# Chapter 23 Inferences About Means

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Chapter 23 Inferences About Means Chapter 23 - Inferences About Means 391 Chapter 23 Solutions to Class Examples 1. See Class Example We want to know if the mean battery lifespan exceeds the 300-minute goal set by the manufacturer. We have 12 battery lifespans in our sample to test the claim. Hypotheses The null hypothesis is that the batteries have a mean lifespan of 300 minutes. The 300-minute goal has not been met. The alternative hypothesis is that the batteries have a mean lifespan greater than 300 minutes. The goal 300-minute goal has been met. H 0 : µ = 300 H A : µ > 300 Model Randomization Condition: This is not a random sample of batteries, but merely 12 batteries produced for preliminary testing. However, it is reasonable to assume that these batteries are representative of all batteries. Nearly Normal Condition: The distribution of battery lifespans is unimodal and symmetric, so it s reasonable to assume that the lifespans of all batteries could be described by a Normal model. Since the conditions have been met, we can do a one sample t-test for the mean, with 11 degrees of freedom. Mechanics - n = 12 df = 11 y = s = Battery Lifespans (min) t = y µ s n t = t P-value = P( y > ) = P(t 11 > ) = Conclusion Since the P-value is high, fail to reject the null hypothesis. There is no evidence to suggest that the mean battery lifespan exceeds 300 minutes. It does not appear that the company has met its goal. Confidence Interval The conditions have been met, so we can create a one-sample t-interval, with 90% confidence.

2 392 Chapter 23 - Inferences About Means y ± t * 11 SE( y ) = ± = (291.05,321.45) I am 90% confident that the mean battery lifespan is between and minutes. Sample Size We want to know how many batteries to test s ME = t * to be 95% sure of estimating the mean lifespan to within n 15 minutes. First, do a preliminary estimate using z * = 1.96 as the 15 = critical value. n Our first estimate is about 15 batteries. Now, do a better estimate, using t * 14 = as the critical value. n = (1.96)(29.31) 15 n We would need to sample about 18 batteries in order to estimate the mean battery lifespan to within 15 minutes, with 95% confidence. ME = t * s n 15 = n n = (2.145)(29.31) 15 n Finally, to estimate the mean battery lifespan to within 5 minutes, you could do the entire process again, perhaps using a critical value with much higher degrees of freedom. We know that it s going to take lots more batteries to cut the margin of error to a third of what it was. Alternatively, we know it will take a sample about 9 times as large, 18(9) = 162 batteries, since the margin of error was decreased to a third of its size. 3. See Class Example 3.

3 Investigative Task Chapter 23 - Inferences About Means 393 This is the first of a pair of tasks that use the same data set. The thrust of the tasks is more important than the particular data you use. We use the SAT scores reported for one college, but other data will work. You may want to find something more relevant to your students and revise the task accordingly. Ideally you need a reasonably large but manageable data set; 250 to 500 cases seems to work well. SAT s (or ACT s) are ideal because each individual has two scores (making paired comparisons possible) and lots of statistics are available. The first part of the task asks students to select a random sample, reviewing use of random numbers and sampling issues from Chapter 12. Based on the various samples, students construct confidence intervals, then use the interval to compare the local performance to state or national results. And this affords you the opportunity to look at all the confidence intervals together, noting that most hit the target while others miss. The second half of this task comes after Chapter 25, asking students to do two hypothesis tests one involving paired data and the other comparing two independent means.

5 Chapter 23 - Inferences About Means 395 Intro Stats - Investigative Task Chapter 23 Components Think Demonstrates clear understanding of construction and interpretation of confidence interval Show Sampling Procedure Randomization is correctly used Method of randomization is described Conditions Random sample Assesses normality Mechanics Identifies procedure Constructs correct interval Tell Interpretation Interprets interval in context Each component is scored as Essentially correct, Partially correct, or Incorrect. 1. The Sample E - Selects a random sample, explains the sampling process clearly. P Selects a random sample, but explanation of the process may be unclear or there may be mistakes in notation, or vocabulary. I Sample is not random or the process is not explained or there are several major mistakes in arithmetic, notation, or vocabulary. 2. The Conditions E Cites randomness, <10% of all possible students, and checks normality with a plot. P Discusses normality but omits the plot or makes minor mistakes in the other conditions. I Misunderstands or omits the conditions, or lists irrelevant issues (np 10). 3. The Mechanics E Identifies the procedure, shows the sample statistics and degrees of freedom, writes the formula using correct critical value and notation, and calculates the correct interval (perhaps with minor arithmetic or rounding errors). P Appears to be doing the proper procedure but omits important information, uses the wrong critical value, uses the wrong notation, or makes major errors in calculations. I Uses the wrong procedure or shows no work or makes several major mistakes. 4. The Interpretation E Correctly interprets the confidence interval in the proper context, and compares the local performance to statewide results. P Writes a conclusion that is correct but not in context or doesn t compare local performance to statewide results. I Does not interpret the confidence interval correctly. Scoring E s count 1 point, P s are 1/2 Score = sum of 4 components; rounding based on quality of P responses Comments Show plot

6 396 Chapter 23 - Inferences About Means Gender Verbal Math Gender Verbal Math Gender Verbal Math F F M F F F M M F M F F M M F M M M F F M M M M F F M F M F M M F F M M F M M M M F M M F M M M F F M M F F F M F M M F M M F M F M F M F F F F F M F M F F M M F F M F F M F M F M F M F F M M M M F F M M M F M F F M F M F F F M M F F M F M M F F M F F M M M M F M M M M F M M F M F F F F F F M F F F

7 Chapter 23 - Inferences About Means 397 Gender Verbal Math Gender Verbal Math Gender Verbal Math F F F F M F M M F F F M F F M M M F F F M F F F M F M M F F M M M F M M F M M M M M F M M M M F M F M F M M M F M F F F M F M F M M F M F M M F F M F M F M M M M M F M F M F M M M F M F M F M M F M M F M F M M F M M M M F F F F F F F M M F M F F M F F M M M F M F M F F M F M F F M M M F F F M M F M

8 398 Chapter 23 - Inferences About Means Chapter 23 Investigative Task Sample Solution SAT Performance I want to determine the mean SAT-Math score at this college. I will take a simple random sample of 25 students by assigning a number to each of the students, and then choosing 25 random numbers , ignoring repeats. Plan I want to find a 95% confidence interval for the mean SAT-Math score of all students at this college. I have data on the scores of 25 students, from a simple random sample of the 300 students at this college. Model Randomization Condition: The students were chosen by a simple random sample. 10% Condition: 25 students represent less than 10% of the population of 300 students Nearly Normal Condition: The distribution of SAT-Math scores in the sample is unimodal and symmetric Math The conditions are satisfied, so I will use a Student s t-model with (n 1) = 25 1 = 24 degrees of freedom to find a one-sample t-interval for the mean. Mechanics From my sample of 25 students: n = 25 scores y = s = y ± t * 24 SE(y ) = ± = ± = (572.13,633.47) Conclusion I am 95% confident that the interval from to contains the true mean SAT-Math score for students at this college. According to my confidence interval, students at this college had a higher mean SAT- Math score than students nationwide. The national mean of 503 was not included in my 95% confidence interval.

9 Chapter 23 - Inferences About Means 399 Intro Stats Quiz A Chapter 23 Name A professor at a large university believes that students take an average of 15 credit hours per term. A random sample of 24 students in her class of 250 students reported the following number of credit hours that they were taking: Does this sample indicate that students are taking more credit hours than the professor believes? Test an appropriate hypothesis and state your conclusion.

10 400 Chapter 23 - Inferences About Means 2. Find a 95% confidence interval for the number of credit hours taken by the students in the professor s class. Interpret your interval.

11 Chapter 23 - Inferences About Means 401 Intro Stats Quiz A Chapter 23 Key A professor at a large university believes that students take an average of 15 credit hours per term. A random sample of 24 students in her class of 250 students reported the following number of credit hours that they were taking: Does this sample indicate that students are taking more credit hours than the professor believes? Test an appropriate hypothesis and state your conclusion. H : 15 0 µ = credit hours; Students in the professor s class are taking an average of 15 credit hours. H A : µ > 15 credit hours; Students in the professor s class are taking more than 15 credit hours, on average. Conditions: * Randomization condition: Students from the class were randomly sampled. * 10% condition: The sample is less than 10% of the class population. * Nearly Normal condition: The histogram of credit hours is unimodal and reasonably symmetric. This is close enough to Normal for our purposes. 5 Histogram of Credit Hours 4 Frequency Credit Hours Under these conditions, the sampling distribution of the mean can be modeled by Student s t with degrees of freedom: df = n 1 = 24 1 = 23. We will use a one-sample t-test for the mean We know: n = 24, y = 16.6, and s = So, SE( y ) = = y µ t = = = The P-value is Pt ( 23 > 3.532) = SE( y) The P-value of says that if the true mean credit hours for students in the professor s class is 15, samples of 24 students can be expected to have an observed mean of 16.6 credit hours or more less than 0.1% of the time. This is rare enough for us to reject the null hypothesis. This sample indicates that the professor s students are taking more than 15 credit hours, on average.

12 402 Chapter 23 - Inferences About Means 2. Find a 95% confidence interval for the number of credit hours taken by the students in the professor s class. Interpret your interval. With the conditions satisfied (from Problem 1), we can find a t-interval for mean credit hours. We know: n = 24, y = 16.6, s = 2.22, and 2.22 SE( y ) = = Our confidence interval has the form n 1 interval is then s y± t. We have t 23 = Our 95% confidence n ± = 16.6 ± 0.94, or to We are 95% confident that the interval to contains the true mean number of credit hours that students in the professor s class are taking.

13 Chapter 23 - Inferences About Means 403 Intro Stats Quiz B Chapter 23 Name Insurance companies track life expectancy information to assist in determining the cost of life insurance policies. The insurance company knows that, last year, the life expectancy of its policyholders was 77 years. They want to know if their clients this year have a longer life expectancy, on average, so the company randomly samples some of the recently paid policies to see if the mean life expectancy of policyholders has increased. The insurance company will only change their premium structure if there is evidence that people who buy their policies are living longer than before Does this sample indicate that the insurance company should change its premiums because life expectancy has increased? Test an appropriate hypothesis and state your conclusion.

14 404 Chapter 23 - Inferences About Means 2. For more accurate cost determination, the insurance companies want to estimate the life expectancy to within one year with 95% confidence. How many randomly selected records would they need to have?

15 Chapter 23 - Inferences About Means 405 Intro Stats Quiz B Chapter 23 Key Insurance companies track life expectancy information to assist in determining the cost of life insurance policies. The insurance company knows that, last year, the life expectancy of its policyholders was 77 years. They want to know if their clients this year have a longer life expectancy, on average, so the company randomly samples some of the recently paid policies to see if the mean life expectancy of policyholders has increased. The insurance company will only change their premium structure if there is evidence that people who buy their policies are living longer than before Does this sample indicate that the insurance company should change its premiums because life expectancy has increased? Test an appropriate hypothesis and state your conclusion. H0 : µ = 77years The average life expectancy of the insurance company s policy holders is 77 years. HA : µ > 77 years The average life expectancy of the insurance company s policy holders is more than 77 years. Conditions: * Randomization condition: The records from the insurance company were randomly sampled. * Nearly Normal condition: The histogram of the ages at death is unimodal and reasonably symmetric. This is close enough to Normal for our purposes. y µ t = = = SE( y) P= P( t > 1.597) = Life Expectancy 4 Under these conditions, the sampling distribution of the mean can be modeled by Student s t with degrees of 1 freedom: df = n 1 = 20 1 = 19. We will use a one-sample t-test for the mean DeathAge We know: n = 20, y = 78.6 years, and s = 4.48 years. SE( y) = s 4.48 = = years. n 20 The P value of says that if the true mean life expectancy for a person had increased to 77 years, samples of 20 records can be expected to have an observed mean life expectancy of 78.6 years or more 6.3% of the time. This is not rare enough for us to reject the null hypothesis. This sample does not indicate that the insurance company needs to increase their premiums because there was not enough evidence to indicate that people who buy their policies are living longer than before. Count 3 2 Histogram

16 406 Chapter 23 - Inferences About Means 2. For more accurate cost determination, the insurance companies want to estimate the life expectancy to within one year with 95% confidence. How many randomly selected records would they need to have? We wish to find the sample size, n, that would allow a 95% confidence level for the mean life expectancy, µ, of a policy holder from the insurance company to have a margin of error of only one year. The 95% critical value of the Student s t-statistic with 19 degrees of freedom (df = n 1) is t * 19 = * 19 ( ) ME = t SE y 2 ( )( ) 2 * t s n = = = ME 1 records. We would need to sample at least 88 records from the insurance company to estimate the life expectancy of a policy holder to within one year with 95% confidence.

17 Chapter 23 - Inferences About Means 407 Intro Stats Quiz C Chapter 23 Name Textbook authors must be careful that the reading level of their book is appropriate for the target audience. Some methods of assessing reading level require estimating the average word length. We ve randomly chosen 20 words from a randomly selected page in Stats: Modeling the World and counted the number of letters in each word: 5, 5, 2, 11, 1, 5, 3, 8, 5, 4, 7, 2, 9, 4, 8, 10, 4, 5, 6, 6 1. Suppose that our editor was hoping that the book would have a mean word length of 6.5 letters. Does this sample indicate that the authors failed to meet this goal? Test an appropriate hypothesis and state your conclusion. 2. For a more definitive evaluation of reading level the editor wants to estimate the text s mean word length to within 0.5 letters with 98% confidence. How many randomly selected words does she need to use?

18 408 Chapter 23 - Inferences About Means Intro Stats Quiz C Chapter 23 Key Textbook authors must be careful that the reading level of their book is appropriate for the target audience. Some methods of assessing reading level require estimating the average word length. We ve randomly chosen 20 words from a randomly selected page in Stats: Modeling the World and counted the number of letters in each word: 5, 5, 2, 11, 1, 5, 3, 8, 5, 4, 7, 2, 9, 4, 8, 10, 4, 5, 6, 6 1. Suppose that our editor was hoping that the book would have a mean word length of 6.5 letters. Does this sample indicate that the authors failed to meet this goal? Test an appropriate hypothesis and state your conclusion. Hypotheses: H 0 :µ = 6.5 H a :µ 6.5 where µ represents the population mean of word lengths Plan: We have a random sample of less than 10% of the words in the book. A histogram of theobserved word lengths looks roughly unimodal and symmetric, so the population of all word lengths may be approximately normal. It is appropriate to use a one sample t-test. Mechanics: n = 20 x = 5.5 s = df = 19 t = P = 2 P(t 19 < 1.67) = = 1.67 Collection Histogram letters Conclusion: Because the P-value is so high we do not reject H 0. This sample does not provide evidence that the average word length differs from the goal of 6.5 letters. 2. For a more definitive evaluation of reading level the editor wants to estimate the text s mean word length to within 0.5 letters with 98% confidence. How many randomly selected words does she need to use? 2 2 zs = = = = n= 157 ME 0.5 First estimate: n ( ) Based on first estimate, 2 2 t157s n= = = ( ) = n= 160 ME 0.5 Table T., using df = 140 from

### AP STATISTICS 2009 SCORING GUIDELINES (Form B)

AP STATISTICS 2009 SCORING GUIDELINES (Form B) Question 5 Intent of Question The primary goals of this question were to assess students ability to (1) state the appropriate hypotheses, (2) identify and

### Name: Date: Use the following to answer questions 3-4:

Name: Date: 1. Determine whether each of the following statements is true or false. A) The margin of error for a 95% confidence interval for the mean increases as the sample size increases. B) The margin

### AP Statistics 2010 Scoring Guidelines

AP Statistics 2010 Scoring Guidelines The College Board The College Board is a not-for-profit membership association whose mission is to connect students to college success and opportunity. Founded in

### Chapter 7 Section 7.1: Inference for the Mean of a Population

Chapter 7 Section 7.1: Inference for the Mean of a Population Now let s look at a similar situation Take an SRS of size n Normal Population : N(, ). Both and are unknown parameters. Unlike what we used

### General Method: Difference of Means. 3. Calculate df: either Welch-Satterthwaite formula or simpler df = min(n 1, n 2 ) 1.

General Method: Difference of Means 1. Calculate x 1, x 2, SE 1, SE 2. 2. Combined SE = SE1 2 + SE2 2. ASSUMES INDEPENDENT SAMPLES. 3. Calculate df: either Welch-Satterthwaite formula or simpler df = min(n

### Chapter 7: Simple linear regression Learning Objectives

Chapter 7: Simple linear regression Learning Objectives Reading: Section 7.1 of OpenIntro Statistics Video: Correlation vs. causation, YouTube (2:19) Video: Intro to Linear Regression, YouTube (5:18) -

### AP Statistics 2011 Scoring Guidelines

AP Statistics 2011 Scoring Guidelines The College Board The College Board is a not-for-profit membership association whose mission is to connect students to college success and opportunity. Founded in

### Hypothesis Test for Mean Using Given Data (Standard Deviation Known-z-test)

Hypothesis Test for Mean Using Given Data (Standard Deviation Known-z-test) A hypothesis test is conducted when trying to find out if a claim is true or not. And if the claim is true, is it significant.

### Chapter Study Guide. Chapter 11 Confidence Intervals and Hypothesis Testing for Means

OPRE504 Chapter Study Guide Chapter 11 Confidence Intervals and Hypothesis Testing for Means I. Calculate Probability for A Sample Mean When Population σ Is Known 1. First of all, we need to find out the

### August 2012 EXAMINATIONS Solution Part I

August 01 EXAMINATIONS Solution Part I (1) In a random sample of 600 eligible voters, the probability that less than 38% will be in favour of this policy is closest to (B) () In a large random sample,

### 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

### Module 5 Hypotheses Tests: Comparing Two Groups

Module 5 Hypotheses Tests: Comparing Two Groups Objective: In medical research, we often compare the outcomes between two groups of patients, namely exposed and unexposed groups. At the completion of this

### Psychology 60 Fall 2013 Practice Exam Actual Exam: Next Monday. Good luck!

Psychology 60 Fall 2013 Practice Exam Actual Exam: Next Monday. Good luck! Name: 1. The basic idea behind hypothesis testing: A. is important only if you want to compare two populations. B. depends on

### Chapter 7 Section 1 Homework Set A

Chapter 7 Section 1 Homework Set A 7.15 Finding the critical value t *. What critical value t * from Table D (use software, go to the web and type t distribution applet) should be used to calculate the

### SPSS on two independent samples. Two sample test with proportions. Paired t-test (with more SPSS)

SPSS on two independent samples. Two sample test with proportions. Paired t-test (with more SPSS) State of the course address: The Final exam is Aug 9, 3:30pm 6:30pm in B9201 in the Burnaby Campus. (One

### ISyE 2028 Basic Statistical Methods - Fall 2015 Bonus Project: Big Data Analytics Final Report: Time spent on social media

ISyE 2028 Basic Statistical Methods - Fall 2015 Bonus Project: Big Data Analytics Final Report: Time spent on social media Abstract: The growth of social media is astounding and part of that success was

### Inferential Statistics

Inferential Statistics Sampling and the normal distribution Z-scores Confidence levels and intervals Hypothesis testing Commonly used statistical methods Inferential Statistics Descriptive statistics are

### Good luck! BUSINESS STATISTICS FINAL EXAM INSTRUCTIONS. Name:

Glo bal Leadership M BA BUSINESS STATISTICS FINAL EXAM Name: INSTRUCTIONS 1. Do not open this exam until instructed to do so. 2. Be sure to fill in your name before starting the exam. 3. You have two hours

### Unit 27: Comparing Two Means

Unit 27: Comparing Two Means Prerequisites Students should have experience with one-sample t-procedures before they begin this unit. That material is covered in Unit 26, Small Sample Inference for One

### MATH 214 (NOTES) Math 214 Al Nosedal. Department of Mathematics Indiana University of Pennsylvania. MATH 214 (NOTES) p. 1/6

MATH 214 (NOTES) Math 214 Al Nosedal Department of Mathematics Indiana University of Pennsylvania MATH 214 (NOTES) p. 1/6 "Pepsi" problem A market research consultant hired by the Pepsi-Cola Co. is interested

### Section 9.3B Inference for Means: Paired Data

Section 9.3B Inference for Means: Paired Data 1 Objective PERFORM significance tests for paired data are called: paired t procedures. Comparative studies (i.e. 2 observations on 1 individual or 1 observation

### t Tests in Excel The Excel Statistical Master By Mark Harmon Copyright 2011 Mark Harmon

t-tests in Excel By Mark Harmon Copyright 2011 Mark Harmon No part of this publication may be reproduced or distributed without the express permission of the author. mark@excelmasterseries.com www.excelmasterseries.com

### AP Statistics 2002 Scoring Guidelines

AP Statistics 2002 Scoring Guidelines The materials included in these files are intended for use by AP teachers for course and exam preparation in the classroom; permission for any other use must be sought

### AP Statistics 2005 Scoring Guidelines

AP Statistics 2005 Scoring Guidelines The College Board: Connecting Students to College Success The College Board is a not-for-profit membership association whose mission is to connect students to college

### 9-3.4 Likelihood ratio test. Neyman-Pearson lemma

9-3.4 Likelihood ratio test Neyman-Pearson lemma 9-1 Hypothesis Testing 9-1.1 Statistical Hypotheses Statistical hypothesis testing and confidence interval estimation of parameters are the fundamental

### LAB 4 INSTRUCTIONS CONFIDENCE INTERVALS AND HYPOTHESIS TESTING

LAB 4 INSTRUCTIONS CONFIDENCE INTERVALS AND HYPOTHESIS TESTING In this lab you will explore the concept of a confidence interval and hypothesis testing through a simulation problem in engineering setting.

### Statistics 151 Practice Midterm 1 Mike Kowalski

Statistics 151 Practice Midterm 1 Mike Kowalski Statistics 151 Practice Midterm 1 Multiple Choice (50 minutes) Instructions: 1. This is a closed book exam. 2. You may use the STAT 151 formula sheets and

### Two-sample t-tests. - Independent samples - Pooled standard devation - The equal variance assumption

Two-sample t-tests. - Independent samples - Pooled standard devation - The equal variance assumption Last time, we used the mean of one sample to test against the hypothesis that the true mean was a particular

### C. The null hypothesis is not rejected when the alternative hypothesis is true. A. population parameters.

Sample Multiple Choice Questions for the material since Midterm 2. Sample questions from Midterms and 2 are also representative of questions that may appear on the final exam.. A randomly selected sample

### Two-sample hypothesis testing, I 9.07 3/09/2004

Two-sample hypothesis testing, I 9.07 3/09/2004 But first, from last time More on the tradeoff between Type I and Type II errors The null and the alternative: Sampling distribution of the mean, m, given

### Online 12 - Sections 9.1 and 9.2-Doug Ensley

Student: Date: Instructor: Doug Ensley Course: MAT117 01 Applied Statistics - Ensley Assignment: Online 12 - Sections 9.1 and 9.2 1. Does a P-value of 0.001 give strong evidence or not especially strong

### Statistics Review PSY379

Statistics Review PSY379 Basic concepts Measurement scales Populations vs. samples Continuous vs. discrete variable Independent vs. dependent variable Descriptive vs. inferential stats Common analyses

### 2013 MBA Jump Start Program. Statistics Module Part 3

2013 MBA Jump Start Program Module 1: Statistics Thomas Gilbert Part 3 Statistics Module Part 3 Hypothesis Testing (Inference) Regressions 2 1 Making an Investment Decision A researcher in your firm just

### MONT 107N Understanding Randomness Solutions For Final Examination May 11, 2010

MONT 07N Understanding Randomness Solutions For Final Examination May, 00 Short Answer (a) (0) How are the EV and SE for the sum of n draws with replacement from a box computed? Solution: The EV is n times

### Confidence Interval: pˆ = E = Indicated decision: < p <

Hypothesis (Significance) Tests About a Proportion Example 1 The standard treatment for a disease works in 0.675 of all patients. A new treatment is proposed. Is it better? (The scientists who created

### Part 3. Comparing Groups. Chapter 7 Comparing Paired Groups 189. Chapter 8 Comparing Two Independent Groups 217

Part 3 Comparing Groups Chapter 7 Comparing Paired Groups 189 Chapter 8 Comparing Two Independent Groups 217 Chapter 9 Comparing More Than Two Groups 257 188 Elementary Statistics Using SAS Chapter 7 Comparing

### Math 58. Rumbos Fall 2008 1. Solutions to Review Problems for Exam 2

Math 58. Rumbos Fall 2008 1 Solutions to Review Problems for Exam 2 1. For each of the following scenarios, determine whether the binomial distribution is the appropriate distribution for the random variable

### Chapter 6: t test for dependent samples

Chapter 6: t test for dependent samples ****This chapter corresponds to chapter 11 of your book ( t(ea) for Two (Again) ). What it is: The t test for dependent samples is used to determine whether the

### Stat 411/511 THE RANDOMIZATION TEST. Charlotte Wickham. stat511.cwick.co.nz. Oct 16 2015

Stat 411/511 THE RANDOMIZATION TEST Oct 16 2015 Charlotte Wickham stat511.cwick.co.nz Today Review randomization model Conduct randomization test What about CIs? Using a t-distribution as an approximation

### STAT 145 (Notes) Al Nosedal anosedal@unm.edu Department of Mathematics and Statistics University of New Mexico. Fall 2013

STAT 145 (Notes) Al Nosedal anosedal@unm.edu Department of Mathematics and Statistics University of New Mexico Fall 2013 CHAPTER 18 INFERENCE ABOUT A POPULATION MEAN. Conditions for Inference about mean

### Introduction to Hypothesis Testing. Point estimation and confidence intervals are useful statistical inference procedures.

Introduction to Hypothesis Testing Point estimation and confidence intervals are useful statistical inference procedures. Another type of inference is used frequently used concerns tests of hypotheses.

### AP STATISTICS (Warm-Up Exercises)

AP STATISTICS (Warm-Up Exercises) 1. Describe the distribution of ages in a city: 2. Graph a box plot on your calculator for the following test scores: {90, 80, 96, 54, 80, 95, 100, 75, 87, 62, 65, 85,

### Chapter 11-12 1 Review

Chapter 11-12 Review Name 1. In formulating hypotheses for a statistical test of significance, the null hypothesis is often a statement of no effect or no difference. the probability of observing the data

### Testing Group Differences using T-tests, ANOVA, and Nonparametric Measures

Testing Group Differences using T-tests, ANOVA, and Nonparametric Measures Jamie DeCoster Department of Psychology University of Alabama 348 Gordon Palmer Hall Box 870348 Tuscaloosa, AL 35487-0348 Phone:

### Hypothesis Testing for a Proportion

Math 122 Intro to Stats Chapter 6 Semester II, 2015-16 Inference for Categorical Data Hypothesis Testing for a Proportion In a survey, 1864 out of 2246 randomly selected adults said texting while driving

### Calculating P-Values. Parkland College. Isela Guerra Parkland College. Recommended Citation

Parkland College A with Honors Projects Honors Program 2014 Calculating P-Values Isela Guerra Parkland College Recommended Citation Guerra, Isela, "Calculating P-Values" (2014). A with Honors Projects.

### Statistical Inference and t-tests

1 Statistical Inference and t-tests Objectives Evaluate the difference between a sample mean and a target value using a one-sample t-test. Evaluate the difference between a sample mean and a target value

### Section 13, Part 1 ANOVA. Analysis Of Variance

Section 13, Part 1 ANOVA Analysis Of Variance Course Overview So far in this course we ve covered: Descriptive statistics Summary statistics Tables and Graphs Probability Probability Rules Probability

### Recall this chart that showed how most of our course would be organized:

Chapter 4 One-Way ANOVA Recall this chart that showed how most of our course would be organized: Explanatory Variable(s) Response Variable Methods Categorical Categorical Contingency Tables Categorical

### THE FIRST SET OF EXAMPLES USE SUMMARY DATA... EXAMPLE 7.2, PAGE 227 DESCRIBES A PROBLEM AND A HYPOTHESIS TEST IS PERFORMED IN EXAMPLE 7.

THERE ARE TWO WAYS TO DO HYPOTHESIS TESTING WITH STATCRUNCH: WITH SUMMARY DATA (AS IN EXAMPLE 7.17, PAGE 236, IN ROSNER); WITH THE ORIGINAL DATA (AS IN EXAMPLE 8.5, PAGE 301 IN ROSNER THAT USES DATA FROM

### AP Statistics 2012 Scoring Guidelines

AP Statistics 2012 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

### The calculations lead to the following values: d 2 = 46, n = 8, s d 2 = 4, s d = 2, SEof d = s d n s d n

EXAMPLE 1: Paired t-test and t-interval DBP Readings by Two Devices The diastolic blood pressures (DBP) of 8 patients were determined using two techniques: the standard method used by medical personnel

### Comparing Means in Two Populations

Comparing Means in Two Populations Overview The previous section discussed hypothesis testing when sampling from a single population (either a single mean or two means from the same population). Now we

### Mind on Statistics. Chapter 13

Mind on Statistics Chapter 13 Sections 13.1-13.2 1. Which statement is not true about hypothesis tests? A. Hypothesis tests are only valid when the sample is representative of the population for the question

### Opgaven Onderzoeksmethoden, Onderdeel Statistiek

Opgaven Onderzoeksmethoden, Onderdeel Statistiek 1. What is the measurement scale of the following variables? a Shoe size b Religion c Car brand d Score in a tennis game e Number of work hours per week

### Statistics 2014 Scoring Guidelines

AP Statistics 2014 Scoring Guidelines College Board, Advanced Placement Program, AP, AP Central, and the acorn logo are registered trademarks of the College Board. AP Central is the official online home

### Understanding Confidence Intervals and Hypothesis Testing Using Excel Data Table Simulation

Understanding Confidence Intervals and Hypothesis Testing Using Excel Data Table Simulation Leslie Chandrakantha lchandra@jjay.cuny.edu Department of Mathematics & Computer Science John Jay College of

### AP Statistics 2001 Solutions and Scoring Guidelines

AP Statistics 2001 Solutions and Scoring Guidelines The materials included in these files are intended for non-commercial use by AP teachers for course and exam preparation; permission for any other use

### CHAPTER 11 CHI-SQUARE: NON-PARAMETRIC COMPARISONS OF FREQUENCY

CHAPTER 11 CHI-SQUARE: NON-PARAMETRIC COMPARISONS OF FREQUENCY The hypothesis testing statistics detailed thus far in this text have all been designed to allow comparison of the means of two or more samples

### Chapter 7 Notes - Inference for Single Samples. You know already for a large sample, you can invoke the CLT so:

Chapter 7 Notes - Inference for Single Samples You know already for a large sample, you can invoke the CLT so: X N(µ, ). Also for a large sample, you can replace an unknown σ by s. You know how to do a

### Hypothesis Testing or How to Decide to Decide Edpsy 580

Hypothesis Testing or How to Decide to Decide Edpsy 580 Carolyn J. Anderson Department of Educational Psychology University of Illinois at Urbana-Champaign Hypothesis Testing or How to Decide to Decide

### Homework 6 Solutions

Math 17, Section 2 Spring 2011 Assignment Chapter 20: 12, 14, 20, 24, 34 Chapter 21: 2, 8, 14, 16, 18 Chapter 20 20.12] Got Milk? The student made a number of mistakes here: Homework 6 Solutions 1. Null

### 1. What is the critical value for this 95% confidence interval? CV = z.025 = invnorm(0.025) = 1.96

1 Final Review 2 Review 2.1 CI 1-propZint Scenario 1 A TV manufacturer claims in its warranty brochure that in the past not more than 10 percent of its TV sets needed any repair during the first two years

### BA 275 Review Problems - Week 6 (10/30/06-11/3/06) CD Lessons: 53, 54, 55, 56 Textbook: pp. 394-398, 404-408, 410-420

BA 275 Review Problems - Week 6 (10/30/06-11/3/06) CD Lessons: 53, 54, 55, 56 Textbook: pp. 394-398, 404-408, 410-420 1. Which of the following will increase the value of the power in a statistical test

### An Introduction to Statistics Course (ECOE 1302) Spring Semester 2011 Chapter 10- TWO-SAMPLE TESTS

The Islamic University of Gaza Faculty of Commerce Department of Economics and Political Sciences An Introduction to Statistics Course (ECOE 130) Spring Semester 011 Chapter 10- TWO-SAMPLE TESTS Practice

### The Goodness-of-Fit Test

on the Lecture 49 Section 14.3 Hampden-Sydney College Tue, Apr 21, 2009 Outline 1 on the 2 3 on the 4 5 Hypotheses on the (Steps 1 and 2) (1) H 0 : H 1 : H 0 is false. (2) α = 0.05. p 1 = 0.24 p 2 = 0.20

### Math 108 Exam 3 Solutions Spring 00

Math 108 Exam 3 Solutions Spring 00 1. An ecologist studying acid rain takes measurements of the ph in 12 randomly selected Adirondack lakes. The results are as follows: 3.0 6.5 5.0 4.2 5.5 4.7 3.4 6.8

### Chapter 23. Inferences for Regression

Chapter 23. Inferences for Regression Topics covered in this chapter: Simple Linear Regression Simple Linear Regression Example 23.1: Crying and IQ The Problem: Infants who cry easily may be more easily

### Wilcoxon Rank Sum or Mann-Whitney Test Chapter 7.11

STAT Non-Parametric tests /0/0 Here s a summary of the tests we will look at: Setting Normal test NonParametric Test One sample One-sample t-test Sign Test Wilcoxon signed-rank test Matched pairs Apply

### Chapter 8. Hypothesis Testing

Chapter 8 Hypothesis Testing Hypothesis In statistics, a hypothesis is a claim or statement about a property of a population. A hypothesis test (or test of significance) is a standard procedure for testing

### COMPARISONS OF CUSTOMER LOYALTY: PUBLIC & PRIVATE INSURANCE COMPANIES.

277 CHAPTER VI COMPARISONS OF CUSTOMER LOYALTY: PUBLIC & PRIVATE INSURANCE COMPANIES. This chapter contains a full discussion of customer loyalty comparisons between private and public insurance companies

### 7.1 Inference for comparing means of two populations

Objectives 7.1 Inference for comparing means of two populations Matched pair t confidence interval Matched pair t hypothesis test http://onlinestatbook.com/2/tests_of_means/correlated.html Overview of

### Answer: C. The strength of a correlation does not change if units change by a linear transformation such as: Fahrenheit = 32 + (5/9) * Centigrade

Statistics Quiz Correlation and Regression -- ANSWERS 1. Temperature and air pollution are known to be correlated. We collect data from two laboratories, in Boston and Montreal. Boston makes their measurements

Chapter 8 Hypothesis Tests Chapter Table of Contents Introduction...157 One-Sample t-test...158 Paired t-test...164 Two-Sample Test for Proportions...169 Two-Sample Test for Variances...172 Discussion

### Two-sample inference: Continuous data

Two-sample inference: Continuous data Patrick Breheny April 5 Patrick Breheny STA 580: Biostatistics I 1/32 Introduction Our next two lectures will deal with two-sample inference for continuous data As

### STAT 360 Probability and Statistics. Fall 2012

STAT 360 Probability and Statistics Fall 2012 1) General information: Crosslisted course offered as STAT 360, MATH 360 Semester: Fall 2012, Aug 20--Dec 07 Course name: Probability and Statistics Number

### 12 Hypothesis Testing

CHAPTER 12 Hypothesis Testing Chapter Outline 12.1 HYPOTHESIS TESTING 12.2 CRITICAL VALUES 12.3 ONE-SAMPLE T TEST 247 12.1. Hypothesis Testing www.ck12.org 12.1 Hypothesis Testing Learning Objectives Develop

### Descriptive Statistics

Descriptive Statistics Primer Descriptive statistics Central tendency Variation Relative position Relationships Calculating descriptive statistics Descriptive Statistics Purpose to describe or summarize

### Introduction. Hypothesis Testing. Hypothesis Testing. Significance Testing

Introduction Hypothesis Testing Mark Lunt Arthritis Research UK Centre for Ecellence in Epidemiology University of Manchester 13/10/2015 We saw last week that we can never know the population parameters

### Hypothesis Testing. Steps for a hypothesis test:

Hypothesis Testing Steps for a hypothesis test: 1. State the claim H 0 and the alternative, H a 2. Choose a significance level or use the given one. 3. Draw the sampling distribution based on the assumption

### 6: Introduction to Hypothesis Testing

6: Introduction to Hypothesis Testing Significance testing is used to help make a judgment about a claim by addressing the question, Can the observed difference be attributed to chance? We break up significance

### Chapter 5 Analysis of variance SPSS Analysis of variance

Chapter 5 Analysis of variance SPSS Analysis of variance Data file used: gss.sav How to get there: Analyze Compare Means One-way ANOVA To test the null hypothesis that several population means are equal,

### Study Guide for the Final Exam

Study Guide for the Final Exam When studying, remember that the computational portion of the exam will only involve new material (covered after the second midterm), that material from Exam 1 will make

### Nonparametric Two-Sample Tests. Nonparametric Tests. Sign Test

Nonparametric Two-Sample Tests Sign test Mann-Whitney U-test (a.k.a. Wilcoxon two-sample test) Kolmogorov-Smirnov Test Wilcoxon Signed-Rank Test Tukey-Duckworth Test 1 Nonparametric Tests Recall, nonparametric

### Paired 2 Sample t-test

Variations of the t-test: Paired 2 Sample 1 Paired 2 Sample t-test Suppose we are interested in the effect of different sampling strategies on the quality of data we recover from archaeological field surveys.

### Two-sample hypothesis testing, II 9.07 3/16/2004

Two-sample hypothesis testing, II 9.07 3/16/004 Small sample tests for the difference between two independent means For two-sample tests of the difference in mean, things get a little confusing, here,

### CHAPTER 13 SIMPLE LINEAR REGRESSION. Opening Example. Simple Regression. Linear Regression

Opening Example CHAPTER 13 SIMPLE LINEAR REGREION SIMPLE LINEAR REGREION! Simple Regression! Linear Regression Simple Regression Definition A regression model is a mathematical equation that descries the

### Summary A Contemporary Study of Factors Influencing Urban and Rural Consumers for Buying Different Life Insurance Policies in Haryana.

Summary The topic of research was A Contemporary Study of Factors Influencing Urban and Rural Consumers for Buying Different Life Insurance Policies in Haryana. Summary of the thesis presents an overview

### Simple Linear Regression Inference

Simple Linear Regression Inference 1 Inference requirements The Normality assumption of the stochastic term e is needed for inference even if it is not a OLS requirement. Therefore we have: Interpretation

### Single sample hypothesis testing, II 9.07 3/02/2004

Single sample hypothesis testing, II 9.07 3/02/2004 Outline Very brief review One-tailed vs. two-tailed tests Small sample testing Significance & multiple tests II: Data snooping What do our results mean?

### Unit 26: Small Sample Inference for One Mean

Unit 26: Small Sample Inference for One Mean Prerequisites Students need the background on confidence intervals and significance tests covered in Units 24 and 25. Additional Topic Coverage Additional coverage

### Data Analysis Tools. Tools for Summarizing Data

Data Analysis Tools This section of the notes is meant to introduce you to many of the tools that are provided by Excel under the Tools/Data Analysis menu item. If your computer does not have that tool

### Sections 4.5-4.7: Two-Sample Problems. Paired t-test (Section 4.6)

Sections 4.5-4.7: Two-Sample Problems Paired t-test (Section 4.6) Examples of Paired Differences studies: Similar subjects are paired off and one of two treatments is given to each subject in the pair.

### Inference for two Population Means

Inference for two Population Means Bret Hanlon and Bret Larget Department of Statistics University of Wisconsin Madison October 27 November 1, 2011 Two Population Means 1 / 65 Case Study Case Study Example

### Math 1070 Exam 2B 22 March, 2013

Math 1070 Exam 2B 22 March, 2013 This exam will last 50 minutes and consists of 13 multiple choice and 6 free response problems. Write your answers in the space provided. All solutions must be sufficiently

### MATH 140 HYBRID INTRODUCTORY STATISTICS COURSE SYLLABUS

MATH 140 HYBRID INTRODUCTORY STATISTICS COURSE SYLLABUS Instructor: Mark Schilling Email: mark.schilling@csun.edu (Note: If your CSUN email address is not one you use regularly, be sure to set up automatic

### Tutorial 5: Hypothesis Testing

Tutorial 5: Hypothesis Testing Rob Nicholls nicholls@mrc-lmb.cam.ac.uk MRC LMB Statistics Course 2014 Contents 1 Introduction................................ 1 2 Testing distributional assumptions....................

### Stats Review Chapters 9-10

Stats Review Chapters 9-10 Created by Teri Johnson Math Coordinator, Mary Stangler Center for Academic Success Examples are taken from Statistics 4 E by Michael Sullivan, III And the corresponding Test