# Hypothesis Testing: Two Means, Paired Data, Two Proportions

Size: px
Start display at page:

Transcription

1 Chapter 10 Hypothesis Testing: Two Means, Paired Data, Two Proportions 10.1 Hypothesis Testing: Two Population Means and Two Population Proportions Student Learning Objectives By the end of this chapter, the student should be able to: Classify hypothesis tests by type. Conduct and interpret hypothesis tests for two population means, population standard deviations known. Conduct and interpret hypothesis tests for two population means, population standard deviations unknown. Conduct and interpret hypothesis tests for two population proportions. Conduct and interpret hypothesis tests for matched or paired samples Introduction Studies often compare two groups. For example, researchers are interested in the effect aspirin has in preventing heart attacks. Over the last few years, newspapers and magazines have reported about various aspirin studies involving two groups. Typically, one group is given aspirin and the other group is given a placebo. Then, the heart attack rate is studied over several years. There are other situations that deal with the comparison of two groups. For example, studies compare various diet and exercise programs. Politicians compare the proportion of individuals from different income brackets who might vote for them. Students are interested in whether SAT or GRE preparatory courses really help raise their scores. In the previous chapter, you learned to conduct hypothesis tests on single means and single proportions. You will expand upon that in this chapter. You will compare two averages or two proportions to each other. The general procedure is still the same, just expanded. 1 This content is available online at < 423

2 424 CHAPTER 10. HYPOTHESIS TESTING: TWO MEANS, PAIRED DATA, TWO PROPORTIONS To compare two averages or two proportions, you work with two groups. The groups are classified either as independent or matched pairs. Independent groups mean that the two samples taken are independent, that is, sample values selected from one population are not related in any way to sample values selected from the other population. Matched pairs consist of two samples that are dependent. The parameter tested using matched pairs is the population mean. The parameters tested using independent groups are either population means or population proportions. NOTE: This chapter relies on either a calculator or a computer to calculate the degrees of freedom, the test statistics, and p-values. TI-83+ and TI-84 instructions are included as well as the test statistic formulas. Because of technology, we do not need to separate two population means, independent groups, population variances unknown into large and small sample sizes. This chapter deals with the following hypothesis tests: Independent groups (samples are independent) Test of two population means. Test of two population proportions. Matched or paired samples (samples are dependent) Becomes a test of one population mean Comparing Two Independent Population Means with Unknown Population Standard Deviations 2 1. The two independent samples are simple random samples from two distinct populations. 2. Both populations are normally distributed with the population means and standard deviations unknown unless the sample sizes are greater than 30. In that case, the populations need not be normally distributed. The comparison of two population means is very common. A difference between the two samples depends on both the means and the standard deviations. Very different means can occur by chance if there is great variation among the individual samples. In order to account for the variation, we take the difference of the sample means, X 1 - X 2, and divide by the standard error (shown below) in order to standardize the difference. The result is a t-score test statistic (shown below). Because we do not know the population standard deviations, we estimate them using the two sample standard deviations from our independent samples. For the hypothesis test, we calculate the estimated standard deviation, or standard error, of the difference in sample means, X 1 - X 2. The standard error is: The test statistic (t-score) is calculated as follows: T-score where: 2 This content is available online at < (S 1 ) 2 + (S 2) 2 n 2 (10.1) (x 1 x 2 ) (µ 1 µ 2 ) (10.2) (S 1 ) 2 + (S 2) 2 n 2

3 425 s 1 and s 2, the sample standard deviations, are estimates of σ 1 and σ 2, respectively. σ 1 and σ 2 are the unknown population standard deviations. x 1 and x 2 are the sample means. µ 1 and µ 2 are the population means. The degrees of freedom (df) is a somewhat complicated calculation. However, a computer or calculator calculates it easily. The dfs are not always a whole number. The test statistic calculated above is approximated by the Student-t distribution with dfs as follows: Degrees of freedom d f = [ ] 2 (s 1 ) 2 + (s 2) 2 n 2 [ ] 2 [ ] 2 (10.3) 1 1 (s 1 ) (s 2 ) 2 n 2 When both sample sizes and n 2 are five or larger, the Student-t approximation is very good. Notice that the sample variances s 1 2 and s 2 2 are not pooled. (If the question comes up, do not pool the variances.) NOTE: It is not necessary to compute this by hand. A calculator or computer easily computes it. Example 10.1: Independent groups The average amount of time boys and girls ages 7 through 11 spend playing sports each day is believed to be the same. An experiment is done, data is collected, resulting in the table below: Sample Size Girls 9 2 hours Average Number of Hours Playing Sports Per Day Boys hours 1.00 Table 10.1 Sample Standard Deviation 0.75 Problem Is there a difference in the average amount of time boys and girls ages 7 through 11 play sports each day? Test at the 5% level of significance. Solution The population standard deviations are not known. Let g be the subscript for girls and b be the subscript for boys. Then, µ g is the population mean for girls and µ b is the population mean for boys. This is a test of two independent groups, two population means. Random variable: X g X b = difference in the average amount of time girls and boys play sports each day. H o : µ g = µ b ( µg µ b = 0 ) H a : µ g = µ b ( µg µ b = 0 ) The words "the same" tell you H o has an "=". Since there are no other words to indicate H a, then assume "is different." This is a two-tailed test. Distribution for the test: Use t d f where d f is calculated using the d f formula for independent groups, two population means. Using a calculator, d f is approximately Do not pool the variances.

4 426 CHAPTER 10. HYPOTHESIS TESTING: TWO MEANS, PAIRED DATA, TWO PROPORTIONS Calculate the p-value using a Student-t distribution: p-value = Graph: Figure 10.1 s g = 0.75 s b = 1 So, x g x b = = 1.2 Half the p-value is below -1.2 and half is above 1.2. Make a decision: Since α > p-value, reject H o. This means you reject µ g = µ b. The means are different. Conclusion: At the 5% level of significance, the sample data show there is sufficient evidence to conclude that the average number of hours that girls and boys aged 7 through 11 play sports per day is different. NOTE: TI-83+ and TI-84: Press STAT. Arrow over to TESTS and press 4:2-SampTTest. Arrow over to Stats and press ENTER. Arrow down and enter 2 for the first sample mean, 0.75 for Sx1, 9 for n1, 3.2 for the second sample mean, 1 for Sx2, and 16 for n2. Arrow down to µ1: and arrow to does not equal µ2. Press ENTER. Arrow down to Pooled: and No. Press ENTER. Arrow down to Calculate and press ENTER. The p-value is p = , the dfs are approximately , and the test statistic is Do the procedure again but instead of Calculate do Draw. Example 10.2 A study is done by a community group in two neighboring colleges to determine which one graduates students with more math classes. College A samples 11 graduates. Their average is 4 math classes with a standard deviation of 1.5 math classes. College B samples 9 graduates. Their average is 3.5 math classes with a standard deviation of 1 math class. The community group believes that a student who graduates from college A has taken more math classes, on the average. Test at a 1% significance level. Answer the following questions.

5 427 Problem 1 (Solution on p. 460.) Is this a test of two means or two proportions? Problem 2 (Solution on p. 460.) Are the populations standard deviations known or unknown? Problem 3 (Solution on p. 460.) Which distribution do you use to perform the test? Problem 4 (Solution on p. 460.) What is the random variable? Problem 5 (Solution on p. 460.) What are the null and alternate hypothesis? Problem 6 (Solution on p. 460.) Is this test right, left, or two tailed? Problem 7 (Solution on p. 460.) What is the p-value? Problem 8 (Solution on p. 460.) Do you reject or not reject the null hypothesis? Conclusion: At the 1% level of significance, from the sample data, there is not sufficient evidence to conclude that a student who graduates from college A has taken more math classes, on the average, than a student who graduates from college B Comparing Two Independent Population Means with Known Population Standard Deviations 3 Even though this situation is not likely (knowing the population standard deviations is not likely), the following example illustrates hypothesis testing for independent means, known population standard deviations. The distribution is Normal and is for the difference of sample means, X 1 X 2. The normal distribution has the following format: Normal distribution (σ X 1 X 2 N u 1 u 2, 1 ) 2 + (σ 2) 2 (10.4) n 2 The standard deviation is: The test statistic (z-score) is: (σ 1 ) 2 + (σ 2) 2 n 2 (10.5) z = (x 1 x 2 ) (µ 1 µ 2 ) (10.6) (σ 1 ) 2 + (σ 2) 2 n 2 3 This content is available online at <

6 428 CHAPTER 10. HYPOTHESIS TESTING: TWO MEANS, PAIRED DATA, TWO PROPORTIONS Example 10.3 independent groups, population standard deviations known: The mean lasting time of 2 competing floor waxes is to be compared. Twenty floors are randomly assigned to test each wax. The following table is the result. Wax Sample Mean Number of Months Floor Wax Last Population Standard Deviation Table 10.2 Problem Does the data indicate that wax 1 is more effective than wax 2? Test at a 5% level of significance. Solution This is a test of two independent groups, two population means, population standard deviations known. Random Variable: X 1 X 2 = difference in the average number of months the competing floor waxes last. H o : µ 1 µ 2 H a : µ 1 > µ 2 The words "is more effective" says that wax 1 lasts longer than wax 2, on the average. "Longer" is a > symbol and goes into H a. Therefore, this is a right-tailed test. Distribution for the test: The population standard deviations are known so the distribution is normal. Using the formula above, the distribution is: ( ) 0.33 X 1 X 2 N 0, Since µ 1 µ 2 then µ 1 µ 2 0 and the mean for the normal distribution is 0. Calculate the p-value using the normal distribution: p-value = Graph:

7 429 Figure 10.2 x 1 x 2 = = 0.1 Compare α and the p-value: α = 0.05 and p-value = Therefore, α < p-value. Make a decision: Since α < p-value, do not reject H o. Conclusion: At the 5% level of significance, from the sample data, there is not sufficient evidence to conclude that wax 1 lasts longer (wax 1 is more effective) than wax 2. NOTE: TI-83+ and TI-84: Press STAT. Arrow over to TESTS and press 3:2-SampZTest. Arrow over to Stats and press ENTER. Arrow down and enter.33 for sigma1,.36 for sigma2, 3 for the first sample mean, 20 for n1, 2.9 for the second sample mean, and 20 for n2. Arrow down to µ1: and arrow to > µ2. Press ENTER. Arrow down to Calculate and press ENTER. The p-value is p = and the test statistic is Do the procedure again but instead of Calculate do Draw Comparing Two Independent Population Proportions 4 1. The two independent samples are simple random samples that are independent. 2. The number of successes is at least five and the number of failures is at least five for each of the samples. Comparing two proportions, like comparing two means, is common. If two estimated proportions are different, it may be due to a difference in the populations or it may be due to chance. A hypothesis test can help determine if a difference in the estimated proportions (P A P B ) reflects a difference in the populations. The difference of two proportions follows an approximate normal distribution. Generally, the null hypothesis states that the two proportions are the same. That is, H o : p A = p B. To conduct the test, we use a pooled proportion, p c. 4 This content is available online at <

8 430 CHAPTER 10. HYPOTHESIS TESTING: TWO MEANS, PAIRED DATA, TWO PROPORTIONS The pooled proportion is calculated as follows: p c = X A + X B n A + n B (10.7) The distribution for the differences is: P A P B N [ 0, ( 1 p c (1 p c ) + 1 ) ] (10.8) n A n B The test statistic (z-score) is: z = (p A p B ) (p A p B ) ( ) (10.9) p c (1 p c ) 1 na + B Example 10.4: Two population proportions Two types of medication for hives are being tested to determine if there is a difference in the percentage of adult patient reactions. Twenty out of a random sample of 200 adults given medication A still had hives 30 minutes after taking the medication. Twelve out of another random sample of 200 adults given medication B still had hives 30 minutes after taking the medication. Test at a 1% level of significance Determining the solution This is a test of 2 population proportions. Problem (Solution on p. 460.) How do you know? Let A and B be the subscripts for medication A and medication B. Then p A and p B are the desired population proportions. Random Variable: P A P B = difference in the percentages of adult patients who did not react after 30 minutes to medication A and medication B. H o : p A = p B p A p B = 0 H a : p A = p B p A p B = 0 The words "is a difference" tell you the test is two-tailed. Distribution for the test: Since this is a test of two binomial population proportions, the distribution is normal: p c = X A+X B n A +n B = = p c = 0.92 Therefore, P A P B N [ ( ) ] 0, (0.08) (0.92) P A P B follows an approximate normal distribution. Calculate the p-value using the normal distribution: p-value =

9 431 Estimated proportion for group A: Estimated proportion for group B: Graph: p A = X A n A = = 0.1 p B = X B n B = = 0.06 Figure 10.3 P A P B = = Half the p-value is below and half is above Compare α and the p-value: α = 0.01 and the p-value = α < p-value. Make a decision: Since α < p-value, you cannot reject H o. Conclusion: At a 1% level of significance, from the sample data, there is not sufficient evidence to conclude that there is a difference in the percentages of adult patients who did not react after 30 minutes to medication A and medication B. NOTE: TI-83+ and TI-84: Press STAT. Arrow over to TESTS and press 6:2-PropZTest. Arrow down and enter 20 for x1, 200 for n1, 12 for x2, and 200 for n2. Arrow down to p1: and arrow to not equal p2. Press ENTER. Arrow down to Calculate and press ENTER. The p-value is p = and the test statistic is Do the procedure again but instead of Calculate do Draw Matched or Paired Samples 5 1. Simple random sampling is used. 2. Sample sizes are often small. 3. Two measurements (samples) are drawn from the same pair of individuals or objects. 4. Differences are calculated from the matched or paired samples. 5. The differences form the sample that is used for the hypothesis test. 5 This content is available online at <

10 432 CHAPTER 10. HYPOTHESIS TESTING: TWO MEANS, PAIRED DATA, TWO PROPORTIONS 6. The matched pairs have differences that either come from a population that is normal or the number of differences is greater than 30 or both. In a hypothesis test for matched or paired samples, subjects are matched in pairs and differences are calculated. The differences are the data. The population mean for the differences, µ d, is then tested using a Student-t test for a single population mean with n 1 degrees of freedom where n is the number of differences. The test statistic (t-score) is: t = x d µ ( ) d (10.10) n sd Example 10.5: Matched or paired samples A study was conducted to investigate the effectiveness of hypnotism in reducing pain. Results for randomly selected subjects are shown in the table. The "before" value is matched to an "after" value. Subject: A B C D E F G H Before After Table 10.3 Problem Are the sensory measurements, on average, lower after hypnotism? Test at a 5% significance level. Solution Corresponding "before" and "after" values form matched pairs. After Data Before Data Difference Table 10.4 The data for the test are the differences: {0.2, -4.1, -1.6, -1.8, -3.2, -2, -2.9, -9.6} The sample mean and sample standard deviation of the differences are: x d = 3.13 and s d = 2.91 Verify these values. Let µ d be the population mean for the differences. We use the subscript d to denote "differences."

11 433 Random Variable: X d = the average difference of the sensory measurements There is no improvement. (µ d is the population mean of the differences.) H o : µ d 0 (10.11) H a : µ d < 0 (10.12) There is improvement. The score should be lower after hypnotism so the difference ought to be negative to indicate improvement. Distribution for the test: The distribution is a student-t with d f = n 1 = 8 1 = 7. Use t 7. (Notice that the test is for a single population mean.) Calculate the p-value using the Student-t distribution: p-value = Graph: Figure 10.4 X d is the random variable for the differences. The sample mean and sample standard deviation of the differences are: x d = 3.13 s d = 2.91 Compare α and the p-value: α = 0.05 and p-value = α > p-value. Make a decision: Since α > p-value, reject H o. This means that µ d < 0 and there is improvement. Conclusion: At a 5% level of significance, from the sample data, there is sufficient evidence to conclude that the sensory measurements, on average, are lower after hypnotism. Hypnotism appears to be effective in reducing pain. NOTE: For the TI-83+ and TI-84 calculators, you can either calculate the differences ahead of time (after - before) and put the differences into a list or you can put the after data into a first list and

12 434 CHAPTER 10. HYPOTHESIS TESTING: TWO MEANS, PAIRED DATA, TWO PROPORTIONS the before data into a second list. Then go to a third list and arrow up to the name. Enter 1st list name - 2nd list name. The calculator will do the subtraction and you will have the differences in the third list. NOTE: TI-83+ and TI-84: Use your list of differences as the data. Press STAT and arrow over to TESTS. Press 2:T-Test. Arrow over to Data and press ENTER. Arrow down and enter 0 for µ 0, the name of the list where you put the data, and 1 for Freq:. Arrow down to µ: and arrow over to < µ 0. Press ENTER. Arrow down to Calculate and press ENTER. The p-value is and the test statistic is Do these instructions again except arrow to Draw (instead of Calculate). Press ENTER. Example 10.6 A college football coach was interested in whether the college s strength development class increased his players maximum lift (in pounds) on the bench press exercise. He asked 4 of his players to participate in a study. The amount of weight they could each lift was recorded before they took the strength development class. After completing the class, the amount of weight they could each lift was again measured. The data are as follows: Weight (in pounds) Player 1 Player 2 Player 3 Player 4 Amount of weighted lifted prior to the class Amount of weight lifted after the class Table 10.5 The coach wants to know if the strength development class makes his players stronger, on average. Problem (Solution on p. 460.) Record the differences data. Calculate the differences by subtracting the amount of weight lifted prior to the class from the weight lifted after completing the class. The data for the differences are: {90, 11, -8, -8} Using the differences data, calculate the sample mean and the sample standard deviation. x d = 21.3 s d = 46.7 Using the difference data, this becomes a test of a single (fill in the blank). Define the random variable: X d = average difference in the maximum lift per player. The distribution for the hypothesis test is t 3. H o : µd 0 H a : µ d > 0 Graph:

13 435 Figure 10.5 Calculate the p-value: The p-value is Decision: If the level of significance is 5%, the decision is to not reject the null hypothesis because α < p-value. What is the conclusion? Example 10.7 Seven eighth graders at Kennedy Middle School measured how far they could push the shot-put with their dominant (writing) hand and their weaker (non-writing) hand. They thought that they could push equal distances with either hand. The following data was collected. Distance (in feet) using Dominant Hand Weaker Hand Student 1 Student 2 Student 3 Student 4 Student 5 Student 6 Student Table 10.6 Problem (Solution on p. 460.) Conduct a hypothesis test to determine whether the differences in distances between the children s dominant versus weaker hands is significant. HINT: use a t-test on the difference data. CHECK: The test statistic is 2.18 and the p-value is What is your conclusion?

14 436 CHAPTER 10. HYPOTHESIS TESTING: TWO MEANS, PAIRED DATA, TWO PROPORTIONS 10.6 Summary of Types of Hypothesis Tests 6 Two Population Means Populations are independent and population standard deviations are unknown. Populations are independent and population standard deviations are known (not likely). Matched or Paired Samples Two samples are drawn from the same set of objects. Samples are dependent. Two Population Proportions Populations are independent. 6 This content is available online at <

### Hypothesis Testing: Two Means, Paired Data, Two Proportions

Chapter 10 Hypothesis Testing: Two Means, Paired Data, Two Proportions 10.1 Hypothesis Testing: Two Population Means and Two Population Proportions 1 10.1.1 Student Learning Objectives By the end of this

### 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

### HYPOTHESIS TESTING: POWER OF THE TEST

HYPOTHESIS TESTING: POWER OF THE TEST The first 6 steps of the 9-step test of hypothesis are called "the test". These steps are not dependent on the observed data values. When planning a research project,

### Section 7.1. Introduction to Hypothesis Testing. Schrodinger s cat quantum mechanics thought experiment (1935)

Section 7.1 Introduction to Hypothesis Testing Schrodinger s cat quantum mechanics thought experiment (1935) Statistical Hypotheses A statistical hypothesis is a claim about a population. Null hypothesis

### 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.

### 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

### 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

### 3.4 Statistical inference for 2 populations based on two samples

3.4 Statistical inference for 2 populations based on two samples Tests for a difference between two population means The first sample will be denoted as X 1, X 2,..., X m. The second sample will be denoted

### 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

### 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

### Introduction to Hypothesis Testing. Hypothesis Testing. Step 1: State the Hypotheses

Introduction to Hypothesis Testing 1 Hypothesis Testing A hypothesis test is a statistical procedure that uses sample data to evaluate a hypothesis about a population Hypothesis is stated in terms of the

### 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

### Hypothesis Testing --- One Mean

Hypothesis Testing --- One Mean A hypothesis is simply a statement that something is true. Typically, there are two hypotheses in a hypothesis test: the null, and the alternative. Null Hypothesis The hypothesis

### 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

### 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)

### Week 3&4: Z tables and the Sampling Distribution of X

Week 3&4: Z tables and the Sampling Distribution of X 2 / 36 The Standard Normal Distribution, or Z Distribution, is the distribution of a random variable, Z N(0, 1 2 ). The distribution of any other normal

### 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

### Difference of Means and ANOVA Problems

Difference of Means and Problems Dr. Tom Ilvento FREC 408 Accounting Firm Study An accounting firm specializes in auditing the financial records of large firm It is interested in evaluating its fee structure,particularly

### Comparing Two Groups. Standard Error of ȳ 1 ȳ 2. Setting. Two Independent Samples

Comparing Two Groups Chapter 7 describes two ways to compare two populations on the basis of independent samples: a confidence interval for the difference in population means and a hypothesis test. The

### 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

### 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

### 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

### Name: (b) Find the minimum sample size you should use in order for your estimate to be within 0.03 of p when the confidence level is 95%.

Chapter 7-8 Exam Name: Answer the questions in the spaces provided. If you run out of room, show your work on a separate paper clearly numbered and attached to this exam. Please indicate which program

### Non-Parametric Tests (I)

Lecture 5: Non-Parametric Tests (I) KimHuat LIM lim@stats.ox.ac.uk http://www.stats.ox.ac.uk/~lim/teaching.html Slide 1 5.1 Outline (i) Overview of Distribution-Free Tests (ii) Median Test for Two Independent

### Lesson 9 Hypothesis Testing

Lesson 9 Hypothesis Testing Outline Logic for Hypothesis Testing Critical Value Alpha (α) -level.05 -level.01 One-Tail versus Two-Tail Tests -critical values for both alpha levels Logic for Hypothesis

### Understand the role that hypothesis testing plays in an improvement project. Know how to perform a two sample hypothesis test.

HYPOTHESIS TESTING Learning Objectives Understand the role that hypothesis testing plays in an improvement project. Know how to perform a two sample hypothesis test. Know how to perform a hypothesis test

### Unit 31 A Hypothesis Test about Correlation and Slope in a Simple Linear Regression

Unit 31 A Hypothesis Test about Correlation and Slope in a Simple Linear Regression Objectives: To perform a hypothesis test concerning the slope of a least squares line To recognize that testing for a

### CHAPTER 14 NONPARAMETRIC TESTS

CHAPTER 14 NONPARAMETRIC TESTS Everything that we have done up until now in statistics has relied heavily on one major fact: that our data is normally distributed. We have been able to make inferences

### Math 251, Review Questions for Test 3 Rough Answers

Math 251, Review Questions for Test 3 Rough Answers 1. (Review of some terminology from Section 7.1) In a state with 459,341 voters, a poll of 2300 voters finds that 45 percent support the Republican candidate,

### 5/31/2013. Chapter 8 Hypothesis Testing. Hypothesis Testing. Hypothesis Testing. Outline. Objectives. Objectives

C H 8A P T E R 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 6 Confidence Intervals and Copyright 2013 The McGraw Hill Companies, Inc.

### 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

### Chapter 8: Hypothesis Testing for One Population Mean, Variance, and Proportion

Chapter 8: Hypothesis Testing for One Population Mean, Variance, and Proportion Learning Objectives Upon successful completion of Chapter 8, you will be able to: Understand terms. State the null and alternative

### 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

### Summary of Formulas and Concepts. Descriptive Statistics (Ch. 1-4)

Summary of Formulas and Concepts Descriptive Statistics (Ch. 1-4) Definitions Population: The complete set of numerical information on a particular quantity in which an investigator is interested. We assume

### 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

### Hypothesis testing - Steps

Hypothesis testing - Steps Steps to do a two-tailed test of the hypothesis that β 1 0: 1. Set up the hypotheses: H 0 : β 1 = 0 H a : β 1 0. 2. Compute the test statistic: t = b 1 0 Std. error of b 1 =

### Confidence Intervals

Section 6.1 75 Confidence Intervals Section 6.1 C H A P T E R 6 4 Example 4 (pg. 284) Constructing a Confidence Interval Enter the data from Example 1 on pg. 280 into L1. In this example, n > 0, so the

### 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,

### 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,

### Experimental Design. Power and Sample Size Determination. Proportions. Proportions. Confidence Interval for p. The Binomial Test

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

### Introduction to Hypothesis Testing

I. Terms, Concepts. Introduction to Hypothesis Testing A. In general, we do not know the true value of population parameters - they must be estimated. However, we do have hypotheses about what the true

### BA 275 Review Problems - Week 5 (10/23/06-10/27/06) CD Lessons: 48, 49, 50, 51, 52 Textbook: pp. 380-394

BA 275 Review Problems - Week 5 (10/23/06-10/27/06) CD Lessons: 48, 49, 50, 51, 52 Textbook: pp. 380-394 1. Does vigorous exercise affect concentration? In general, the time needed for people to complete

### 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

### 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.

### 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

### How Does My TI-84 Do That

How Does My TI-84 Do That A guide to using the TI-84 for statistics Austin Peay State University Clarksville, Tennessee How Does My TI-84 Do That A guide to using the TI-84 for statistics Table of Contents

### WHERE DOES THE 10% CONDITION COME FROM?

1 WHERE DOES THE 10% CONDITION COME FROM? The text has mentioned The 10% Condition (at least) twice so far: p. 407 Bernoulli trials must be independent. If that assumption is violated, it is still okay

### Unit 26 Estimation with Confidence Intervals

Unit 26 Estimation with Confidence Intervals Objectives: To see how confidence intervals are used to estimate a population proportion, a population mean, a difference in population proportions, or a difference

### Stats on the TI 83 and TI 84 Calculator

Stats on the TI 83 and TI 84 Calculator Entering the sample values STAT button Left bracket { Right bracket } Store (STO) List L1 Comma Enter Example: Sample data are {5, 10, 15, 20} 1. Press 2 ND 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

### 8 6 X 2 Test for a Variance or Standard Deviation

Section 8 6 x 2 Test for a Variance or Standard Deviation 437 This test uses the P-value method. Therefore, it is not necessary to enter a significance level. 1. Select MegaStat>Hypothesis Tests>Proportion

### 6.4 Normal Distribution

Contents 6.4 Normal Distribution....................... 381 6.4.1 Characteristics of the Normal Distribution....... 381 6.4.2 The Standardized Normal Distribution......... 385 6.4.3 Meaning of Areas under

### WISE Power Tutorial All Exercises

ame Date Class WISE Power Tutorial All Exercises Power: The B.E.A.. Mnemonic Four interrelated features of power can be summarized using BEA B Beta Error (Power = 1 Beta Error): Beta error (or Type II

### 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

### 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

### TI 83/84 Calculator The Basics of Statistical Functions

What you want to do How to start What to do next Put Data in Lists STAT EDIT 1: EDIT ENTER Clear numbers already in a list: Arrow up to L1, then hit CLEAR, ENTER. Then just type the numbers into the appropriate

### Part 2: Analysis of Relationship Between Two Variables

Part 2: Analysis of Relationship Between Two Variables Linear Regression Linear correlation Significance Tests Multiple regression Linear Regression Y = a X + b Dependent Variable Independent Variable

### 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

### 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

### Descriptive Statistics

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

### Chapter 7 TEST OF HYPOTHESIS

Chapter 7 TEST OF HYPOTHESIS In a certain perspective, we can view hypothesis testing just like a jury in a court trial. In a jury trial, the null hypothesis is similar to the jury making a decision of

### Chapter 26: Tests of Significance

Chapter 26: Tests of Significance Procedure: 1. State the null and alternative in words and in terms of a box model. 2. Find the test statistic: z = observed EV. SE 3. Calculate the P-value: The area under

### 1 Hypothesis Testing. H 0 : population parameter = hypothesized value:

1 Hypothesis Testing In Statistics, a hypothesis proposes a model for the world. Then we look at the data. If the data are consistent with that model, we have no reason to disbelieve the hypothesis. Data

### SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

Ch. 10 Chi SquareTests and the F-Distribution 10.1 Goodness of Fit 1 Find Expected Frequencies Provide an appropriate response. 1) The frequency distribution shows the ages for a sample of 100 employees.

### z-scores AND THE NORMAL CURVE MODEL

z-scores AND THE NORMAL CURVE MODEL 1 Understanding z-scores 2 z-scores A z-score is a location on the distribution. A z- score also automatically communicates the raw score s distance from the mean A

### Normal distribution. ) 2 /2σ. 2π σ

Normal distribution The normal distribution is the most widely known and used of all distributions. Because the normal distribution approximates many natural phenomena so well, it has developed into a

### How To Calculate Confidence Intervals In A Population Mean

Chapter 8 Confidence Intervals 8.1 Confidence Intervals 1 8.1.1 Student Learning Objectives By the end of this chapter, the student should be able to: Calculate and interpret confidence intervals for one

### Standard Deviation Estimator

CSS.com Chapter 905 Standard Deviation Estimator Introduction Even though it is not of primary interest, an estimate of the standard deviation (SD) is needed when calculating the power or sample size of

### Statistiek II. John Nerbonne. October 1, 2010. Dept of Information Science j.nerbonne@rug.nl

Dept of Information Science j.nerbonne@rug.nl October 1, 2010 Course outline 1 One-way ANOVA. 2 Factorial ANOVA. 3 Repeated measures ANOVA. 4 Correlation and regression. 5 Multiple regression. 6 Logistic

### Non-Inferiority Tests for Two Means using Differences

Chapter 450 on-inferiority Tests for Two Means using Differences Introduction This procedure computes power and sample size for non-inferiority tests in two-sample designs in which the outcome is a continuous

### 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

### Testing Research and Statistical Hypotheses

Testing Research and Statistical Hypotheses Introduction In the last lab we analyzed metric artifact attributes such as thickness or width/thickness ratio. Those were continuous variables, which as you

### Introduction. Statistics Toolbox

Introduction A hypothesis test is a procedure for determining if an assertion about a characteristic of a population is reasonable. For example, suppose that someone says that the average price of a gallon

### Tests of Hypotheses Using Statistics

Tests of Hypotheses Using Statistics Adam Massey and Steven J. Miller Mathematics Department Brown University Providence, RI 0292 Abstract We present the various methods of hypothesis testing that one

### 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

### TI-Inspire manual 1. I n str uctions. Ti-Inspire for statistics. General Introduction

TI-Inspire manual 1 I n str uctions Ti-Inspire for statistics General Introduction TI-Inspire manual 2 General instructions Press the Home Button to go to home page Pages you will use the most #1 is a

### Introduction to Hypothesis Testing OPRE 6301

Introduction to Hypothesis Testing OPRE 6301 Motivation... The purpose of hypothesis testing is to determine whether there is enough statistical evidence in favor of a certain belief, or hypothesis, about

### Confidence Intervals for the Difference Between Two Means

Chapter 47 Confidence Intervals for the Difference Between Two Means Introduction This procedure calculates the sample size necessary to achieve a specified distance from the difference in sample means

### Hypothesis Testing: Single Mean and Single Proportion

Chapter 9 Hypothesis Testing: Single Mean and Single Proportion 9.1 Hypothesis Testing: Single Mean and Single Proportion 1 9.1.1 Student Learning Objectives By the end of this chapter, the student should

### 5.1 Identifying the Target Parameter

University of California, Davis Department of Statistics Summer Session II Statistics 13 August 20, 2012 Date of latest update: August 20 Lecture 5: Estimation with Confidence intervals 5.1 Identifying

### 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

### research/scientific includes the following: statistical hypotheses: you have a null and alternative you accept one and reject the other

1 Hypothesis Testing Richard S. Balkin, Ph.D., LPC-S, NCC 2 Overview When we have questions about the effect of a treatment or intervention or wish to compare groups, we use hypothesis testing Parametric

### " Y. Notation and Equations for Regression Lecture 11/4. Notation:

Notation: Notation and Equations for Regression Lecture 11/4 m: The number of predictor variables in a regression Xi: One of multiple predictor variables. The subscript i represents any number from 1 through

### 1.5 Oneway Analysis of Variance

Statistics: Rosie Cornish. 200. 1.5 Oneway Analysis of Variance 1 Introduction Oneway analysis of variance (ANOVA) is used to compare several means. This method is often used in scientific or medical experiments

### 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

### 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.

### 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

### 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

### NONPARAMETRIC STATISTICS 1. depend on assumptions about the underlying distribution of the data (or on the Central Limit Theorem)

NONPARAMETRIC STATISTICS 1 PREVIOUSLY parametric statistics in estimation and hypothesis testing... construction of confidence intervals computing of p-values classical significance testing depend on assumptions

### AP Statistics Solutions to Packet 2

AP Statistics Solutions to Packet 2 The Normal Distributions Density Curves and the Normal Distribution Standard Normal Calculations HW #9 1, 2, 4, 6-8 2.1 DENSITY CURVES (a) Sketch a density curve that

### 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,

### The Wilcoxon Rank-Sum Test

1 The Wilcoxon Rank-Sum Test The Wilcoxon rank-sum test is a nonparametric alternative to the twosample t-test which is based solely on the order in which the observations from the two samples fall. We

### ELEMENTARY STATISTICS

ELEMENTARY STATISTICS Study Guide Dr. Shinemin Lin Table of Contents 1. Introduction to Statistics. Descriptive Statistics 3. Probabilities and Standard Normal Distribution 4. Estimates and Sample Sizes

### 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

### Two-Sample T-Tests Assuming Equal Variance (Enter Means)

Chapter 4 Two-Sample T-Tests Assuming Equal Variance (Enter Means) Introduction This procedure provides sample size and power calculations for one- or two-sided two-sample t-tests when the variances of

### 4. Continuous Random Variables, the Pareto and Normal Distributions

4. Continuous Random Variables, the Pareto and Normal Distributions A continuous random variable X can take any value in a given range (e.g. height, weight, age). The distribution of a continuous random