Chapter 9 Inferences from Two Samples
|
|
- Delphia Shields
- 5 years ago
- Views:
Transcription
1 Chapter 9 Inferences from Two Samples 9-1 Review and Preview 9-2 Two Proportions 9-3 Two Means: Independent Samples 9-4 Two Dependent Samples (Matched Pairs) 9-5 Two Variances or Standard Deviations
2 Review In Chapters 7 and 8 we introduced methods of inferential statistics. In Chapter 7: confidence interval. In Chapter 8: hypothesis testing. Chapters 7 and 8 both involved methods for dealing with a sample from a single population.
3 Preview The objective: testing hypotheses to situations involving two sets of sample data instead of just one.
4 Chapter 9 Inferences from Two Samples 9-1 Review and Preview 9-2 Two Proportions 9-3 Two Means: Independent Samples 9-4 Two Dependent Samples (Matched Pairs) 9-5 Two Variances or Standard Deviations
5 Key Concept 1. testing a claim made about two population proportions 2. constructing a confidence interval estimate of the difference between the two population proportions. This section is based on proportions, but can also deal with probabilities or percentages.
6 Notation for Two Proportions For population 1, we let: p 1 n 1 x 1 pˆ qˆ 1 = population proportion = size of the sample = number of successes in the sample = x n 1 1 = (the sample proportion) pˆ The corresponding notations apply to which come from population 2. p, n, x pˆ and qˆ, 2 2 2, 2 2
7 Pooled Sample Proportion The pooled sample proportion is given by: p = x n + x n 1 2 q = 1 p
8 Requirements 1. We have proportions from two independent simple random samples. 2. For each of the two samples, the number of successes is at least 5 and the number of failures is at least 5. (so that the binomial distribution can be approximated by the normal distribution)
9 Test Statistic for Two Proportions For H : p = p z = ( pˆ pˆ ) ( p p ) pq n + pq n 1 2 where (assumed in the null hypothesis) p 1 p2 = 0
10 Confidence Interval Estimate of p 1 p 2 ( pˆ pˆ ) E < ( p p ) < ( pˆ pˆ ) + E with margin of error E pq ˆˆ = zα /2 + n pq ˆˆ n
11 Example Do people having different spending habits depending on the type of money they have? 89 undergraduates were randomly assigned to two groups and were given a choice of keeping the money or buying gum or mints. The claim is that money in large denominations is less likely to be spent relative to an equivalent amount in many smaller denominations. Let s test the claim at the 0.05 significance level.
12 Example Below are the sample data and summary statistics: Group 1 Group 2 Subjects Given $1 Bill Subjects Given 4 Quarters Spent the money x 1 = 12 x 2 = 27 Subjects in group n 1 = 46 n 2 = 43 pˆ p = 12 pˆ = = =
13 Requirement Check: Example 1.The 89 subjects were randomly assigned to two groups, so we consider these independent random samples. 2.The subjects given the $1 bill include 12 who spent it and 34 who did not. The subjects given the quarters include 27 who spent it and 16 who did not. All counts are above 5, so the requirements are all met.
14 Example Step 1: The claim that money in large denominations is less likely to be spent can be expressed as p 1 < p 2. Step 2: If p 1 < p 2 is false, then p 1 p 2. Step 3: The hypotheses can be written as: H : p = p H : p < p 1 1 2
15 Example Step 4: The significance level is α = Step 5: We will use the normal distribution to run the test with: pˆ 12 ˆ 27 1 = p2 = p = = q = 1 p = =
16 Example Step 6: Calculate the value of the test statistic: z = = ( pˆ pˆ ) ( p p ) pq n + pq n ( )( ) ( )( ) = 3.49
17 Example Step 6: This is a left-tailed test, so the P-value is the area to the left of the test statistic z = 3.49, or The critical value is also shown below.
18 Example Step 7: reject the null hypothesis. Step 8: There is sufficient evidence to support the claim that people with money in large denominations are less likely to spend relative to people with money in smaller denominations.
19 Example We can also construct a confidence interval to estimate the difference between the population proportions. Caution: A conclusion based on a confidence interval might be different from a conclusion based on a hypothesis test.
20 Example Construct a 90% confidence interval estimate of the difference between the two population proportions. What does the result suggest about our claim about people spending large denominations relative to spending small denominations?
21 Example E pq ˆˆ = z + α /2 n pq ˆˆ n = =
22 Example ( pˆ pˆ ) E < ( p p ) < ( pˆ pˆ ) + E < ( p p ) < < ( p p ) < The confidence interval limits do not include 0, implying that there is a significant difference between the two proportions. There does appear to be sufficient evidence to support the claim that money in large denominations is less likely to be spent relative to an equivalent amount in many smaller denominations.
23 Chapter 9 Inferences from Two Samples 9-1 Review and Preview 9-2 Two Proportions 9-3 Two Means: Independent Samples 9-4 Two Dependent Samples (Matched Pairs) 9-5 Two Variances or Standard Deviations
24 Key Concept Using sample data from two independent samples to test hypotheses made about two population means or to construct confidence interval estimates of the difference between two population means.
25 Key Concept Part 1: the standard deviations of the two populations are unknown and are not assumed to be equal. Part 2: two other situations: (1) The two population standard deviations are both known; (2) the two population standard deviations are unknown but are assumed to be equal (compare to part 1) Because σ is typically unknown in real situations, most attention should be given to the methods in Part 1.
26 Part 1: Independent Samples with σ 1 and σ 2 Unknown and Not Assumed Equal
27 Definitions Two samples are independent if the sample values selected from one population are not related to or somehow paired or matched with the sample values from the other population. Two samples are dependent if the sample values are paired. (such as before/after data), or each pair of sample values consists of matched pairs (such as husband/wife data)
28 Notation μ 1 σ 1 n 1 x 1 s 1 = population mean = population standard deviation = size of the first sample = sample mean = sample standard deviation Corresponding notations apply to population 2.
29 Requirements 1. σ 1 and σ 2 are unknown and no assumption is made about the equality of σ 1 and σ The two samples are independent. 3. Both samples are simple random samples. 4. Either or both of these conditions are satisfied: The two sample sizes are both large (over 30) or both samples come from populations having normal distributions.
30 Hypothesis Test for Two Means: Independent Samples t = ( x x ) ( µ µ ) s 2 s n + n 1 2 (where μ 1 μ 2 is often assumed to be 0)
31 Confidence Interval Estimate of μ 1 μ 2 Independent Samples ( x x ) E < ( µ µ ) < ( x x ) + E where E 2 s s = t + α /2 n n 1 2 df = smaller of n 1 1 or n 2 1.
32 Example Researchers conducted trials to investigate the effects of color on creativity. Subjects with a red background were asked to think of creative uses for a brick; other subjects with a blue background were given the same task. Responses were given by a panel of judges. Researchers make the claim that blue enhances performance on a creative task. Test the claim using a 0.01 significance level.
33 Example The data: Red Background Blue Background Sample size Sample mean score x = 3.39 Sample standard deviation n = 35 s = 0.97 x = 3.97 n = 36 s = 0.63
34 Example Requirement check: 1.The values of the two population standard deviations are unknown and assumed not equal. 2.The subject groups are independent. 3.The samples are simple random samples. 4.Both sample sizes exceed 30. The requirements are all satisfied.
35 Example Step 1: μ 1 < μ 2. Step 2: If the original claim is false, then μ 1 μ 2. Step 3: H 0 : μ 1 = μ 2 H 1 : μ 1 < μ 2 (claim, left tail) Step 4: The significance level is α = 0.01 Step 5: we have two independent samples and we are testing a claim about two population means, σ is unknown, we use a t distribution.
36 Example Step 6: Calculate the test statistic t = ( x x ) ( µ µ ) s 2 s n + n 1 2 ( ) 0 = =
37 Example Step 6: Because we are using a t distribution, the critical value of t = is found from Table A-3. We use 34 degrees of freedom.
38 Example Step 7: reject the null hypothesis μ 1 μ 2 = 0. P-Value Method: P-value = reject the null hypothesis. Step 8. There is sufficient evidence to support the claim that the red background group has a lower mean creativity score than the blue background group.
39 Example Using the data from this color creativity example, construct a 98% confidence interval estimate for the difference between the mean creativity score for those with a red background and the mean creativity score for those with a blue background.
40 Example E s1 s = tα /2 + = = n n x = 3.39 and x = ( x x ) E < ( µ µ ) < ( x x ) + E < ( µ µ ) <
41 Example We are 98% confident that the limits 1.05 and 0.11 actually do contain the difference between the two population means. Because those limits do not include 0, our interval suggests that there is a significant difference between the two means.
42 Part 2: Alternative Methods These methods are rarely used in practice because the underlying assumptions are usually not met. 1.The two population standard deviations are both known the test statistic will be a z instead of a t and use the standard normal model. 2.The two population standard deviations are unknown but assumed to be equal pool the sample variances.
43 Hypothesis Test for Two Means: Independent Samples with σ 1 and σ 2 Known The test statistic will be: z = ( x x ) ( µ µ ) σ n σ + n
44 Confidence Interval Estimate for μ 1 μ 2 with σ 1 and σ 2 Known ( x x ) E < ( µ µ ) < ( x x ) + E E σ 2 σ = z + α /2 n n 1 2
45 Inference About Two Means, Assuming σ 1 = σ 2 The test statistic will be t = ( x x ) ( µ µ ) s n s 2 2 p p + n 1 2 where s ( n 1) s + ( n 1) s p = ( n1 1) + ( n2 1) df = n + n 2 1 2
46 Confidence Interval Estimate for μ 1 μ 2 Assuming σ 1 = σ 2 ( x x ) E < ( µ µ ) < ( x x ) + E E 2 s 2 p p s = t + α /2 n n 1 2 df = n + n 2 1 2
47 Methods for Inferences About Two Independent Means
48 Chapter 9 Inferences from Two Samples 9-1 Review and Preview 9-2 Two Proportions 9-3 Two Means: Independent Samples 9-4 Two Dependent Samples (Matched Pairs) 9-5 Two Variances or Standard Deviations
49 Key Concept Develop methods for testing hypotheses and constructing confidence intervals involving the mean of the differences of the values from two dependent populations.
50 Dependent Samples Example Matched pairs of heights of U.S. presidents and heights of their main opponents. Since there is a relationship as a basis for matching the pairs of data, this data consists of dependent samples. Height (cm) of President Height (cm) of Main Opponent
51 Good Experimental Design Using dependent samples with paired data is generally better than using two independent samples. The advantage of using matched pairs is that we reduce extraneous variation, which could occur if each experimental unit were treated independently.
52 Notation for Dependent Samples d = individual difference between the two values of a single matched pair µ d = mean value of the differences d for the population of all matched pairs of data d = mean value of the differences d for the paired sample data s d = standard deviation of the differences d for the paired sample data n = number of pairs of sample data
53 Requirements 1. The sample data are dependent. 2. The samples are simple random samples. 3. Either or both of these conditions is satisfied: The number of pairs of sample data is large ( n > 30) or the pairs of values have differences that are from a population having a distribution that is approximately normal.
54 Hypothesis Test Statistic for Matched Pairs t = d µ d s d n where degrees of freedom = n 1
55 Confidence Intervals for Matched Pairs d E < µ < d + E d E = t α /2 s d n Critical values of t α/2 : Use Table A-3 with df = n 1 degrees of freedom.
56 Example At α = 0.05, use the data below to test the claim that for the population of heights of presidents and their main opponents, the differences have a mean greater than 0 cm (so presidents tend to be taller than their opponents). Height (cm) of President Height (cm) of Main Opponent Difference d
57 Example Requirement Check: 1. The samples are dependent because the values are paired. 2. The pairs of data are randomly selected. 3. The number of data points is 5, so normality should be checked (and it is assumed the condition is met).
58 Example Step 1: The claim is that µ d > 0 cm. Step 2: If the original claim is not true, we have µ d 0 cm. Step 3: The hypotheses can be written as: H H 0 0 : µ = 0 cm d : µ > 0 cm d
59 Example Step 4: The significance level is α = Step 5: We use the Student t distribution. The summary statistics are: d = 3.2 s = 11.4
60 Example Step 6: Determine the value of the test statistic: t d µ = d = = s 11.4 d n 5 with df = 5 1 = 4
61 Example Step 6: Using technology, the P-value is Using the critical value method:
62 Example Step 7: Fail to reject the H 0. Conclusion: There is not sufficient evidence to support the claim that for the population of heights of presidents and their main opponent, the differences have a mean greater than 0 cm. In other words, presidents do not appear to be taller than their opponents.
63 Example Confidence Interval: Support the conclusions with a 90% confidence interval estimate for µ d. E = t s d α / 2 = = n d E < µ < d + E < µ < < µ < 14.1 d d d
64 Example The confidence interval does contain the value of 0 cm, so it is very possible that the mean of the differences is equal to 0 cm, indicating that there is no significant difference between the heights.
65 Chapter 9 Inferences from Two Samples 9-1 Review and Preview 9-2 Two Proportions 9-3 Two Means: Independent Samples 9-4 Two Dependent Samples (Matched Pairs) 9-5 Two Variances or Standard Deviations
66 Key Concept This section presents the F test for comparing two population variances (or standard deviations). Note that the F test is very sensitive to departures from normal distributions.
67 Part 1 F Test as a Hypothesis Test with Two Variances or Standard Deviations
68 Notation for Hypothesis Tests with Two Variances or Standard Deviations 2 s 1 n 1 2 σ 1 = larger of two sample variances = size of the sample with the larger variance = variance of the population from which the sample with the larger variance is drawn s, n, and σ are used for the other sample and population
69 Requirements 1. The two populations are independent. 2. The two samples are simple random samples. 3. The two populations are each normally distributed
70 Test Statistic for Hypothesis Tests with Two Variances F = s 2 1 s s 1 Where is the larger of the two sample variances P-values: Automatically provided by technology or estimated using Table A-5. Critical values: Using Table A-5, we obtain critical F values that are determined by the following three values: 1. The significance level α. 2. Numerator degrees of freedom = n Denominator degrees of freedom = n 2 1
71 Properties of the F Distribution The F distribution is not symmetric. Values of the F distribution cannot be negative. The exact shape of the F distribution depends on the two different degrees of freedom.
72 Finding Critical F Values
73 Properties of the F Distribution If the two populations do have equal variances, then F = s 2 1 s 2 2 s 2 1 s 2 2 will be close to 1 because and are close in value.
74 Properties of the F Distribution Consequently, a value of F near 1 will be evidence in favor of the conclusion that: σ 2 = σ But a large value of F will be evidence against the conclusion of equality of the population variances.
75 Example Subjects with a red background were asked to think of creative uses for a brick, while other subjects were given the same task with a blue background. Creativity score responses were scored by a panel of judges. Test the claim that those tested with a red background have creativity scores with a standard deviation equal to the standard deviation for those tested with a blue background. Use a 0.05 significance level. Creativity Score n s Red Background Blue Background x
76 Requirement Check: Example 1. The two populations are independent of each other. 2. simple random samples. 3. We have no knowledge of the distribution of the data, so we proceed assuming the samples are from normally distributed populations, but we note that the validity of the results depends on that assumption.
77 Example Step 1: The claim of equal standard deviations can be expressed as: σ = σ Step 2: If the original claim is false, then: σ σ Step 3: We can write the hypotheses as: H H : σ : σ = σ σ
78 Step 4: The significance level is α = Step 5: Because this test involves two population variances, we use the F distribution. Step 6: The test statistic is Example s 0.97 F = = = s The degrees of freedom for the numerator are n 1 1 = 34. The degrees of freedom for the denominator are n 2 1 = 35.
79 Step 6: Example From Table A-5, the critical F value is between and Technology can be used to find the exact critical F value of The P-value can be determined using technology: P-value =
80 Example Step 7: Since the P-value = is below the significance level of 0.05, we can reject the null hypothesis. Also, the figure below shows the test statistic is in the critical region, so reject.
81 Example There is sufficient evidence to warrant rejection of the claim that the two standard deviations are equal. The variation among creativity scores for those with a red background appears to be different from the variation among creativity scores for those with a blue background.
82 Part 2 Alternative Methods Two Methods that are not so Sensitive to Departures from Normality.
Lecture Slides. Elementary Statistics Eleventh Edition. by Mario F. Triola. and the Triola Statistics Series 9.1-1
Lecture Slides Elementary Statistics Eleventh Edition and the Triola Statistics Series by Mario F. Triola Copyright 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 9.1-1 Chapter 9 Inferences
More informationSingle Sample Means. SOCY601 Alan Neustadtl
Single Sample Means SOCY601 Alan Neustadtl The Central Limit Theorem If we have a population measured by a variable with a mean µ and a standard deviation σ, and if all possible random samples of size
More informationSTA Module 10 Comparing Two Proportions
STA 2023 Module 10 Comparing Two Proportions Learning Objectives Upon completing this module, you should be able to: 1. Perform large-sample inferences (hypothesis test and confidence intervals) to compare
More informationAMS7: WEEK 7. CLASS 1. More on Hypothesis Testing Monday May 11th, 2015
AMS7: WEEK 7. CLASS 1 More on Hypothesis Testing Monday May 11th, 2015 Testing a Claim about a Standard Deviation or a Variance We want to test claims about or 2 Example: Newborn babies from mothers taking
More informationCIVL /8904 T R A F F I C F L O W T H E O R Y L E C T U R E - 8
CIVL - 7904/8904 T R A F F I C F L O W T H E O R Y L E C T U R E - 8 Chi-square Test How to determine the interval from a continuous distribution I = Range 1 + 3.322(logN) I-> Range of the class interval
More informationTwo-Sample Inferential Statistics
The t Test for Two Independent Samples 1 Two-Sample Inferential Statistics In an experiment there are two or more conditions One condition is often called the control condition in which the treatment is
More informationSections 7.1 and 7.2. This chapter presents the beginning of inferential statistics. The two major applications of inferential statistics
Sections 7.1 and 7.2 This chapter presents the beginning of inferential statistics. The two major applications of inferential statistics Estimate the value of a population parameter Test some claim (or
More informationSection 9.4. Notation. Requirements. Definition. Inferences About Two Means (Matched Pairs) Examples
Objective Section 9.4 Inferences About Two Means (Matched Pairs) Compare of two matched-paired means using two samples from each population. Hypothesis Tests and Confidence Intervals of two dependent means
More informationReview 6. n 1 = 85 n 2 = 75 x 1 = x 2 = s 1 = 38.7 s 2 = 39.2
Review 6 Use the traditional method to test the given hypothesis. Assume that the samples are independent and that they have been randomly selected ) A researcher finds that of,000 people who said that
More informationChapter 6 Estimation and Sample Sizes
Chapter 6 Estimation and Sample Sizes This chapter presents the beginning of inferential statistics.! The two major applications of inferential statistics! Estimate the value of a population parameter!
More informationFinal Exam Review (Math 1342)
Final Exam Review (Math 1342) 1) 5.5 5.7 5.8 5.9 6.1 6.1 6.3 6.4 6.5 6.6 6.7 6.7 6.7 6.9 7.0 7.0 7.0 7.1 7.2 7.2 7.4 7.5 7.7 7.7 7.8 8.0 8.1 8.1 8.3 8.7 Min = 5.5 Q 1 = 25th percentile = middle of first
More informationChapter 8. Inferences Based on a Two Samples Confidence Intervals and Tests of Hypothesis
Chapter 8 Inferences Based on a Two Samples Confidence Intervals and Tests of Hypothesis Copyright 2018, 2014, and 2011 Pearson Education, Inc. Slide - 1 Content 1. Identifying the Target Parameter 2.
More informationLecture Slides. Elementary Statistics. by Mario F. Triola. and the Triola Statistics Series
Lecture Slides Elementary Statistics Tenth Edition and the Triola Statistics Series by Mario F. Triola Slide 1 Chapter 9 Inferences from Two Samples 9-1 Overview 9-2 Inferences About Two Proportions 9-3
More informationSTAT Chapter 9: Two-Sample Problems. Paired Differences (Section 9.3)
STAT 515 -- Chapter 9: Two-Sample Problems Paired Differences (Section 9.3) Examples of Paired Differences studies: Similar subjects are paired off and one of two treatments is given to each subject in
More information2011 Pearson Education, Inc
Statistics for Business and Economics Chapter 7 Inferences Based on Two Samples: Confidence Intervals & Tests of Hypotheses Content 1. Identifying the Target Parameter 2. Comparing Two Population Means:
More informationPractice Questions: Statistics W1111, Fall Solutions
Practice Questions: Statistics W, Fall 9 Solutions Question.. The standard deviation of Z is 89... P(=6) =..3. is definitely inside of a 95% confidence interval for..4. (a) YES (b) YES (c) NO (d) NO Questions
More informationHYPOTHESIS TESTING. Hypothesis Testing
MBA 605 Business Analytics Don Conant, PhD. HYPOTHESIS TESTING Hypothesis testing involves making inferences about the nature of the population on the basis of observations of a sample drawn from the population.
More informationMATH 240. Chapter 8 Outlines of Hypothesis Tests
MATH 4 Chapter 8 Outlines of Hypothesis Tests Test for Population Proportion p Specify the null and alternative hypotheses, ie, choose one of the three, where p is some specified number: () H : p H : p
More informationPsychology 282 Lecture #4 Outline Inferences in SLR
Psychology 282 Lecture #4 Outline Inferences in SLR Assumptions To this point we have not had to make any distributional assumptions. Principle of least squares requires no assumptions. Can use correlations
More informationSTAT Chapter 8: Hypothesis Tests
STAT 515 -- Chapter 8: Hypothesis Tests CIs are possibly the most useful forms of inference because they give a range of reasonable values for a parameter. But sometimes we want to know whether one particular
More informationSampling distribution of t. 2. Sampling distribution of t. 3. Example: Gas mileage investigation. II. Inferential Statistics (8) t =
2. The distribution of t values that would be obtained if a value of t were calculated for each sample mean for all possible random of a given size from a population _ t ratio: (X - µ hyp ) t s x The result
More informationLecture Slides. Elementary Statistics Tenth Edition. by Mario F. Triola. and the Triola Statistics Series
Lecture Slides Elementary Statistics Tenth Edition and the Triola Statistics Series by Mario F. Triola Slide 1 Chapter 7 Estimates and Sample Sizes 7-1 Overview 7-2 Estimating a Population Proportion 7-3
More information10/31/2012. One-Way ANOVA F-test
PSY 511: Advanced Statistics for Psychological and Behavioral Research 1 1. Situation/hypotheses 2. Test statistic 3.Distribution 4. Assumptions One-Way ANOVA F-test One factor J>2 independent samples
More informationChapter 24. Comparing Means
Chapter 4 Comparing Means!1 /34 Homework p579, 5, 7, 8, 10, 11, 17, 31, 3! /34 !3 /34 Objective Students test null and alternate hypothesis about two!4 /34 Plot the Data The intuitive display for comparing
More informationMathematical Notation Math Introduction to Applied Statistics
Mathematical Notation Math 113 - Introduction to Applied Statistics Name : Use Word or WordPerfect to recreate the following documents. Each article is worth 10 points and should be emailed to the instructor
More informationChapter 5 Confidence Intervals
Chapter 5 Confidence Intervals Confidence Intervals about a Population Mean, σ, Known Abbas Motamedi Tennessee Tech University A point estimate: a single number, calculated from a set of data, that is
More informationLECTURE 12 CONFIDENCE INTERVAL AND HYPOTHESIS TESTING
LECTURE 1 CONFIDENCE INTERVAL AND HYPOTHESIS TESTING INTERVAL ESTIMATION Point estimation of : The inference is a guess of a single value as the value of. No accuracy associated with it. Interval estimation
More informationEC2001 Econometrics 1 Dr. Jose Olmo Room D309
EC2001 Econometrics 1 Dr. Jose Olmo Room D309 J.Olmo@City.ac.uk 1 Revision of Statistical Inference 1.1 Sample, observations, population A sample is a number of observations drawn from a population. Population:
More informationIntroduction to Business Statistics QM 220 Chapter 12
Department of Quantitative Methods & Information Systems Introduction to Business Statistics QM 220 Chapter 12 Dr. Mohammad Zainal 12.1 The F distribution We already covered this topic in Ch. 10 QM-220,
More informationQuestions 3.83, 6.11, 6.12, 6.17, 6.25, 6.29, 6.33, 6.35, 6.50, 6.51, 6.53, 6.55, 6.59, 6.60, 6.65, 6.69, 6.70, 6.77, 6.79, 6.89, 6.
Chapter 7 Reading 7.1, 7.2 Questions 3.83, 6.11, 6.12, 6.17, 6.25, 6.29, 6.33, 6.35, 6.50, 6.51, 6.53, 6.55, 6.59, 6.60, 6.65, 6.69, 6.70, 6.77, 6.79, 6.89, 6.112 Introduction In Chapter 5 and 6, we emphasized
More informationT.I.H.E. IT 233 Statistics and Probability: Sem. 1: 2013 ESTIMATION AND HYPOTHESIS TESTING OF TWO POPULATIONS
ESTIMATION AND HYPOTHESIS TESTING OF TWO POPULATIONS In our work on hypothesis testing, we used the value of a sample statistic to challenge an accepted value of a population parameter. We focused only
More informationAn inferential procedure to use sample data to understand a population Procedures
Hypothesis Test An inferential procedure to use sample data to understand a population Procedures Hypotheses, the alpha value, the critical region (z-scores), statistics, conclusion Two types of errors
More informationChapter 9. Inferences from Two Samples. Objective. Notation. Section 9.2. Definition. Notation. q = 1 p. Inferences About Two Proportions
Chapter 9 Inferences from Two Samples 9. Inferences About Two Proportions 9.3 Inferences About Two s (Independent) 9.4 Inferences About Two s (Matched Pairs) 9.5 Comparing Variation in Two Samples Objective
More information# of 6s # of times Test the null hypthesis that the dice are fair at α =.01 significance
Practice Final Exam Statistical Methods and Models - Math 410, Fall 2011 December 4, 2011 You may use a calculator, and you may bring in one sheet (8.5 by 11 or A4) of notes. Otherwise closed book. The
More informationCHAPTER 9: HYPOTHESIS TESTING
CHAPTER 9: HYPOTHESIS TESTING THE SECOND LAST EXAMPLE CLEARLY ILLUSTRATES THAT THERE IS ONE IMPORTANT ISSUE WE NEED TO EXPLORE: IS THERE (IN OUR TWO SAMPLES) SUFFICIENT STATISTICAL EVIDENCE TO CONCLUDE
More informationExam 2 (KEY) July 20, 2009
STAT 2300 Business Statistics/Summer 2009, Section 002 Exam 2 (KEY) July 20, 2009 Name: USU A#: Score: /225 Directions: This exam consists of six (6) questions, assessing material learned within Modules
More informationChapter 6. Estimates and Sample Sizes
Chapter 6 Estimates and Sample Sizes Lesson 6-1/6-, Part 1 Estimating a Population Proportion This chapter begins the beginning of inferential statistics. There are two major applications of inferential
More informationThe t-test: A z-score for a sample mean tells us where in the distribution the particular mean lies
The t-test: So Far: Sampling distribution benefit is that even if the original population is not normal, a sampling distribution based on this population will be normal (for sample size > 30). Benefit
More informationSociology 6Z03 Review II
Sociology 6Z03 Review II John Fox McMaster University Fall 2016 John Fox (McMaster University) Sociology 6Z03 Review II Fall 2016 1 / 35 Outline: Review II Probability Part I Sampling Distributions Probability
More informationSampling Distributions: Central Limit Theorem
Review for Exam 2 Sampling Distributions: Central Limit Theorem Conceptually, we can break up the theorem into three parts: 1. The mean (µ M ) of a population of sample means (M) is equal to the mean (µ)
More informationInferential statistics
Inferential statistics Inference involves making a Generalization about a larger group of individuals on the basis of a subset or sample. Ahmed-Refat-ZU Null and alternative hypotheses In hypotheses testing,
More informationChapter 7 Comparison of two independent samples
Chapter 7 Comparison of two independent samples 7.1 Introduction Population 1 µ σ 1 1 N 1 Sample 1 y s 1 1 n 1 Population µ σ N Sample y s n 1, : population means 1, : population standard deviations N
More informationInference for Proportions, Variance and Standard Deviation
Inference for Proportions, Variance and Standard Deviation Sections 7.10 & 7.6 Cathy Poliak, Ph.D. cathy@math.uh.edu Office Fleming 11c Department of Mathematics University of Houston Lecture 12 Cathy
More informationDETERMINE whether the conditions for performing inference are met. CONSTRUCT and INTERPRET a confidence interval to compare two proportions.
Section 0. Comparing Two Proportions Learning Objectives After this section, you should be able to DETERMINE whether the conditions for performing inference are met. CONSTRUCT and INTERPRET a confidence
More informationStatistics for IT Managers
Statistics for IT Managers 95-796, Fall 2012 Module 2: Hypothesis Testing and Statistical Inference (5 lectures) Reading: Statistics for Business and Economics, Ch. 5-7 Confidence intervals Given the sample
More informationChapter 12: Inference about One Population
Chapter 1: Inference about One Population 1.1 Introduction In this chapter, we presented the statistical inference methods used when the problem objective is to describe a single population. Sections 1.
More informationME3620. Theory of Engineering Experimentation. Spring Chapter IV. Decision Making for a Single Sample. Chapter IV
Theory of Engineering Experimentation Chapter IV. Decision Making for a Single Sample Chapter IV 1 4 1 Statistical Inference The field of statistical inference consists of those methods used to make decisions
More informationClassroom Activity 7 Math 113 Name : 10 pts Intro to Applied Stats
Classroom Activity 7 Math 113 Name : 10 pts Intro to Applied Stats Materials Needed: Bags of popcorn, watch with second hand or microwave with digital timer. Instructions: Follow the instructions on the
More informationA proportion is the fraction of individuals having a particular attribute. Can range from 0 to 1!
Proportions A proportion is the fraction of individuals having a particular attribute. It is also the probability that an individual randomly sampled from the population will have that attribute Can range
More informationStatistical Inference for Means
Statistical Inference for Means Jamie Monogan University of Georgia February 18, 2011 Jamie Monogan (UGA) Statistical Inference for Means February 18, 2011 1 / 19 Objectives By the end of this meeting,
More informationStatistics for Business and Economics
Statistics for Business and Economics Chapter 6 Sampling and Sampling Distributions Ch. 6-1 6.1 Tools of Business Statistics n Descriptive statistics n Collecting, presenting, and describing data n Inferential
More informationp = q ˆ = 1 -ˆp = sample proportion of failures in a sample size of n x n Chapter 7 Estimates and Sample Sizes
Chapter 7 Estimates and Sample Sizes 7-1 Overview 7-2 Estimating a Population Proportion 7-3 Estimating a Population Mean: σ Known 7-4 Estimating a Population Mean: σ Not Known 7-5 Estimating a Population
More information1 MA421 Introduction. Ashis Gangopadhyay. Department of Mathematics and Statistics. Boston University. c Ashis Gangopadhyay
1 MA421 Introduction Ashis Gangopadhyay Department of Mathematics and Statistics Boston University c Ashis Gangopadhyay 1.1 Introduction 1.1.1 Some key statistical concepts 1. Statistics: Art of data analysis,
More informationStat 529 (Winter 2011) Experimental Design for the Two-Sample Problem. Motivation: Designing a new silver coins experiment
Stat 529 (Winter 2011) Experimental Design for the Two-Sample Problem Reading: 2.4 2.6. Motivation: Designing a new silver coins experiment Sample size calculations Margin of error for the pooled two sample
More informationStatistics Boot Camp. Dr. Stephanie Lane Institute for Defense Analyses DATAWorks 2018
Statistics Boot Camp Dr. Stephanie Lane Institute for Defense Analyses DATAWorks 2018 March 21, 2018 Outline of boot camp Summarizing and simplifying data Point and interval estimation Foundations of statistical
More information10/4/2013. Hypothesis Testing & z-test. Hypothesis Testing. Hypothesis Testing
& z-test Lecture Set 11 We have a coin and are trying to determine if it is biased or unbiased What should we assume? Why? Flip coin n = 100 times E(Heads) = 50 Why? Assume we count 53 Heads... What could
More informationClass 24. Daniel B. Rowe, Ph.D. Department of Mathematics, Statistics, and Computer Science. Marquette University MATH 1700
Class 4 Daniel B. Rowe, Ph.D. Department of Mathematics, Statistics, and Computer Science Copyright 013 by D.B. Rowe 1 Agenda: Recap Chapter 9. and 9.3 Lecture Chapter 10.1-10.3 Review Exam 6 Problem Solving
More informationChapter 23. Inferences About Means. Monday, May 6, 13. Copyright 2009 Pearson Education, Inc.
Chapter 23 Inferences About Means Sampling Distributions of Means Now that we know how to create confidence intervals and test hypotheses about proportions, we do the same for means. Just as we did before,
More informationInferences for Regression
Inferences for Regression An Example: Body Fat and Waist Size Looking at the relationship between % body fat and waist size (in inches). Here is a scatterplot of our data set: Remembering Regression In
More information7.2 One-Sample Correlation ( = a) Introduction. Correlation analysis measures the strength and direction of association between
7.2 One-Sample Correlation ( = a) Introduction Correlation analysis measures the strength and direction of association between variables. In this chapter we will test whether the population correlation
More informationChapter 10. Chapter 10. Multinomial Experiments and. Multinomial Experiments and Contingency Tables. Contingency Tables.
Chapter 10 Multinomial Experiments and Contingency Tables 1 Chapter 10 Multinomial Experiments and Contingency Tables 10-1 1 Overview 10-2 2 Multinomial Experiments: of-fitfit 10-3 3 Contingency Tables:
More informationInferences for Proportions and Count Data
Inferences for Proportions and Count Data Corresponds to Chapter 9 of Tamhane and Dunlop Slides prepared by Elizabeth Newton (MIT), with some slides by Ramón V. León (University of Tennessee) 1 Inference
More informationThe Components of a Statistical Hypothesis Testing Problem
Statistical Inference: Recall from chapter 5 that statistical inference is the use of a subset of a population (the sample) to draw conclusions about the entire population. In chapter 5 we studied one
More informationMathematical Notation Math Introduction to Applied Statistics
Mathematical Notation Math 113 - Introduction to Applied Statistics Name : Use Word or WordPerfect to recreate the following documents. Each article is worth 10 points and can be printed and given to the
More information11-2 Multinomial Experiment
Chapter 11 Multinomial Experiments and Contingency Tables 1 Chapter 11 Multinomial Experiments and Contingency Tables 11-11 Overview 11-2 Multinomial Experiments: Goodness-of-fitfit 11-3 Contingency Tables:
More informationChapter. Hypothesis Testing with Two Samples. Copyright 2015, 2012, and 2009 Pearson Education, Inc. 1
Chapter 8 Hypothesis Testing with Two Samples Copyright 2015, 2012, and 2009 Pearson Education, Inc 1 Two Sample Hypothesis Test Compares two parameters from two populations Sampling methods: Independent
More informationChapter 24. Comparing Means. Copyright 2010 Pearson Education, Inc.
Chapter 24 Comparing Means Copyright 2010 Pearson Education, Inc. Plot the Data The natural display for comparing two groups is boxplots of the data for the two groups, placed side-by-side. For example:
More informationM(t) = 1 t. (1 t), 6 M (0) = 20 P (95. X i 110) i=1
Math 66/566 - Midterm Solutions NOTE: These solutions are for both the 66 and 566 exam. The problems are the same until questions and 5. 1. The moment generating function of a random variable X is M(t)
More informationChapter 23: Inferences About Means
Chapter 3: Inferences About Means Sample of Means: number of observations in one sample the population mean (theoretical mean) sample mean (observed mean) is the theoretical standard deviation of the population
More informationPopulation 1 Population 2
Two Population Case Testing the Difference Between Two Population Means Sample of Size n _ Sample mean = x Sample s.d.=s x Sample of Size m _ Sample mean = y Sample s.d.=s y Pop n mean=μ x Pop n s.d.=
More informationTwo Sample Problems. Two sample problems
Two Sample Problems Two sample problems The goal of inference is to compare the responses in two groups. Each group is a sample from a different population. The responses in each group are independent
More informationUnit 9: Inferences for Proportions and Count Data
Unit 9: Inferences for Proportions and Count Data Statistics 571: Statistical Methods Ramón V. León 12/15/2008 Unit 9 - Stat 571 - Ramón V. León 1 Large Sample Confidence Interval for Proportion ( pˆ p)
More informationEstimating a Population Mean
Estimating a Population Mean MATH 130, Elements of Statistics I J. Robert Buchanan Department of Mathematics Fall 2017 Objectives At the end of this lesson we will be able to: obtain a point estimate for
More informationAP Statistics Ch 12 Inference for Proportions
Ch 12.1 Inference for a Population Proportion Conditions for Inference The statistic that estimates the parameter p (population proportion) is the sample proportion p ˆ. p ˆ = Count of successes in the
More informationEXAM 3 Math 1342 Elementary Statistics 6-7
EXAM 3 Math 1342 Elementary Statistics 6-7 Name Date ********************************************************************************************************************************************** MULTIPLE
More informationFormulas and Tables. for Elementary Statistics, Tenth Edition, by Mario F. Triola Copyright 2006 Pearson Education, Inc. ˆp E p ˆp E Proportion
Formulas and Tables for Elementary Statistics, Tenth Edition, by Mario F. Triola Copyright 2006 Pearson Education, Inc. Ch. 3: Descriptive Statistics x Sf. x x Sf Mean S(x 2 x) 2 s Å n 2 1 n(sx 2 ) 2 (Sx)
More informationLecture 10: Comparing two populations: proportions
Lecture 10: Comparing two populations: proportions Problem: Compare two sets of sample data: e.g. is the proportion of As in this semester 152 the same as last Fall? Methods: Extend the methods introduced
More informationStatistics for Business and Economics: Confidence Intervals for Proportions
Statistics for Business and Economics: Confidence Intervals for Proportions STT 315: Section 107 Acknowledgement: I d like to thank Dr. Ashoke Sinha for allowing me to use and edit the slides. Statistical
More informationIntroduction to Statistical Data Analysis III
Introduction to Statistical Data Analysis III JULY 2011 Afsaneh Yazdani Preface Major branches of Statistics: - Descriptive Statistics - Inferential Statistics Preface What is Inferential Statistics? The
More informationChapter 22. Comparing Two Proportions. Bin Zou STAT 141 University of Alberta Winter / 15
Chapter 22 Comparing Two Proportions Bin Zou (bzou@ualberta.ca) STAT 141 University of Alberta Winter 2015 1 / 15 Introduction In Ch.19 and Ch.20, we studied confidence interval and test for proportions,
More informationEcon 325: Introduction to Empirical Economics
Econ 325: Introduction to Empirical Economics Chapter 9 Hypothesis Testing: Single Population Ch. 9-1 9.1 What is a Hypothesis? A hypothesis is a claim (assumption) about a population parameter: population
More informationPurposes of Data Analysis. Variables and Samples. Parameters and Statistics. Part 1: Probability Distributions
Part 1: Probability Distributions Purposes of Data Analysis True Distributions or Relationships in the Earths System Probability Distribution Normal Distribution Student-t Distribution Chi Square Distribution
More informationNominal Data. Parametric Statistics. Nonparametric Statistics. Parametric vs Nonparametric Tests. Greg C Elvers
Nominal Data Greg C Elvers 1 Parametric Statistics The inferential statistics that we have discussed, such as t and ANOVA, are parametric statistics A parametric statistic is a statistic that makes certain
More informationCHAPTER 9, 10. Similar to a courtroom trial. In trying a person for a crime, the jury needs to decide between one of two possibilities:
CHAPTER 9, 10 Hypothesis Testing Similar to a courtroom trial. In trying a person for a crime, the jury needs to decide between one of two possibilities: The person is guilty. The person is innocent. To
More information10.1. Comparing Two Proportions. Section 10.1
/6/04 0. Comparing Two Proportions Sectio0. Comparing Two Proportions After this section, you should be able to DETERMINE whether the conditions for performing inference are met. CONSTRUCT and INTERPRET
More informationStatistical Inference. Why Use Statistical Inference. Point Estimates. Point Estimates. Greg C Elvers
Statistical Inference Greg C Elvers 1 Why Use Statistical Inference Whenever we collect data, we want our results to be true for the entire population and not just the sample that we used But our sample
More informationChapter 12 - Lecture 2 Inferences about regression coefficient
Chapter 12 - Lecture 2 Inferences about regression coefficient April 19th, 2010 Facts about slope Test Statistic Confidence interval Hypothesis testing Test using ANOVA Table Facts about slope In previous
More informationChapter 1 Statistical Inference
Chapter 1 Statistical Inference causal inference To infer causality, you need a randomized experiment (or a huge observational study and lots of outside information). inference to populations Generalizations
More information10.4 Hypothesis Testing: Two Independent Samples Proportion
10.4 Hypothesis Testing: Two Independent Samples Proportion Example 3: Smoking cigarettes has been known to cause cancer and other ailments. One politician believes that a higher tax should be imposed
More informationAP Statistics Cumulative AP Exam Study Guide
AP Statistics Cumulative AP Eam Study Guide Chapters & 3 - Graphs Statistics the science of collecting, analyzing, and drawing conclusions from data. Descriptive methods of organizing and summarizing statistics
More informationThis gives us an upper and lower bound that capture our population mean.
Confidence Intervals Critical Values Practice Problems 1 Estimation 1.1 Confidence Intervals Definition 1.1 Margin of error. The margin of error of a distribution is the amount of error we predict when
More information1 Hypothesis testing for a single mean
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike License. Your use of this material constitutes acceptance of that license and the conditions of use of materials on this
More informationFormulas and Tables. for Essentials of Statistics, by Mario F. Triola 2002 by Addison-Wesley. ˆp E p ˆp E Proportion.
Formulas and Tables for Essentials of Statistics, by Mario F. Triola 2002 by Addison-Wesley. Ch. 2: Descriptive Statistics x Sf. x x Sf Mean S(x 2 x) 2 s Å n 2 1 n(sx 2 ) 2 (Sx) 2 s Å n(n 2 1) Mean (frequency
More informationData Analysis and Statistical Methods Statistics 651
Data Analysis and Statistical Methods Statistics 651 http://www.stat.tamu.edu/~suhasini/teaching.html Suhasini Subba Rao Motivations for the ANOVA We defined the F-distribution, this is mainly used in
More informationProbability and Statistics
The big picture Probability and Statistics Sample Population 1) Data Collection Data ) Explanatory Data Analysis (EDA) Inference on Relationship Between two Variables 4) Inference 3) Probability The Big
More informationECO220Y Review and Introduction to Hypothesis Testing Readings: Chapter 12
ECO220Y Review and Introduction to Hypothesis Testing Readings: Chapter 12 Winter 2012 Lecture 13 (Winter 2011) Estimation Lecture 13 1 / 33 Review of Main Concepts Sampling Distribution of Sample Mean
More informationHarvard University. Rigorous Research in Engineering Education
Statistical Inference Kari Lock Harvard University Department of Statistics Rigorous Research in Engineering Education 12/3/09 Statistical Inference You have a sample and want to use the data collected
More informationCHAPTER 10 Comparing Two Populations or Groups
CHAPTER 10 Comparing Two Populations or Groups 10.1 Comparing Two Proportions The Practice of Statistics, 5th Edition Starnes, Tabor, Yates, Moore Bedford Freeman Worth Publishers Comparing Two Proportions
More informationIntroduction to Statistics
MTH4106 Introduction to Statistics Notes 15 Spring 2013 Testing hypotheses about the mean Earlier, we saw how to test hypotheses about a proportion, using properties of the Binomial distribution It is
More information10.2: The Chi Square Test for Goodness of Fit
10.2: The Chi Square Test for Goodness of Fit We can perform a hypothesis test to determine whether the distribution of a single categorical variable is following a proposed distribution. We call this
More information