# Solution to HW - 1. Problem 1. [Points = 3] In September, Chapel Hill s daily high temperature has a mean

Save this PDF as:

Size: px
Start display at page:

Download "Solution to HW - 1. Problem 1. [Points = 3] In September, Chapel Hill s daily high temperature has a mean"

## Transcription

1 Problem 1. [Points = 3] In September, Chapel Hill s daily high temperature has a mean of 81 degree F and a standard deviation of 10 degree F. What is the mean, standard deviation and variance in terms of Celsius? Answer 1: Note that C = (5/9) (F 32) = a + b F where a = 160/9 and b = 5/9. Therefore, the mean temperature is a+b degree Celsius. The standard deviation is b degree Celsius. Problem 2. [Points = 3] In a given population of two-earner male/female couples, male earnings have a mean of 40,000 USD per year and a standard deviation of 12,000 USD. Female earnings have a mean of 45,000 USD per year and a standard deviation of 18,000 USD. The correlation between male and female earnings for a couple is Let C denote the combined earnings for a randomly selected couple. (a) What is the mean of C? (b) What is the covariance between male and female earnings? (c) What is the standard deviation of C? Answer 2: Let us denote male earnings by m and female earnings by f. (a) C = m + f, and hence C = m + f = 85, 000 USD. (b) Cov(m, f) = Corr(m, f) sd(m) sd(f) =.8 12, , 000 = (c) sd(c) = sd(m) 2 + sd(f) Cov(m, f) = USD. Problem 3. [Points = 3] X and Y are discrete random variables with the following joint distribution: That is, P r[x = 1, Y = 14] = 0.02, and so forth. 1

2 Joint Probability Value of Y Distribution Value of X (a) Calculate the probability distribution, mean, and variance of Y. (b) Calculate the probability distribution, mean, and variance of Y given X = 8. (c) Calculate the covariance and correlation between X and Y. Answer 3: (a) The probability distribution of Y is given by P [Y = 14] = P [X = 1, Y = 14] + P [X = 5, Y = 14] + P [X = 8, Y = 14] =.21 P [Y = 22] = P [X = 1, Y = 22] + P [X = 5, Y = 22] + P [X = 8, Y = 22] =.23 P [Y = 30] = P [X = 1, Y = 30] + P [X = 5, Y = 30] + P [X = 8, Y = 30] =.30 P [Y = 40] = P [X = 1, Y = 40] + P [X = 5, Y = 40] + P [X = 8, Y = 40] =.15 P [Y = 65] = P [X = 1, Y = 65] + P [X = 5, Y = 65] + P [X = 8, Y = 65] =.11. Now note that E[Y ] = 14 P [Y = 14]+22 P [Y = 22]+30 P [Y = 30]+40 P [Y = 40] + 65 P [Y = 65] = and that E[Y 2 ] = 14 2 P [Y = 14] P [Y = 22] P [Y = 30] P [Y = 40] P [Y = 65] = Therefore, V (Y ) = E[Y 2 ] E 2 [Y ] = (b) The probability distribution of Y conditional on X = 8 is given by P [Y = 14 X = 8] = P [X = 8, Y = 14]/P [X = 8] =.05 P [Y = 22 X = 8] = P [X = 8, Y = 22]/P [X = 8] =.08 2

3 P [Y = 30 X = 8] = P [X = 8, Y = 30]/P [X = 8] =.38 P [Y = 40 X = 8] = P [X = 8, Y = 40]/P [X = 8] =.26 P [Y = 65 X = 8] = P [X = 8, Y = 65]/P [X = 8] =.23, where I use P [X = 8] from the following computations P [X = 1] = P [X = 1, Y = 14] + P [X = 1, Y = 22] + P [X = 1, Y = 30] + P [X = 1, Y = 40] + P [X = 1, Y = 65] =.21 P [X = 5] = P [X = 5, Y = 14] + P [X = 5, Y = 22] + P [X = 5, Y = 30] + P [X = 5, Y = 40] + P [X = 5, Y = 65] =.4 P [X = 8] = P [X = 8, Y = 14] + P [X = 8, Y = 22] + P [X = 8, Y = 30] + P [X = 8, Y = 40] + P [X = 8, Y = 65] =.39. Now note that E[Y X = 8] = 14 P [Y = 14 X = 8] + 22 P [Y = 22 X = 8] + 30 P [Y = 30 X = 8] + 40 P [Y = 40 X = 8] + 65 P [Y = 65 X = 8] = and that E[Y 2 X = 8] = 14 2 P [Y = 14 X = 8] P [Y = 22 X = 8] P [Y = 30 X = 8] P [Y = 40 X = 8] P [Y = 65 X = 8] = Therefore, V (Y X = 8) = E[Y 2 X = 8] E 2 [Y X = 8] = (c) Cov(X, Y ) = E[XY ] E[X]E[Y ] and Corr(X, Y ) = Cov(X, Y )/ V (X)V (Y ). So we will further need to compute E[X], V (X) and E[XY ]. Note that E[X] = 1 P [X = 1] + 5 P [X = 5] + 8 P [X = 8] = 5.33, E[X 2 ] = 1 2 P [X = 1] P [X = 5]+8 2 P [X = 8] = 35.7 and hence V (X) = E[X 2 ] E 2 [X] = Also, by a similar computation we get E[XY ] = Therefore, Cov(X, Y ) = E[XY ] E[X]E[Y ] = and Corr(X, Y ) =.29. Problem 4. [Points = 4] Let X and Z be independently distributed standard normal random variables, and let Y = X 2 + Z. 3

4 (a) Show that E[Y X] = X 2. (b) Show that E[Y ] = 1. (c) Show that E[XY ] = 0. (Hint: Use the fact that the odd moments of a standard normal random variable are all zero.) (d) Show that Cov(X, Y ) = 0. Answer 4: X N(0, 1) E[X] = 0, V (X) = E[X 2 ] E 2 [X] = 1, and the same for Z. Also X and Z independent implies E[Z X] = E[Z] = 0 and Cov(X, Z) = E[XZ] E[X]E[Z] = 0 0 = 0. (a) E[Y X] = E[X 2 + Z X] = E[X 2 X] + E[Z X] = X = X 2. (b) By the law of iterated expectations, we have E[Y ] = E X [E[Y X]] = E X [X 2 ] = 1. (c) Cov(X, Y ) = Cov(X, X 2 + Z) = Cov(X, X 2 ) + Cov(X, Z) = Cov(X, X 2 ) = E[X 3 ] E[X]E[X 2 ] = E[X 3 ] = 0 (because the odd moments of N(0, 1) are zero). Problem 5. [Points = 7] Grades on a standardized test are known to have a mean of 1000 for students in the United States. The test is administered to 453 randomly selected students in Florida. In this sample, the mean is 1013 and the standard deviation is 108. (a) Construct a 95 % confidence interval for the average test score for Florida students. (b) Is there statistically significant evidence that Florida students perform differently than other students in the United States? (c) Another 503 students are selected at random from Florida. They are given a threehour preparation course before the test is administered. Their average test score is 1019 with a standard deviation of 95. 4

5 (i) Construct a 95 % confidence interval for the change in average test score associated with the prep course. (ii) Is there a statistically significant evidence that the prep course helped? (d) The original 453 students are given the prep course and then asked to take the test a second time. The average change in their test scores is 9 points and the standard deviation of the change is 60 points. (i) Construct a 95 % confidence interval for the change in average test scores. (ii) Is there statistically significant evidence that students will perform better on their second attempt after taking the prep course? (iii) Students may have performed better in their second attempt because of the prep course or because they gained test-taking experience in their first attempt. Describe an experiment that would quantify these two effects. Answer 5: (a) A 95 % confidence interval will be [(Florida mean) ± 1.96 (Florida sd)/ n] = [ , ]. (b) Note that the National average 1000 is not included in the 95 % confidence for Florida. So we can say that we reject at the 5 percent level the hypothesis that the Florida students perform the same as the students from all over the nation. (c) (i) Note that the students with the prep course are different from the original Florida students and it is hence reasonable to assume that there is no correlation between the two groups. Let E[old: score w/o prep] = µ wo and E[new: score with prep] = µ w. We want to compute a 95 % confidence interval for = µ w µ wo, i.e. the 5

6 change in the average test score. Note that = X new:w X old:wo = = 6 and since we can assume the scores of the two groups to be uncorrelated, sd( ) = sd 2 new:w/n new:w + sd 2 old:wo /n old:wo = / / Therefore, a 95 % confidence interval for the change is [ ± 1.96 sd( )] = [ 6.96, 18.96]. (ii) Note that 0 in inside this confidence interval (and so are some negative values). However, the confidence interval also contains positive values, i.e. an increase in average score. Hence we cannot reject at the 5 % level the null hypothesis that the prep course had no effect. But we also cannot reject at the 5 % level that the increase in average score is equal to some positive value ranging from 0 to (d) (i) A 95 % confidence interval will be [(change mean) ± 1.96 (change sd)/ n] = [3.47, 14.53]. (ii) Of course. Note that no-negative or zero value is included in the confidence interval. (iii) It will be more effective to divide the 453 students randomly into two groups (of roughly equal size) and ask one group to take the test the second time without the prep course and the other group to take the test after a prep course. We will discuss more on this in the class. 6

### Covariance and Correlation. Consider the joint probability distribution f XY (x, y).

Chapter 5: JOINT PROBABILITY DISTRIBUTIONS Part 2: Section 5-2 Covariance and Correlation Consider the joint probability distribution f XY (x, y). Is there a relationship between X and Y? If so, what kind?

### Joint Probability Distributions and Random Samples. Week 5, 2011 Stat 4570/5570 Material from Devore s book (Ed 8), and Cengage

5 Joint Probability Distributions and Random Samples Week 5, 2011 Stat 4570/5570 Material from Devore s book (Ed 8), and Cengage Two Discrete Random Variables The probability mass function (pmf) of a single

### 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

### Covariance and Correlation

Covariance and Correlation ( c Robert J. Serfling Not for reproduction or distribution) We have seen how to summarize a data-based relative frequency distribution by measures of location and spread, such

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

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

### Topic 4: Multivariate random variables. Multiple random variables

Topic 4: Multivariate random variables Joint, marginal, and conditional pmf Joint, marginal, and conditional pdf and cdf Independence Expectation, covariance, correlation Conditional expectation Two jointly

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

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

### Paired vs. Pooled Example Simulation to Compare Power

Math 3070 1. Treibergs Paired vs. Pooled Example Simulation to Compare Power Name: Example July 30, 2011 This is Problem 6.15.10 of Navidi, Statistics for Engineers and Scientists, 2nd ed., Mc Graw Hill,

### Examination 110 Probability and Statistics Examination

Examination 0 Probability and Statistics Examination Sample Examination Questions The Probability and Statistics Examination consists of 5 multiple-choice test questions. The test is a three-hour examination

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

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

### Topic 8 The Expected Value

Topic 8 The Expected Value Functions of Random Variables 1 / 12 Outline Names for Eg(X ) Variance and Standard Deviation Independence Covariance and Correlation 2 / 12 Names for Eg(X ) If g(x) = x, then

### How to Conduct a Hypothesis Test

How to Conduct a Hypothesis Test The idea of hypothesis testing is relatively straightforward. In various studies we observe certain events. We must ask, is the event due to chance alone, or is there some

### Homework Zero. Max H. Farrell Chicago Booth BUS41100 Applied Regression Analysis. Complete before the first class, do not turn in

Homework Zero Max H. Farrell Chicago Booth BUS41100 Applied Regression Analysis Complete before the first class, do not turn in This homework is intended as a self test of your knowledge of the basic statistical

### Expectation Discrete RV - weighted average Continuous RV - use integral to take the weighted average

PHP 2510 Expectation, variance, covariance, correlation Expectation Discrete RV - weighted average Continuous RV - use integral to take the weighted average Variance Variance is the average of (X µ) 2

### Joint Distribution and Correlation

Joint Distribution and Correlation Michael Ash Lecture 3 Reminder: Start working on the Problem Set Mean and Variance of Linear Functions of an R.V. Linear Function of an R.V. Y = a + bx What are the properties

### ST 371 (VIII): Theory of Joint Distributions

ST 371 (VIII): Theory of Joint Distributions So far we have focused on probability distributions for single random variables. However, we are often interested in probability statements concerning two or

### Solutions for the exam for Matematisk statistik och diskret matematik (MVE050/MSG810). Statistik för fysiker (MSG820). December 15, 2012.

Solutions for the exam for Matematisk statistik och diskret matematik (MVE050/MSG810). Statistik för fysiker (MSG8). December 15, 12. 1. (3p) The joint distribution of the discrete random variables X and

### Correlation in Random Variables

Correlation in Random Variables Lecture 11 Spring 2002 Correlation in Random Variables Suppose that an experiment produces two random variables, X and Y. What can we say about the relationship between

### 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

### Probability, Binomial Distributions and Hypothesis Testing Vartanian, SW 540

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

### Chapter 16 Multiple Choice Questions (The answers are provided after the last question.)

Chapter 16 Multiple Choice Questions (The answers are provided after the last question.) 1. Which of the following symbols represents a population parameter? a. SD b. σ c. r d. 0 2. If you drew all possible

### Random Variables, Expectation, Distributions

Random Variables, Expectation, Distributions CS 5960/6960: Nonparametric Methods Tom Fletcher January 21, 2009 Review Random Variables Definition A random variable is a function defined on a probability

### Statistics for Management II-STAT 362-Final Review

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

### E205 Final: Version B

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

### Simple Linear Regression Chapter 11

Simple Linear Regression Chapter 11 Rationale Frequently decision-making situations require modeling of relationships among business variables. For instance, the amount of sale of a product may be related

### Exclusive OR (XOR) and hardware random number generators

Exclusive OR (XOR) and hardware random number generators Robert B Davies February 28, 2002 1 Introduction The exclusive or (XOR) operation is commonly used to reduce the bias from the bits generated by

### For a partition B 1,..., B n, where B i B j = for i. A = (A B 1 ) (A B 2 ),..., (A B n ) and thus. P (A) = P (A B i ) = P (A B i )P (B i )

Probability Review 15.075 Cynthia Rudin A probability space, defined by Kolmogorov (1903-1987) consists of: A set of outcomes S, e.g., for the roll of a die, S = {1, 2, 3, 4, 5, 6}, 1 1 2 1 6 for the roll

### Notes for STA 437/1005 Methods for Multivariate Data

Notes for STA 437/1005 Methods for Multivariate Data Radford M. Neal, 26 November 2010 Random Vectors Notation: Let X be a random vector with p elements, so that X = [X 1,..., X p ], where denotes transpose.

### Math 141. Lecture 7: Variance, Covariance, and Sums. Albyn Jones 1. 1 Library 304. jones/courses/141

Math 141 Lecture 7: Variance, Covariance, and Sums Albyn Jones 1 1 Library 304 jones@reed.edu www.people.reed.edu/ jones/courses/141 Last Time Variance: expected squared deviation from the mean: Standard

### Less Stress More Success Maths Leaving Cert Higher Level Paper 2

Less Stress More Success Maths Leaving Cert Higher Level Paper 2 Revised pages for Chapter 13 Statistics IV: The Normal Curve, z -Scores, Hypothesis Testing and Simulation 228 LESS STRESS MORE SUCCESS

### Calculate the holding period return for this investment. It is approximately

1. An investor purchases 100 shares of XYZ at the beginning of the year for \$35. The stock pays a cash dividend of \$3 per share. The price of the stock at the time of the dividend is \$30. The dividend

### The Bivariate Normal Distribution

The Bivariate Normal Distribution This is Section 4.7 of the st edition (2002) of the book Introduction to Probability, by D. P. Bertsekas and J. N. Tsitsiklis. The material in this section was not included

### Chapters 5. Multivariate Probability Distributions

Chapters 5. Multivariate Probability Distributions Random vectors are collection of random variables defined on the same sample space. Whenever a collection of random variables are mentioned, they are

Expectations Expectations. (See also Hays, Appendix B; Harnett, ch. 3). A. The expected value of a random variable is the arithmetic mean of that variable, i.e. E() = µ. As Hays notes, the idea of the

### Definition The covariance of X and Y, denoted by cov(x, Y ) is defined by. cov(x, Y ) = E(X µ 1 )(Y µ 2 ).

Correlation Regression Bivariate Normal Suppose that X and Y are r.v. s with joint density f(x y) and suppose that the means of X and Y are respectively µ 1 µ 2 and the variances are 1 2. Definition The

### 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

### Worked examples Multiple Random Variables

Worked eamples Multiple Random Variables Eample Let X and Y be random variables that take on values from the set,, } (a) Find a joint probability mass assignment for which X and Y are independent, and

### Chapter 11: Linear Regression - Inference in Regression Analysis - Part 2

Chapter 11: Linear Regression - Inference in Regression Analysis - Part 2 Note: Whether we calculate confidence intervals or perform hypothesis tests we need the distribution of the statistic we will use.

### Exercises with solutions (1)

Exercises with solutions (). Investigate the relationship between independence and correlation. (a) Two random variables X and Y are said to be correlated if and only if their covariance C XY is not equal

### Simple Linear Regression in SPSS STAT 314

Simple Linear Regression in SPSS STAT 314 1. Ten Corvettes between 1 and 6 years old were randomly selected from last year s sales records in Virginia Beach, Virginia. The following data were obtained,

### Chapter Additional: Standard Deviation and Chi- Square

Chapter Additional: Standard Deviation and Chi- Square Chapter Outline: 6.4 Confidence Intervals for the Standard Deviation 7.5 Hypothesis testing for Standard Deviation Section 6.4 Objectives Interpret

### Power & Effect Size power Effect Size

Power & Effect Size Until recently, researchers were primarily concerned with controlling Type I errors (i.e. finding a difference when one does not truly exist). Although it is important to make sure

### Sample Size Determination

Sample Size Determination Population A: 10,000 Population B: 5,000 Sample 10% Sample 15% Sample size 1000 Sample size 750 The process of obtaining information from a subset (sample) of a larger group (population)

### p1^ = 0.18 p2^ = 0.12 A) 0.150 B) 0.387 C) 0.300 D) 0.188 3) n 1 = 570 n 2 = 1992 x 1 = 143 x 2 = 550 A) 0.270 B) 0.541 C) 0.520 D) 0.

Practice for chapter 9 and 10 Disclaimer: the actual exam does not mirror this. This is meant for practicing questions only. The actual exam in not multiple choice. Find the number of successes x suggested

### Joint Distributions. Tieming Ji. Fall 2012

Joint Distributions Tieming Ji Fall 2012 1 / 33 X : univariate random variable. (X, Y ): bivariate random variable. In this chapter, we are going to study the distributions of bivariate random variables

### Lecture 16: Expected value, variance, independence and Chebyshev inequality

Lecture 16: Expected value, variance, independence and Chebyshev inequality Expected value, variance, and Chebyshev inequality. If X is a random variable recall that the expected value of X, E[X] is the

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

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

### Variables and Data A variable contains data about anything we measure. For example; age or gender of the participants or their score on a test.

The Analysis of Research Data The design of any project will determine what sort of statistical tests you should perform on your data and how successful the data analysis will be. For example if you decide

### Wording of Final Conclusion. Slide 1

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

### Lecture 2: Simple Linear Regression

DMBA: Statistics Lecture 2: Simple Linear Regression Least Squares, SLR properties, Inference, and Forecasting Carlos Carvalho The University of Texas McCombs School of Business mccombs.utexas.edu/faculty/carlos.carvalho/teaching

### Chicago Booth BUSINESS STATISTICS 41000 Final Exam Fall 2011

Chicago Booth BUSINESS STATISTICS 41000 Final Exam Fall 2011 Name: Section: I pledge my honor that I have not violated the Honor Code Signature: This exam has 34 pages. You have 3 hours to complete this

### CHAPTER 6: Continuous Uniform Distribution: 6.1. Definition: The density function of the continuous random variable X on the interval [A, B] is.

Some Continuous Probability Distributions CHAPTER 6: Continuous Uniform Distribution: 6. Definition: The density function of the continuous random variable X on the interval [A, B] is B A A x B f(x; A,

### 13 Two-Sample T Tests

www.ck12.org CHAPTER 13 Two-Sample T Tests Chapter Outline 13.1 TESTING A HYPOTHESIS FOR DEPENDENT AND INDEPENDENT SAMPLES 270 www.ck12.org Chapter 13. Two-Sample T Tests 13.1 Testing a Hypothesis for

### N Mean Std. Deviation Std. Error of Mean

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

### 3.6: General Hypothesis Tests

3.6: General Hypothesis Tests The χ 2 goodness of fit tests which we introduced in the previous section were an example of a hypothesis test. In this section we now consider hypothesis tests more generally.

### Descriptive Statistics

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

### 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

### Math 431 An Introduction to Probability. Final Exam Solutions

Math 43 An Introduction to Probability Final Eam Solutions. A continuous random variable X has cdf a for 0, F () = for 0 <

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

Practice for Chapter 9 and 10 The acutal exam differs. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Find the number of successes x suggested by the

### Module 5 Hypotheses Tests: Comparing Two Groups

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

### Lesson 5 Chapter 4: Jointly Distributed Random Variables

Lesson 5 Chapter 4: Jointly Distributed Random Variables Department of Statistics The Pennsylvania State University 1 Marginal and Conditional Probability Mass Functions The Regression Function Independence

### where b is the slope of the line and a is the intercept i.e. where the line cuts the y axis.

Least Squares Introduction We have mentioned that one should not always conclude that because two variables are correlated that one variable is causing the other to behave a certain way. However, sometimes

### The Scalar Algebra of Means, Covariances, and Correlations

3 The Scalar Algebra of Means, Covariances, and Correlations In this chapter, we review the definitions of some key statistical concepts: means, covariances, and correlations. We show how the means, variances,

### MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Open book and note Calculator OK Multiple Choice 1 point each MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the mean for the given sample data.

### The correlation coefficient

The correlation coefficient Clinical Biostatistics The correlation coefficient Martin Bland Correlation coefficients are used to measure the of the relationship or association between two quantitative

### STA218 Introduction to Hypothesis Testing

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

### Research Variables. Measurement. Scales of Measurement. Chapter 4: Data & the Nature of Measurement

Chapter 4: Data & the Nature of Graziano, Raulin. Research Methods, a Process of Inquiry Presented by Dustin Adams Research Variables Variable Any characteristic that can take more than one form or value.

### 4. Joint Distributions of Two Random Variables

4. Joint Distributions of Two Random Variables 4.1 Joint Distributions of Two Discrete Random Variables Suppose the discrete random variables X and Y have supports S X and S Y, respectively. The joint

### Chapter 11: Two Variable Regression Analysis

Department of Mathematics Izmir University of Economics Week 14-15 2014-2015 In this chapter, we will focus on linear models and extend our analysis to relationships between variables, the definitions

### Chapter 5. Random variables

Random variables random variable numerical variable whose value is the outcome of some probabilistic experiment; we use uppercase letters, like X, to denote such a variable and lowercase letters, like

### II. DISTRIBUTIONS distribution normal distribution. standard scores

Appendix D Basic Measurement And Statistics The following information was developed by Steven Rothke, PhD, Department of Psychology, Rehabilitation Institute of Chicago (RIC) and expanded by Mary F. Schmidt,

### 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

### Bivariate Distributions

Chapter 4 Bivariate Distributions 4.1 Distributions of Two Random Variables In many practical cases it is desirable to take more than one measurement of a random observation: (brief examples) 1. What is

### SYSM 6304: Risk and Decision Analysis Lecture 3 Monte Carlo Simulation

SYSM 6304: Risk and Decision Analysis Lecture 3 Monte Carlo Simulation M. Vidyasagar Cecil & Ida Green Chair The University of Texas at Dallas Email: M.Vidyasagar@utdallas.edu September 19, 2015 Outline

### ECE302 Spring 2006 HW7 Solutions March 11, 2006 1

ECE32 Spring 26 HW7 Solutions March, 26 Solutions to HW7 Note: Most of these solutions were generated by R. D. Yates and D. J. Goodman, the authors of our textbook. I have added comments in italics where

### Joint Exam 1/P Sample Exam 1

Joint Exam 1/P Sample Exam 1 Take this practice exam under strict exam conditions: Set a timer for 3 hours; Do not stop the timer for restroom breaks; Do not look at your notes. If you believe a question

### 7 Hypothesis testing - one sample tests

7 Hypothesis testing - one sample tests 7.1 Introduction Definition 7.1 A hypothesis is a statement about a population parameter. Example A hypothesis might be that the mean age of students taking MAS113X

### The alternative hypothesis,, is the statement that the parameter value somehow differs from that claimed by the null hypothesis. : 0.5 :>0.5 :<0.

Section 8.2-8.5 Null and Alternative Hypotheses... The null hypothesis,, is a statement that the value of a population parameter is equal to some claimed value. :=0.5 The alternative hypothesis,, is the

### 6. Duality between confidence intervals and statistical tests

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

### Chapter 6 Continuous Probability Distributions

Continuous Probability Distributions Learning Objectives 1. Understand the difference between how probabilities are computed for discrete and continuous random variables. 2. Know how to compute probability

### 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

### Hypothesis testing. Power of a test. Alternative is greater than Null. Probability

Probability February 14, 2013 Debdeep Pati Hypothesis testing Power of a test 1. Assuming standard deviation is known. Calculate power based on one-sample z test. A new drug is proposed for people with

### Chapter 11-12 1 Review

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

### Chapter 2, part 2. Petter Mostad

Chapter 2, part 2 Petter Mostad mostad@chalmers.se Parametrical families of probability distributions How can we solve the problem of learning about the population distribution from the sample? Usual procedure:

### Math 62 Statistics Sample Exam Questions

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

### 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

### 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,

### Notes 11 Autumn 2005

MAS 08 Probabilit I Notes Autumn 005 Two discrete random variables If X and Y are discrete random variables defined on the same sample space, then events such as X = and Y = are well defined. The joint

Name: University of Chicago Graduate School of Business Business 41000: Business Statistics Special Notes: 1. This is a closed-book exam. You may use an 8 11 piece of paper for the formulas. 2. Throughout

### 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

### When σ Is Known: Recall the Mystery Mean Activity where x bar = 240.79 and we have an SRS of size 16

8.3 ESTIMATING A POPULATION MEAN When σ Is Known: Recall the Mystery Mean Activity where x bar = 240.79 and we have an SRS of size 16 Task was to estimate the mean when we know that the situation is Normal

### Homework #3 is due Friday by 5pm. Homework #4 will be posted to the class website later this week. It will be due Friday, March 7 th, at 5pm.

Homework #3 is due Friday by 5pm. Homework #4 will be posted to the class website later this week. It will be due Friday, March 7 th, at 5pm. Political Science 15 Lecture 12: Hypothesis Testing Sampling

### Dr. Peter Tröger Hasso Plattner Institute, University of Potsdam. Software Profiling Seminar, Statistics 101

Dr. Peter Tröger Hasso Plattner Institute, University of Potsdam Software Profiling Seminar, 2013 Statistics 101 Descriptive Statistics Population Object Object Object Sample numerical description Object

### 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,

### Section 12.2, Lesson 3. What Can Go Wrong in Hypothesis Testing: The Two Types of Errors and Their Probabilities

Today: Section 2.2, Lesson 3: What can go wrong with hypothesis testing Section 2.4: Hypothesis tests for difference in two proportions ANNOUNCEMENTS: No discussion today. Check your grades on eee and

### Introduction to Statistics for Computer Science Projects

Introduction Introduction to Statistics for Computer Science Projects Peter Coxhead Whole modules are devoted to statistics and related topics in many degree programmes, so in this short session all I

### Lecture 8. Confidence intervals and the central limit theorem

Lecture 8. Confidence intervals and the central limit theorem Mathematical Statistics and Discrete Mathematics November 25th, 2015 1 / 15 Central limit theorem Let X 1, X 2,... X n be a random sample of