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

Size: px
Start display at page:

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

Transcription

1 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 in Chapter 6, when we knew the population standard deviation Confidence interval for the population mean : x z* X n Hypothesis test statistic for the population mean : x 0 z0 / n Used the standard normal distribution. In this instance, we have to use an estimate of, the sample standard deviation s. Using s instead of means that we are no longer able to use the standard normal distribution. Instead, we will have to use the student s t distribution. The student s t distribution is completely determined by the number of degrees of freedom. When looking at the distribution of X, we will use the t distribution with n 1 degrees of freedom, the t(n 1) distribution. Using the t distribution Suppose that an SRS of size n is drawn from a N(, ) population. There is a different t distribution for each sample size, so t(k) stands for the t distribution with k degrees of freedom. Degrees of freedom = k = n 1 = sample size 1 As k increases, the t distribution looks more like the normal distribution (because as n increases, s ). t(k) distributions are symmetric about 0 and are bell shaped, they are just a bit wider than the normal distribution. The t table shows upper tails only, so o if t * is negative, P(t < t * ) = P(t > t * ). o if you have a 2 sided test, multiply the P(t > t * ) by 2 to get the area in both tails. o The normal table showed lower tails only, so the t table is backwards. 1

2 2

3 The One Sample t Confidence Interval: xt* s n where t* is the value for the t(n 1) density curve with area C between t* and t*. Finding t* on the table: Start at the bottom line to get the right column for your confidence level, and then work up to the correct row for your degrees of freedom. What happens if your degrees of freedom isn t on the table, for example df = 79? Always round DOWN to the next lowest degrees of freedom to be conservative. Example 1 a) Find t* for an 80% confidence interval if the sample size is 20. b) Find t* for an 98% confidence interval if the sample size is 35. c) Find a 95% confidence interval for the population mean if the sample mean is 42, the sample standard deviation is 1.9, and the sample size is 50. 3

4 The One Sample t test: State the Null and Alternative hypothesis. Find the test statistic: 0 t x s/ n Calculate the p value. In terms of a random variable T having the t(n 1) distribution, the P value for a test of H 0 against Ha : 0 is PT ( t) Ha : 0 is P( T t) Ha : 0 is 2 P( T t ) These P values are exact if the population distribution is normal and are approximately correct for large n in other cases. Compare the P value to the α level. If P value α, then reject H 0 (significant results) If P value > α, then fail to reject H 0 (non significant results) State conclusions in terms of the problem. Reject / Do not reject the null hypothesis. There is / is not enough evidence that the population average is. Robustness of the t Procedures A confidence interval or statistical test is called robust if the confidence level or P value does not change very much when the assumptions of the procedure are violated. The t procedures are robust against non normality of the population when there are no outliers, especially when the distribution is roughly symmetric and unimodal. When is it appropriate to use the t procedures? Unless a small sample is used, the assumption that the data comes from a SRS is more important than the assumption that the population distribution is normal. n<15: Use t procedures only if the data are close to normal with no outliers. n>15: The t procedure can be used except in the presence of outliers or strong skewness. n is large (n 40): The t procedure can be used even for clearly skewed distributions. You can check to see if data is normally distributed using a histogram or normal quantile plot. 4

5 Example 2 a. An agricultural expert performs a study to measure yield on a tomato field. Studying 10 plots of land, she finds the mean yield is 34 bushels with a sample standard deviation of Find a 95% confidence interval for the unknown mean yield of tomatoes. b. Conduct a hypothesis test with = 0.05 to determine if the mean yield of tomatoes is less than 42 bushels. State your conclusion in terms of the story. c. Draw a picture of the t curve with the number and symbol for the mean you used in your null s hypothesis ( 0 ), the sample mean ( x ), the standard error ( ˆ x ), t = 0, and the test statistic n (t 0 ). Shade the appropriate part of the curve which shows the P value. 5

6 Example 3 (Exercise 7.37) How accurate are radon detectors of a type sold to homeowners? To answer this question, university researchers placed 12 detectors in a chamber that exposed them to 105 picocuries per liter of radon. The detector readings were as follows: a) Is there convincing evidence that the mean reading of all detectors of this type differs from the true value of 105? Use = 0.10 for the test. Carry out a test in detail and write a brief conclusion. (SPSS tells us the mean and standard deviation of the sample data are and 9.40, respectively.) b) Find a 90% confidence interval for the population mean. Now re do the above example using SPSS completely. To do just a confidence interval: enter data, then AnalyzeDescriptive Statistics Explore. Click on Statistics and change the CI to 90%. Then hit OK. If you need to do a hypothesis test and a CI, go to AnalyzeCompare Means One sample T test. Change the test value to 105 (since that is our H 0 ), change options to 90%, and hit OK. (This will give you the output below.) One Sample Test Test Value = 105 t df Sig. (2 tailed) Mean Difference 90% Confidence Interval of the Difference Lower Upper radon detector readings

7 H : H : 105 a H : H : 105 a H : H : 105 a You must choose your hypotheses BEFORE you examine the data. When in doubt, do a two sided test. 7

8 Normal quantile plots In SPSS, go to Graphs Q Q. Move your variable into variable column and hit OK. Normal Q-Q Plot of Radon Detector Reading 120 Expected Normal Value Observed Value Look to see how closely the data points (dots) follow the diagonal line. The line will always be a 45 degree line. Only the data points will change. The closer they follow the line, the more normally distributed the data is. What happens if the t procedure is not appropriate? What if you have outliers or skewness with a smaller sample size (n < 40)? Outliers: Investigate the cause of the outlier(s). o o Was the data recorded correctly? Is there any reason why that data might be invalid (an equipment malfunction, a person lying in their response, etc.)? If there is a good reason why that point could be disregarded, try taking it out and compare the new confidence interval or hypothesis test results to the old ones. If you don t have a valid reason for disregarding the outlier, you have to the outlier in and not use the t procedures. Skewness: o If the skewness is not too extreme, the t procedures are still appropriate if the sample size is bigger than 15. If the skewness is extreme or if the sample size is less than 15, you can use nonparametric procedures. One type of nonparametric test is similar to the t procedures except it uses the median instead of the mean. Another possibility would be to transform the data, possibly using logarithms. A statistician should be consulted if you have data which doesn t fit the t procedures requirements. We won t cover nonparametric procedures or transformations for non normal data in this course, but your book has supplementary chapters (14 and 15) on these topics online if you need them later in your own research. They are also discussed on pages of your book. 8

9 What do you do when you have 2 lists of data instead of 1? First decide whether you have 1 sample with 2 measurements OR 2 independent samples with one measurement each. 1. Matched Pairs (covered in 7.1) One group of individuals with 2 different measurements on them Same individuals, different measurements Examples: pre and post tests, before and after measurements Based on the difference obtained between the 2 measurements 1) Find the difference = post test pre test (or before after, etc.), in the individual measurements. 2) Find the sample mean d and sample standard deviation s of these differences. 3) Use the t distribution because the standard deviation is estimated from the data. Confidence interval: d t * s n Hypothesis testing: H 0 : diff = 0 d 0 t test statistic: t0 s/ n Conclusion: Reject / Do Not Reject H 0. There is / There is not enough evidence that the population mean difference in is. 9

10 Example 4 (Exercise 7.31) Researchers are interested in whether Vitamin C is lost when wheat soy blend (CSB) is cooked as gruel. Samples of gruel were collected, and the vitamin C content was measured (in mg per 100 grams of gruel) before and after cooking. Here are the results: Sample Mean St. Dev. Before After Before After a) Set up an appropriate hypothesis test and carry it out for these data. State your conclusions in a sentence. Use α=.10. b) Find a 90% confidence interval for the mean vitamin C content loss. 10

11 Example 5 In an effort to determine whether sensitivity training for nurses would improve the quality of nursing provided at an area hospital, eight different nurses were selected and their nursing skills were given a score from After this initial screening, a training program was administered, then the same nurses were rated again. Below is a table of their pre and post training scores. Conduct a test to determine whether training improved the quality of nursing provided. Individuals Pre training score Post training score Enter the pre and post training scores to SPSS. Then AnalyzeCompare MeansPaired Samples T test. Then input both variable names and hit the arrow key. If you need to change the confidence interval, go to Options. SPSS will always do the left column of data the right column of data for the order of the difference. If this bothers you, just be careful how you enter the data into the program Paired Samples Statistics Pair 1 Post-training score Pre-training score Std. Error Mean N Std. Deviation Mean

12 Data entered as written above with pre training in left column and post training in right column: Paired Samples Test Paired Differences t df Sig. (2 tailed) Mean Std. Deviation Std. Error Mean 95% Confidence Interval of the Difference Lower Upper Pair 1 pretraining posttraining Data entered backwards from how it is written above with post training in left column and pre training in right column: Paired Samples Test Pair 1 Post-training score - Pre-training score Mean Paired Differences 95% Confidence Interval of the Std. Error Difference Std. Deviation Mean Lower Upper t df Sig. (2-tailed) What s different? What s the same? Which one matches the way that you defined diff? a. What are your hypotheses? b. What is the test statistic? c. What is the P value? d. What is your conclusion in terms of the story if α=.05? e. What is the 95% confidence interval of the difference in nursing scores? 12

13 2 Sample Comparison of Means (covered in 7.2) A group of individuals is divided into 2 different experimental groups Each group has different individuals who may receive different treatments Responses from each sample are independent of each other. Examples: treatment vs. control groups, male vs. female, 2 groups of different women Hypothesis Test H 0 : A = B (same as H 0 : A B = 0) H a : A > B or H a : A < B or H a : A B (pick one) 2 Sample t Test Statistic is used for hypothesis testing when the standard deviations are ESTIMATED from the data (these are approximately t distributions, but not exact) ( xa xb) t0 ~ t distribution with df = min ( n 1, 1) 2 2 A nb sa sb n n A B Conclusion: Reject / Do Not Reject H 0. There is / There is not enough evidence that the difference between the population means for and is. Confidence Interval for A B : 2 2 * sa sb * ( x A x ) B t where t ~t distribution with df = min na 1, nb 1 n A n B ***Equal sample sizes are recommended, but not required. Assumptions for Comparing Two Means 1. Two independent random samples from two distinct populations or two treatment groups from randomized comparative experiments are compared. The same variable is measured on both samples. The assumption of independence says that one sample has no influence on the other. 2. Both populations are normally distributed. 3. The means 1 and 2 and standard deviations 1 and 2 of both populations are unknown. 13

14 Robustness of the Two Sample t Procedures The two sample t procedures are more robust than the one sample t methods, particularly when the distributions are not symmetric. They are robust in the following circumstances: If two samples are of equal size and the two populations that the samples come from have similar distributions, then the t distribution is accurate for a variety of distributions, even when the sample sizes are as small as n n When the two population distributions are different, larger samples are needed. If n 1 n2 15 : Use two sample t procedures if the data are close to normal. If the data are clearly non normal or if outliers are present, do not use t. If 15 n 1 n2 40 : The t procedures can be used except in the presence of outliers or strong skewness. If n n 40 : The t procedures can be used even for clearly skewed distributions 1 2 Example 6 A group of 15 college seniors are selected to participate in a manual dexterity skill test against a group of 20 industrial workers. Skills are assessed by scores obtained on a test taken by both groups. Conduct a 5% alpha hypothesis test to determine whether the industrial workers had better manual dexterity skills than the students. Descriptive statistics are listed below. Also construct a 95% confidence interval for this problem. group n x s df students workers

15 Example 7 (Exercise 7.84) The SSHA is a psychological test designed to measure the motivation, study habits, and attitudes towards learning of college students. These factors, along with ability, are important in explaining success in school. A selective private college gives the SSHA to an SRS of both male and female first year students. The data for the women are as follows: Here are the scores for the men: a) Test whether the mean SSHA score for men is different than the mean score for women. State your hypotheses, carry out the test using SPSS, obtain a P value, and give your conclusions. Use a 10% significance level. When you enter your data into SPSS, have 2 variables: gender (type: string) and score (numeric). In the gender column, state whether a score is from a man or a woman, and in the score column, state all 38 scores. AnalyzeCompare MeansIndependent Samples T Test. Move score into Test Variable(s) box. Move gender into Grouping Variable box, and then click Define Groups and state which woman and man as group 1 and group 2, hit Continue. We will need a 90% confidence interval in part c, so go to Options to change it. Group Statistics gender N Mean Std. Deviation Std. Error Mean score woman man

16 score Equal variances assumed Equal variances not assumed Independent Samples Test Levene's Test for Equality of Variances t-test for Equality of Means 90% Confidence Interval of the Mean Std. Error Difference F Sig. t df Sig. (2-tailed) Difference Difference Lower Upper What do we do with this Equal variances assumed and Equal variances not assumed? Always go with the bottom row, Equal variances not assumed. This is the more conservative approach. b.) Most studies have found that the mean SSHA score for men is lower than the mean score in a comparable group of women. Test this supposition here. c.) Give a 90% confidence interval for the difference in means of SSHA scores of male and female first year students at this college. 16

### Independent t- Test (Comparing Two Means)

Independent t- Test (Comparing Two Means) The objectives of this lesson are to learn: the definition/purpose of independent t-test when to use the independent t-test the use of SPSS to complete an independent

### SCHOOL OF HEALTH AND HUMAN SCIENCES DON T FORGET TO RECODE YOUR MISSING VALUES

SCHOOL OF HEALTH AND HUMAN SCIENCES Using SPSS Topics addressed today: 1. Differences between groups 2. Graphing Use the s4data.sav file for the first part of this session. DON T FORGET TO RECODE YOUR

### 7. Comparing Means Using t-tests.

7. Comparing Means Using t-tests. Objectives Calculate one sample t-tests Calculate paired samples t-tests Calculate independent samples t-tests Graphically represent mean differences In this chapter,

### 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

### Two Related Samples t Test

Two Related Samples t Test In this example 1 students saw five pictures of attractive people and five pictures of unattractive people. For each picture, the students rated the friendliness of the person

### HYPOTHESIS TESTING WITH SPSS:

HYPOTHESIS TESTING WITH SPSS: A NON-STATISTICIAN S GUIDE & TUTORIAL by Dr. Jim Mirabella SPSS 14.0 screenshots reprinted with permission from SPSS Inc. Published June 2006 Copyright Dr. Jim Mirabella CHAPTER

### 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

### UNDERSTANDING THE INDEPENDENT-SAMPLES t TEST

UNDERSTANDING The independent-samples t test evaluates the difference between the means of two independent or unrelated groups. That is, we evaluate whether the means for two independent groups are significantly

### 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

### Chapter 2 Probability Topics SPSS T tests

Chapter 2 Probability Topics SPSS T tests Data file used: gss.sav In the lecture about chapter 2, only the One-Sample T test has been explained. In this handout, we also give the SPSS methods to perform

### 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

### 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,

### 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

### Chapter 23 Inferences About Means

Chapter 23 Inferences About Means Chapter 23 - Inferences About Means 391 Chapter 23 Solutions to Class Examples 1. See Class Example 1. 2. We want to know if the mean battery lifespan exceeds the 300-minute

### INTERPRETING THE ONE-WAY ANALYSIS OF VARIANCE (ANOVA)

INTERPRETING THE ONE-WAY ANALYSIS OF VARIANCE (ANOVA) As with other parametric statistics, we begin the one-way ANOVA with a test of the underlying assumptions. Our first assumption is the assumption of

### Data analysis process

Data analysis process Data collection and preparation Collect data Prepare codebook Set up structure of data Enter data Screen data for errors Exploration of data Descriptive Statistics Graphs Analysis

### Introduction to Analysis of Variance (ANOVA) Limitations of the t-test

Introduction to Analysis of Variance (ANOVA) The Structural Model, The Summary Table, and the One- Way ANOVA Limitations of the t-test Although the t-test is commonly used, it has limitations Can only

### 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

### 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

### 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

### Chapter 7. One-way ANOVA

Chapter 7 One-way ANOVA One-way ANOVA examines equality of population means for a quantitative outcome and a single categorical explanatory variable with any number of levels. The t-test of Chapter 6 looks

### Density Curve. A density curve is the graph of a continuous probability distribution. It must satisfy the following properties:

Density Curve A density curve is the graph of a continuous probability distribution. It must satisfy the following properties: 1. The total area under the curve must equal 1. 2. Every point on the curve

### TI-Inspire manual 1. Instructions. Ti-Inspire for statistics. General Introduction

TI-Inspire manual 1 General Introduction Instructions Ti-Inspire for statistics TI-Inspire manual 2 TI-Inspire manual 3 Press the On, Off button to go to Home page TI-Inspire manual 4 Use the to navigate

### NCSS Statistical Software

Chapter 06 Introduction This procedure provides several reports for the comparison of two distributions, including confidence intervals for the difference in means, two-sample t-tests, the z-test, the

### 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

### Lesson 1: Comparison of Population Means Part c: Comparison of Two- Means

Lesson : Comparison of Population Means Part c: Comparison of Two- Means Welcome to lesson c. This third lesson of lesson will discuss hypothesis testing for two independent means. Steps in Hypothesis

### Chapter 7. Comparing Means in SPSS (t-tests) Compare Means analyses. Specifically, we demonstrate procedures for running Dependent-Sample (or

1 Chapter 7 Comparing Means in SPSS (t-tests) This section covers procedures for testing the differences between two means using the SPSS Compare Means analyses. Specifically, we demonstrate procedures

### Introduction. Chapter 14: Nonparametric Tests

2 Chapter 14: Nonparametric Tests Introduction robustness outliers transforming data other standard distributions nonparametric methods rank tests The most commonly used methods for inference about the

### 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

### 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

### The Dummy s Guide to Data Analysis Using SPSS

The Dummy s Guide to Data Analysis Using SPSS Mathematics 57 Scripps College Amy Gamble April, 2001 Amy Gamble 4/30/01 All Rights Rerserved TABLE OF CONTENTS PAGE Helpful Hints for All Tests...1 Tests

### 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:

### An analysis method for a quantitative outcome and two categorical explanatory variables.

Chapter 11 Two-Way ANOVA An analysis method for a quantitative outcome and two categorical explanatory variables. If an experiment has a quantitative outcome and two categorical explanatory variables that

### Examining Differences (Comparing Groups) using SPSS Inferential statistics (Part I) Dwayne Devonish

Examining Differences (Comparing Groups) using SPSS Inferential statistics (Part I) Dwayne Devonish Statistics Statistics are quantitative methods of describing, analysing, and drawing inferences (conclusions)

### 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

### How To Test For Significance On A Data Set

Non-Parametric Univariate Tests: 1 Sample Sign Test 1 1 SAMPLE SIGN TEST A non-parametric equivalent of the 1 SAMPLE T-TEST. ASSUMPTIONS: Data is non-normally distributed, even after log transforming.

### 13: Additional ANOVA Topics. Post hoc Comparisons

13: Additional ANOVA Topics Post hoc Comparisons ANOVA Assumptions Assessing Group Variances When Distributional Assumptions are Severely Violated Kruskal-Wallis Test Post hoc Comparisons In the prior

### Chapter 9. Two-Sample Tests. Effect Sizes and Power Paired t Test Calculation

Chapter 9 Two-Sample Tests Paired t Test (Correlated Groups t Test) Effect Sizes and Power Paired t Test Calculation Summary Independent t Test Chapter 9 Homework Power and Two-Sample Tests: Paired Versus

### Descriptive Statistics

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

### p ˆ (sample mean and sample

Chapter 6: Confidence Intervals and Hypothesis Testing When analyzing data, we can t just accept the sample mean or sample proportion as the official mean or proportion. When we estimate the statistics

### 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

### 4. Descriptive Statistics: Measures of Variability and Central Tendency

4. Descriptive Statistics: Measures of Variability and Central Tendency Objectives Calculate descriptive for continuous and categorical data Edit output tables Although measures of central tendency and

### A POPULATION MEAN, CONFIDENCE INTERVALS AND HYPOTHESIS TESTING

CHAPTER 5. A POPULATION MEAN, CONFIDENCE INTERVALS AND HYPOTHESIS TESTING 5.1 Concepts When a number of animals or plots are exposed to a certain treatment, we usually estimate the effect of the treatment

### Normality Testing in Excel

Normality Testing 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

### 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.

### Final Exam Practice Problem Answers

Final Exam Practice Problem Answers The following data set consists of data gathered from 77 popular breakfast cereals. The variables in the data set are as follows: Brand: The brand name of the cereal

### Draft 1, Attempted 2014 FR Solutions, AP Statistics Exam

Free response questions, 2014, first draft! Note: Some notes: Please make critiques, suggest improvements, and ask questions. This is just one AP stats teacher s initial attempts at solving these. I, as

### THE KRUSKAL WALLLIS TEST

THE KRUSKAL WALLLIS TEST TEODORA H. MEHOTCHEVA Wednesday, 23 rd April 08 THE KRUSKAL-WALLIS TEST: The non-parametric alternative to ANOVA: testing for difference between several independent groups 2 NON

### 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

### Projects Involving Statistics (& SPSS)

Projects Involving Statistics (& SPSS) Academic Skills Advice Starting a project which involves using statistics can feel confusing as there seems to be many different things you can do (charts, graphs,

### Simple Regression Theory II 2010 Samuel L. Baker

SIMPLE REGRESSION THEORY II 1 Simple Regression Theory II 2010 Samuel L. Baker Assessing how good the regression equation is likely to be Assignment 1A gets into drawing inferences about how close the

### November 08, 2010. 155S8.6_3 Testing a Claim About a Standard Deviation or Variance

Chapter 8 Hypothesis Testing 8 1 Review and Preview 8 2 Basics of Hypothesis Testing 8 3 Testing a Claim about a Proportion 8 4 Testing a Claim About a Mean: σ Known 8 5 Testing a Claim About a Mean: σ

### The Chi-Square Test. STAT E-50 Introduction to Statistics

STAT -50 Introduction to Statistics The Chi-Square Test The Chi-square test is a nonparametric test that is used to compare experimental results with theoretical models. That is, we will be comparing observed

### DDBA 8438: The t Test for Independent Samples Video Podcast Transcript

DDBA 8438: The t Test for Independent Samples Video Podcast Transcript JENNIFER ANN MORROW: Welcome to The t Test for Independent Samples. My name is Dr. Jennifer Ann Morrow. In today's demonstration,

### One-Way Analysis of Variance

One-Way Analysis of Variance Note: Much of the math here is tedious but straightforward. We ll skim over it in class but you should be sure to ask questions if you don t understand it. I. Overview A. We

### KSTAT MINI-MANUAL. Decision Sciences 434 Kellogg Graduate School of Management

KSTAT MINI-MANUAL Decision Sciences 434 Kellogg Graduate School of Management Kstat is a set of macros added to Excel and it will enable you to do the statistics required for this course very easily. To

### NCSS Statistical Software

Chapter 06 Introduction This procedure provides several reports for the comparison of two distributions, including confidence intervals for the difference in means, two-sample t-tests, the z-test, the

### Introduction to Statistics with SPSS (15.0) Version 2.3 (public)

Babraham Bioinformatics Introduction to Statistics with SPSS (15.0) Version 2.3 (public) Introduction to Statistics with SPSS 2 Table of contents Introduction... 3 Chapter 1: Opening SPSS for the first

### P(every one of the seven intervals covers the true mean yield at its location) = 3.

1 Let = number of locations at which the computed confidence interval for that location hits the true value of the mean yield at its location has a binomial(7,095) (a) P(every one of the seven intervals

### Week 4: Standard Error and Confidence Intervals

Health Sciences M.Sc. Programme Applied Biostatistics Week 4: Standard Error and Confidence Intervals Sampling Most research data come from subjects we think of as samples drawn from a larger population.

### Lecture Notes Module 1

Lecture Notes Module 1 Study Populations A study population is a clearly defined collection of people, animals, plants, or objects. In psychological research, a study population usually consists of a specific

### In the past, the increase in the price of gasoline could be attributed to major national or global

Chapter 7 Testing Hypotheses Chapter Learning Objectives Understanding the assumptions of statistical hypothesis testing Defining and applying the components in hypothesis testing: the research and null

### 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

### HYPOTHESIS TESTING: CONFIDENCE INTERVALS, T-TESTS, ANOVAS, AND REGRESSION

HYPOTHESIS TESTING: CONFIDENCE INTERVALS, T-TESTS, ANOVAS, AND REGRESSION HOD 2990 10 November 2010 Lecture Background This is a lightning speed summary of introductory statistical methods for senior undergraduate

### Class 19: Two Way Tables, Conditional Distributions, Chi-Square (Text: Sections 2.5; 9.1)

Spring 204 Class 9: Two Way Tables, Conditional Distributions, Chi-Square (Text: Sections 2.5; 9.) Big Picture: More than Two Samples In Chapter 7: We looked at quantitative variables and compared the

### UNDERSTANDING THE DEPENDENT-SAMPLES t TEST

UNDERSTANDING THE DEPENDENT-SAMPLES t TEST A dependent-samples t test (a.k.a. matched or paired-samples, matched-pairs, samples, or subjects, simple repeated-measures or within-groups, or correlated groups)

### UNDERSTANDING THE TWO-WAY ANOVA

UNDERSTANDING THE e have seen how the one-way ANOVA can be used to compare two or more sample means in studies involving a single independent variable. This can be extended to two independent variables

### One-Way ANOVA using SPSS 11.0. SPSS ANOVA procedures found in the Compare Means analyses. Specifically, we demonstrate

1 One-Way ANOVA using SPSS 11.0 This section covers steps for testing the difference between three or more group means using the SPSS ANOVA procedures found in the Compare Means analyses. Specifically,

### How To Run Statistical Tests in Excel

How To Run Statistical Tests in Excel Microsoft Excel is your best tool for storing and manipulating data, calculating basic descriptive statistics such as means and standard deviations, and conducting

### Chapter 1: Looking at Data Section 1.1: Displaying Distributions with Graphs

Types of Variables Chapter 1: Looking at Data Section 1.1: Displaying Distributions with Graphs Quantitative (numerical)variables: take numerical values for which arithmetic operations make sense (addition/averaging)

### 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....................

### How To Check For Differences In The One Way Anova

MINITAB ASSISTANT WHITE PAPER This paper explains the research conducted by Minitab statisticians to develop the methods and data checks used in the Assistant in Minitab 17 Statistical Software. One-Way

### Introduction to. Hypothesis Testing CHAPTER LEARNING OBJECTIVES. 1 Identify the four steps of hypothesis testing.

Introduction to Hypothesis Testing CHAPTER 8 LEARNING OBJECTIVES After reading this chapter, you should be able to: 1 Identify the four steps of hypothesis testing. 2 Define null hypothesis, alternative

### 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

### 12: Analysis of Variance. Introduction

1: Analysis of Variance Introduction EDA Hypothesis Test Introduction In Chapter 8 and again in Chapter 11 we compared means from two independent groups. In this chapter we extend the procedure to consider

### EPS 625 INTERMEDIATE STATISTICS FRIEDMAN TEST

EPS 625 INTERMEDIATE STATISTICS The Friedman test is an extension of the Wilcoxon test. The Wilcoxon test can be applied to repeated-measures data if participants are assessed on two occasions or conditions

### 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

### Results from the 2014 AP Statistics Exam. Jessica Utts, University of California, Irvine Chief Reader, AP Statistics jutts@uci.edu

Results from the 2014 AP Statistics Exam Jessica Utts, University of California, Irvine Chief Reader, AP Statistics jutts@uci.edu The six free-response questions Question #1: Extracurricular activities

### CALCULATIONS & STATISTICS

CALCULATIONS & STATISTICS CALCULATION OF SCORES Conversion of 1-5 scale to 0-100 scores When you look at your report, you will notice that the scores are reported on a 0-100 scale, even though respondents

### Why Taking This Course? Course Introduction, Descriptive Statistics and Data Visualization. Learning Goals. GENOME 560, Spring 2012

Why Taking This Course? Course Introduction, Descriptive Statistics and Data Visualization GENOME 560, Spring 2012 Data are interesting because they help us understand the world Genomics: Massive Amounts

### Chi Square Tests. Chapter 10. 10.1 Introduction

Contents 10 Chi Square Tests 703 10.1 Introduction............................ 703 10.2 The Chi Square Distribution.................. 704 10.3 Goodness of Fit Test....................... 709 10.4 Chi Square

### Regression Analysis: A Complete Example

Regression Analysis: A Complete Example This section works out an example that includes all the topics we have discussed so far in this chapter. A complete example of regression analysis. PhotoDisc, Inc./Getty

### January 26, 2009 The Faculty Center for Teaching and Learning

THE BASICS OF DATA MANAGEMENT AND ANALYSIS A USER GUIDE January 26, 2009 The Faculty Center for Teaching and Learning THE BASICS OF DATA MANAGEMENT AND ANALYSIS Table of Contents Table of Contents... i

### CONTINGENCY TABLES ARE NOT ALL THE SAME David C. Howell University of Vermont

CONTINGENCY TABLES ARE NOT ALL THE SAME David C. Howell University of Vermont To most people studying statistics a contingency table is a contingency table. We tend to forget, if we ever knew, that contingency

### 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

### 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

### Main Effects and Interactions

Main Effects & Interactions page 1 Main Effects and Interactions So far, we ve talked about studies in which there is just one independent variable, such as violence of television program. You might randomly

### Stata Walkthrough 4: Regression, Prediction, and Forecasting

Stata Walkthrough 4: Regression, Prediction, and Forecasting Over drinks the other evening, my neighbor told me about his 25-year-old nephew, who is dating a 35-year-old woman. God, I can t see them getting

### Point Biserial Correlation Tests

Chapter 807 Point Biserial Correlation Tests Introduction The point biserial correlation coefficient (ρ in this chapter) is the product-moment correlation calculated between a continuous random variable

### = \$96 = \$24. (b) The degrees of freedom are. s n. 7.3. For the mean monthly rent, the 95% confidence interval for µ is

Chapter 7 Solutions 71 (a) The standard error of the mean is df = n 1 = 15 s n = \$96 = \$24 (b) The degrees of freedom are 16 72 In each case, use df = n 1; if that number is not in Table D, drop to the

### IBM SPSS Statistics for Beginners for Windows

ISS, NEWCASTLE UNIVERSITY IBM SPSS Statistics for Beginners for Windows A Training Manual for Beginners Dr. S. T. Kometa A Training Manual for Beginners Contents 1 Aims and Objectives... 3 1.1 Learning

### Using Excel for inferential statistics

FACT SHEET Using Excel for inferential statistics Introduction When you collect data, you expect a certain amount of variation, just caused by chance. A wide variety of statistical tests can be applied

### Unit 31: One-Way ANOVA

Unit 31: One-Way ANOVA Summary of Video A vase filled with coins takes center stage as the video begins. Students will be taking part in an experiment organized by psychology professor John Kelly in which

### STAT 350 Practice Final Exam Solution (Spring 2015)

PART 1: Multiple Choice Questions: 1) A study was conducted to compare five different training programs for improving endurance. Forty subjects were randomly divided into five groups of eight subjects

### Testing for differences I exercises with SPSS

Testing for differences I exercises with SPSS Introduction The exercises presented here are all about the t-test and its non-parametric equivalents in their various forms. In SPSS, all these tests can

### 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

### The Normal Distribution

Chapter 6 The Normal Distribution 6.1 The Normal Distribution 1 6.1.1 Student Learning Objectives By the end of this chapter, the student should be able to: Recognize the normal probability distribution