorov Smirnov Test & Power of Tests Lecture 13: Kolmogorov Smirnov Test & Power of Tests S. Massa, Department of Statistics, University of Oxford 2 February 2016 An example Suppose you are given the following 100 observations. -0.16-0.68-0.32-0.85

More information

CHAPTERS 4-6: Hypothesis Tests Read sections 4.3, 4.5, 5.1.5, Confidence Interval vs. Hypothesis Test (4.3):

CHAPTERS 4-6: Hypothesis Tests Read sections 4.3, 4.5, 5.1.5, Confidence Interval vs. Hypothesis Test (4.3): CHAPTERS 4-6: Hypothesis Tests Read sections 4.3, 4.5, 5.1.5, 6.1.3 Confidence Interval vs. Hypothesis Test (4.3): The purpose of a confidence interval is to estimate the value of a parameter. The purpose

More information

Water Quality Problem. Hypothesis Testing of Means. Water Quality Example. Water Quality Example. Water quality example. Water Quality Example

Water Quality Problem. Hypothesis Testing of Means. Water Quality Example. Water Quality Example. Water quality example. Water Quality Example Water Quality Problem Hypothesis Testing of Means Dr. Tom Ilvento FREC 408 Suppose I am concerned about the quality of drinking water for people who use wells in a particular geographic area I will test

More information

Lecture 1: t tests and CLT

Lecture 1: t tests and CLT Lecture 1: t tests and CLT http://www.stats.ox.ac.uk/ winkel/phs.html Dr Matthias Winkel 1 Outline I. z test for unknown population mean - review II. Limitations of the z test III. t test for unknown population

More information

Chapter 8: Hypothesis Testing of a Single Population Parameter

Chapter 8: Hypothesis Testing of a Single Population Parameter Chapter 8: Hypothesis Testing of a Single Population Parameter THE LANGUAGE OF STATISTICAL DECISION MAKING DEFINITIONS: The population is the entire group of objects or individuals under study, about which

More information

Hypothesis Tests for a Population Proportion

Hypothesis Tests for a Population Proportion Hypothesis Tests for a Population Proportion MATH 130, Elements of Statistics I J. Robert Buchanan Department of Mathematics Fall 2015 Review: Steps of Hypothesis Testing 1. A statement is made regarding

More information

AP Statistics 1998 Scoring Guidelines

AP Statistics 1998 Scoring Guidelines AP Statistics 1998 Scoring Guidelines These materials are intended for non-commercial use by AP teachers for course and exam preparation; permission for any other use must be sought from the Advanced Placement

More information

Chi-Square & F Distributions

Chi-Square & F Distributions Chi-Square & F Distributions Carolyn J. Anderson EdPsych 580 Fall 2005 Chi-Square & F Distributions p. 1/55 Chi-Square & F Distributions... and Inferences about Variances The Chi-square Distribution Definition,

More information

A Quick Guide to Confidence Intervals and Hypotheses Tests Using the TI-Calc AP Statistics

A Quick Guide to Confidence Intervals and Hypotheses Tests Using the TI-Calc AP Statistics Example: Confidence Intervals for One Proportion In January 2007, Consumer Reports conducted a study of bacteria in frozen chicken sold in the US. They purchased a random selection of 525 packages of frozen

More information

Inferences Based on a Single Sample Tests of Hypothesis

Inferences Based on a Single Sample Tests of Hypothesis Inferences Based on a Single Sample Tests of Hypothesis 11 Inferences Based on a Single Sample Tests of Hypothesis Chapter 8 8. used to decide whether or not to reject the null hypothesis in favor of the

More information

We know from STAT.1030 that the relevant test statistic for equality of proportions is:

We know from STAT.1030 that the relevant test statistic for equality of proportions is: 2. Chi 2 -tests for equality of proportions Introduction: Two Samples Consider comparing the sample proportions p 1 and p 2 in independent random samples of size n 1 and n 2 out of two populations which

More information

STATISTICS 8, FINAL EXAM. Last six digits of Student ID#: Circle your Discussion Section: 1 2 3 4

STATISTICS 8, FINAL EXAM. Last six digits of Student ID#: Circle your Discussion Section: 1 2 3 4 STATISTICS 8, FINAL EXAM NAME: KEY Seat Number: Last six digits of Student ID#: Circle your Discussion Section: 1 2 3 4 Make sure you have 8 pages. You will be provided with a table as well, as a separate

More information

11. CONFIDENCE INTERVALS FOR THE MEAN; KNOWN VARIANCE

11. CONFIDENCE INTERVALS FOR THE MEAN; KNOWN VARIANCE 11. CONFIDENCE INTERVALS FOR THE MEAN; KNOWN VARIANCE We assume here that the population variance σ 2 is known. This is an unrealistic assumption, but it allows us to give a simplified presentation which

More information

Lecture 7: Binomial Test, Chisquare

Lecture 7: Binomial Test, Chisquare Lecture 7: Binomial Test, Chisquare Test, and ANOVA May, 01 GENOME 560, Spring 01 Goals ANOVA Binomial test Chi square test Fisher s exact test Su In Lee, CSE & GS suinlee@uw.edu 1 Whirlwind Tour of One/Two

More information

Statistics - Written Examination MEC Students - BOVISA

Statistics - Written Examination MEC Students - BOVISA Statistics - Written Examination MEC Students - BOVISA Prof.ssa A. Guglielmi 26.0.2 All rights reserved. Legal action will be taken against infringement. Reproduction is prohibited without prior consent.

More information

LAB 4 ASSIGNMENT CONFIDENCE INTERVALS AND HYPOTHESIS TESTING. Using Data to Make Decisions

LAB 4 ASSIGNMENT CONFIDENCE INTERVALS AND HYPOTHESIS TESTING. Using Data to Make Decisions LAB 4 ASSIGNMENT CONFIDENCE INTERVALS AND HYPOTHESIS TESTING This lab assignment will give you the opportunity to explore the concept of a confidence interval and hypothesis testing in the context of a

More information

Introduction to the Practice of Statistics Fifth Edition Moore, McCabe Section 8.1 Homework Answers

Introduction to the Practice of Statistics Fifth Edition Moore, McCabe Section 8.1 Homework Answers Introduction to the Practice of Statistics Fifth Edition Moore, McCabe Section 8.1 Homework Answers 8.1 In each of the following circumstances state whether you would use the large sample confidence interval,

More information

AP STATISTICS 2012 SCORING GUIDELINES

AP STATISTICS 2012 SCORING GUIDELINES 2012 SCORING GUIDELINES Question 4 Intent of Question The primary goal of this question was to assess students ability to identify, set up, perform, and interpret the results of an appropriate hypothesis

More information

Statistical Significance and Bivariate Tests

Statistical Significance and Bivariate Tests Statistical Significance and Bivariate Tests BUS 735: Business Decision Making and Research 1 1.1 Goals Goals Specific goals: Re-familiarize ourselves with basic statistics ideas: sampling distributions,

More information

Elements of Hypothesis Testing (Summary from lecture notes)

Elements of Hypothesis Testing (Summary from lecture notes) Statistics-20090 MINITAB - Lab 1 Large Sample Tests of Hypothesis About a Population Mean We use hypothesis tests to make an inference about some population parameter of interest, for example the mean

More information

Sampling (cont d) and Confidence Intervals Lecture 9 8 March 2006 R. Ryznar

Sampling (cont d) and Confidence Intervals Lecture 9 8 March 2006 R. Ryznar Sampling (cont d) and Confidence Intervals 11.220 Lecture 9 8 March 2006 R. Ryznar Census Surveys Decennial Census Every (over 11 million) household gets the short form and 17% or 1/6 get the long form

More information

CHAPTER 15: Tests of Significance: The Basics

CHAPTER 15: Tests of Significance: The Basics CHAPTER 15: Tests of Significance: The Basics The Basic Practice of Statistics 6 th Edition Moore / Notz / Fligner Lecture PowerPoint Slides Chapter 15 Concepts 2 The Reasoning of Tests of Significance

More information

Chi-Square Tests. In This Chapter BONUS CHAPTER

Chi-Square Tests. In This Chapter BONUS CHAPTER BONUS CHAPTER Chi-Square Tests In the previous chapters, we explored the wonderful world of hypothesis testing as we compared means and proportions of one, two, three, and more populations, making an educated

More information

FINAL EXAM REVIEW - Fa 13

FINAL EXAM REVIEW - Fa 13 FINAL EXAM REVIEW - Fa 13 Determine which of the four levels of measurement (nominal, ordinal, interval, ratio) is most appropriate. 1) The temperatures of eight different plastic spheres. 2) The sample

More information

4) The role of the sample mean in a confidence interval estimate for the population mean is to: 4)

4) The role of the sample mean in a confidence interval estimate for the population mean is to: 4) MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) Assume that the change in daily closing prices for stocks on the New York Stock Exchange is a random

More information

AP STATISTICS 2011 SCORING GUIDELINES

AP STATISTICS 2011 SCORING GUIDELINES AP STATISTICS 2011 SCORING GUIDELINES Question 6 Intent of Question The primary goals of this question were to assess students ability to (1) construct and interpret a confidence interval for a population

More information

Stats for Strategy Exam 1 In-Class Practice Questions DIRECTIONS

Stats for Strategy Exam 1 In-Class Practice Questions DIRECTIONS Stats for Strategy Exam 1 In-Class Practice Questions DIRECTIONS Choose the single best answer for each question. Discuss questions with classmates, TAs and Professor Whitten. Raise your hand to check

More information

STAT303 Spring 2014 Exam #2 Form A

STAT303 Spring 2014 Exam #2 Form A 1 STAT303 Spring 2014 Exam #2 Form A March 6, 2014 1. Don t even open this until you are told to do so. 2. Remember to turn your phone off now. 3. Please turn your hats around backwards or take them off.

More information

Two-Sample T-Test from Means and SD s

Two-Sample T-Test from Means and SD s Chapter 07 Two-Sample T-Test from Means and SD s Introduction This procedure computes the two-sample t-test and several other two-sample tests directly from the mean, standard deviation, and sample size.

More information

Hypothesis Testing. Concept of Hypothesis Testing

Hypothesis Testing. Concept of Hypothesis Testing Quantitative Methods 2013 Hypothesis Testing with One Sample 1 Concept of Hypothesis Testing Testing Hypotheses is another way to deal with the problem of making a statement about an unknown population

More information

4. Sum the results of the calculation described in step 3 for all classes of progeny

4. Sum the results of the calculation described in step 3 for all classes of progeny F09 Biol 322 chi square notes 1. Before proceeding with the chi square calculation, clearly state the genetic hypothesis concerning the data. This hypothesis is an interpretation of the data that gives

More information

Math 225. Chi-Square Tests. 1. A Chi-Square Goodness-of-Fit Test. Conditions. Chi-Square Distribution. The Test Statistic

Math 225. Chi-Square Tests. 1. A Chi-Square Goodness-of-Fit Test. Conditions. Chi-Square Distribution. The Test Statistic Chi-Square Tests Math 5 Chapter 11 In this chapter, you will learn about two chi-square tests: Goodness of fit. Are the proportions of the different outcomes in this population equal to the hypothesized

More information

Chapter Five: Paired Samples Methods 1/38

Chapter Five: Paired Samples Methods 1/38 Chapter Five: Paired Samples Methods 1/38 5.1 Introduction 2/38 Introduction Paired data arise with some frequency in a variety of research contexts. Patients might have a particular type of laser surgery

More information

8.2 Confidence Intervals for One Population Mean When σ is Known

8.2 Confidence Intervals for One Population Mean When σ is Known 8.2 Confidence Intervals for One Population Mean When σ is Known Tom Lewis Fall Term 2009 8.2 Confidence Intervals for One Population Mean When σ isfall Known Term 2009 1 / 6 Outline 1 An example 2 Finding

More information

ACTM Regional Statistics Multiple Choice Questions

ACTM Regional Statistics Multiple Choice Questions ACTM Regional Statistics Multiple Choice Questions This exam includes 2 multiple- choice items and three constructed- response items that may be used as tie- breakers. Record your answer to each of the

More information

Hypothesis Testing. Chapter Introduction

Hypothesis Testing. Chapter Introduction Contents 9 Hypothesis Testing 553 9.1 Introduction............................ 553 9.2 Hypothesis Test for a Mean................... 557 9.2.1 Steps in Hypothesis Testing............... 557 9.2.2 Diagrammatic

More information

Chapter 8 Hypothesis Testing Chapter 8 Hypothesis Testing 8-1 Overview 8-2 Basics of Hypothesis Testing

Chapter 8 Hypothesis Testing Chapter 8 Hypothesis Testing 8-1 Overview 8-2 Basics of Hypothesis Testing Chapter 8 Hypothesis Testing 1 Chapter 8 Hypothesis Testing 8-1 Overview 8-2 Basics of Hypothesis Testing 8-3 Testing a Claim About a Proportion 8-5 Testing a Claim About a Mean: s Not Known 8-6 Testing

More information

HYPOTHESIS TESTING (ONE SAMPLE) - CHAPTER 7 1. used confidence intervals to answer questions such as...

HYPOTHESIS TESTING (ONE SAMPLE) - CHAPTER 7 1. used confidence intervals to answer questions such as... HYPOTHESIS TESTING (ONE SAMPLE) - CHAPTER 7 1 PREVIOUSLY used confidence intervals to answer questions such as... You know that 0.25% of women have red/green color blindness. You conduct a study of men

More information

Sections 7.1 and 7.2. This chapter presents the beginning of inferential statistics. The two major applications of inferential statistics

Sections 7.1 and 7.2. This chapter presents the beginning of inferential statistics. The two major applications of inferential statistics Sections 7.1 and 7.2 This chapter presents the beginning of inferential statistics. The two major applications of inferential statistics Estimate a population parameter: proportion, mean Test some claim

More information

Null Hypothesis Significance Testing Signifcance Level, Power, t-tests Spring 2014 Jeremy Orloff and Jonathan Bloom

Null Hypothesis Significance Testing Signifcance Level, Power, t-tests Spring 2014 Jeremy Orloff and Jonathan Bloom Null Hypothesis Significance Testing Signifcance Level, Power, t-tests 18.05 Spring 2014 Jeremy Orloff and Jonathan Bloom Simple and composite hypotheses Simple hypothesis: the sampling distribution is

More information

Hypothesis Testing COMP 245 STATISTICS. Dr N A Heard. 1 Hypothesis Testing 2 1.1 Introduction... 2 1.2 Error Rates and Power of a Test...

Hypothesis Testing COMP 245 STATISTICS. Dr N A Heard. 1 Hypothesis Testing 2 1.1 Introduction... 2 1.2 Error Rates and Power of a Test... Hypothesis Testing COMP 45 STATISTICS Dr N A Heard Contents 1 Hypothesis Testing 1.1 Introduction........................................ 1. Error Rates and Power of a Test.............................

More information

Terminology. 2 There is no mathematical difference between the errors, however. The bottom line is that we choose one type

Terminology. 2 There is no mathematical difference between the errors, however. The bottom line is that we choose one type Hypothesis Testing 10.2.1 Terminology The null hypothesis H 0 is a nothing hypothesis, whose interpretation could be that nothing has changed, there is no difference, there is nothing special taking place,

More information

Chapter 7. Hypothesis Testing with One Sample

Chapter 7. Hypothesis Testing with One Sample Chapter 7 Hypothesis Testing with One Sample 7.1 Introduction to Hypothesis Testing Hypothesis Tests A hypothesis test is a process that uses sample statistics to test a claim about the value of a population

More information

Hypothesis Testing. April 21, 2009

Hypothesis Testing. April 21, 2009 Hypothesis Testing April 21, 2009 Your Claim is Just a Hypothesis I ve never made a mistake. Once I thought I did, but I was wrong. Your Claim is Just a Hypothesis Confidence intervals quantify how sure

More information

Ch. 8 Hypothesis Testing

Ch. 8 Hypothesis Testing Ch. 8 Hypothesis Testing 8.1 Foundations of Hypothesis Testing Definitions In statistics, a hypothesis is a claim about a property of a population. A hypothesis test is a standard procedure for testing

More information

Outline of Topics. Statistical Methods I. Types of Data. Descriptive Statistics

Outline of Topics. Statistical Methods I. Types of Data. Descriptive Statistics Statistical Methods I Tamekia L. Jones, Ph.D. (tjones@cog.ufl.edu) Research Assistant Professor Children s Oncology Group Statistics & Data Center Department of Biostatistics Colleges of Medicine and Public

More information

Biostatistics Lab Notes

Biostatistics Lab Notes Biostatistics Lab Notes Page 1 Lab 1: Measurement and Sampling Biostatistics Lab Notes Because we used a chance mechanism to select our sample, each sample will differ. My data set (GerstmanB.sav), looks

More information

Hence, multiplying by 12, the 95% interval for the hourly rate is (965, 1435)

Hence, multiplying by 12, the 95% interval for the hourly rate is (965, 1435) Confidence Intervals for Poisson data For an observation from a Poisson distribution, we have σ 2 = λ. If we observe r events, then our estimate ˆλ = r : N(λ, λ) If r is bigger than 20, we can use this

More information

HYPOTHESIS TESTING (TWO SAMPLE) - CHAPTER 8 1. how can a sample be used to estimate the unknown parameters of a population

HYPOTHESIS TESTING (TWO SAMPLE) - CHAPTER 8 1. how can a sample be used to estimate the unknown parameters of a population HYPOTHESIS TESTING (TWO SAMPLE) - CHAPTER 8 1 PREVIOUSLY estimation how can a sample be used to estimate the unknown parameters of a population use confidence intervals around point estimates of central

More information

The Paired t-test and Hypothesis Testing. John McGready Johns Hopkins University

The Paired t-test and Hypothesis Testing. John McGready Johns Hopkins University This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike License. Your use of this material constitutes acceptance of that license and the conditions of use of materials on this

More information

Principles of Hypothesis

Principles of Hypothesis Principles of Hypothesis Testing for Public Health Laura Lee Johnson, Ph.D. Statistician National Center for Complementary and Alternative Medicine johnslau@mail.nih.gov Fall 2011 Answers to Questions

More information

Probability, Binomial Distributions and Hypothesis Testing Vartanian, SW 540

Probability, Binomial Distributions and Hypothesis Testing Vartanian, SW 540 Probability, Binomial Distributions and Hypothesis Testing Vartanian, SW 540 1. Assume you are tossing a coin 11 times. The following distribution gives the likelihoods of getting a particular number of

More information

For eg:- The yield of a new paddy variety will be 3500 kg per hectare scientific hypothesis. In Statistical language if may be stated as the random

For eg:- The yield of a new paddy variety will be 3500 kg per hectare scientific hypothesis. In Statistical language if may be stated as the random Lecture.9 Test of significance Basic concepts null hypothesis alternative hypothesis level of significance Standard error and its importance steps in testing Test of Significance Objective To familiarize

More information

FALL 2005 EXAM C SOLUTIONS

FALL 2005 EXAM C SOLUTIONS FALL 005 EXAM C SOLUTIONS Question #1 Key: D S ˆ(300) = 3/10 (there are three observations greater than 300) H ˆ (300) = ln[ S ˆ (300)] = ln(0.3) = 1.0. Question # EX ( λ) = VarX ( λ) = λ µ = v = E( λ)

More information

Hypothesis. Testing Examples and Case Studies. Chapter 23. Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.

Hypothesis. Testing Examples and Case Studies. Chapter 23. Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc. Hypothesis Chapter 23 Testing Examples and Case Studies Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc. 23.1 How Hypothesis Tests Are Reported in the News 1. Determine the null hypothesis

More information

AP * Statistics Review

AP * Statistics Review AP * Statistics Review Confidence Intervals Teacher Packet AP* is a trademark of the College Entrance Examination Board. The College Entrance Examination Board was not involved in the production of this

More information

Hypothesis Testing. Jungmo Yoon CMC, Claremont McKeena College. Jungmo Yoon (CMC) Testing CMC, / 7

Hypothesis Testing. Jungmo Yoon CMC, Claremont McKeena College. Jungmo Yoon (CMC) Testing CMC, / 7 Hypothesis Testing Jungmo Yoon Claremont McKeena College CMC, 2009 Jungmo Yoon (CMC) Testing CMC, 2009 1 / 7 Concepts of Hypothesis Testing A criminal trial as an example of hypothesis testing. A jury

More information

Comparing Two Populations OPRE 6301

Comparing Two Populations OPRE 6301 Comparing Two Populations OPRE 6301 Introduction... In many applications, we are interested in hypotheses concerning differences between the means of two populations. For example, we may wish to decide

More information

STATISTICS 151 SECTION 1 FINAL EXAM MAY

STATISTICS 151 SECTION 1 FINAL EXAM MAY STATISTICS 151 SECTION 1 FINAL EXAM MAY 2 2009 This is an open book exam. Course text, personal notes and calculator are permitted. You have 3 hours to complete the test. Personal computers and cellphones

More information