Mock Exam - 2 hours - use of basic (non-programmable) calculator is allowed - all exercises carry the same marks - exam is strictly individual
|
|
- Ashlee Holmes
- 5 years ago
- Views:
Transcription
1 Mock Exam - 2 hours - use of basic (non-programmable) calculator is allowed - all exercises carry the same marks - exam is strictly individual Question 1. Suppose you want to estimate the percentage of videos on YouTube that are cat videos. It is impossible for you to watch all videos on YouTube so you use a random video picker to select 1000 videos for you. You find that 2% of these videos are cat videos. Determine which of the following is an observation, a variable, a sample statistic, or a population parameter. (a) Percentage of all videos on YouTube that are cat videos. (b) 2%. (c) A video in your sample. (d) Whether or not a video is a cat video. (a) Population parameter (b) Sample statistics (c) Observation (d) Variable Question 2. Suppose you have used a Normal distribution to compute the 95% confidence interval for the mean of a population with known variance, using a sample of size 25 and obtained the interval [ 1; 3]. (a) What is the variance of the population? (b) What is the sample size needed to reduce the length of such interval by half? (a) The computation of a 95% confidence interval is ˆx ± z SE, thus ˆx = 3+( 1) = 1 and 2 z SE = 2. We know that SE = σ n, and from the z-table in order to leave area in each tail, we have z = 1.96: z σ SE = 1.96 = 2 σ = = σ2 = (b) We want z SE = n = 1, therefore n = 100. Question 3. In a given one-sample t-test against null hypothesis H 0 of zero mean, the test statistic is found to be The student who conducted the test concludes that she cannot reject the null hypothesis. Which of the following scenarios does not justify the student s decision (df means degrees of freedom; alpha is the significance level of the test)? (a) df = 10, alpha = 0.01, and a one-sided greater than alternative hypothesis. (b) df = 60, alpha = 0.05, and a two-sided alternative hypothesis. (c) df = 10, alpha = 0.05, and a one-sided greater than alternative hypothesis. (d) df = 500, alpha = 0.01, and two-sided alternative hypothesis. (e) all of these scenarios would justify the failure to reject H 0. (a) T = 1.80, df = 10, one-side greater than as alternative, implies 0.05 < pvalue < 0.1 (from t-table with one-tail). So student is justified. (b) T = 1.80, df = 60, two-sided alternative, implies 0.05 < pvalue < 0.1 (from t-table with
2 two-tails, df = 30 as approximation, or could use Normal table since df is large, would result in pvalue=0.0718). So student is justified. (c) T = 1.80, df = 10, one-side greater than as alternative, implies 0.05 < pvalue < 0.1 (from t-table with one-tail). So student is justified. (d) T = 1.80, df = 500, two-sided alternative, imples 0.05 < pvalue < 0.1 (from t-table with two-tails, df = 30, or could use Normal table since df is large, would result in pvalue=0.0718). So student is justified. (e) This is the correct answer. Question 4. Thirty randomly chosen people were asked in Utrecht if they ever had their bike stolen, and 20 answered yes. Two hundred random people were asked in Amsterdam, and 76 answered yes. (a) Construct a 90% confidence interval for the difference in percentage of people who had a bike stolen in Amsterdam minus Utrecht. (b) With a significance level of 0.05, check if the percentage of people who had a bike stolen in Amsterdam is higher than in Utrecht. Proportion in Utrecht is p 1 = 20/n 1 = , with n 1 = 30. Proportion in Amsterdam is p 2 = 76/n 2 = 0.38, with n 2 = 200. We have at least 10 people in each group, so we will approximate with the Normal distribution. (a) The standard error for a confidence interval of proportions of two groups is: SE = p 1 (1 p 1 ) + p 2(1 p 2 ) = n 1 n 2 The 90% confidence interval is (from z-table we have z = 1.65): (p 2 p 1 ) ± z SE = ± = ( , ). (b) The null hypothesis would be H 0 : percentage of people in Amsterdam less than or equal to in Utrecht (p 2 p 1, or equally ok is p 2 = p 1 ); alternative H A : percentage of people in Amsterdam is higher than in Utrecht (that is, p 2 > p 1 ). There is no reason to run a test, since the difference is already towards more cases in Utrecht, so we are sure that we cannot reject the null hypothesis. Hypothetically, let us assume that the question was check if the number in Amsterdam is lower than in Utrecht. Then H 0 : percentage of people in Amsterdam greater than or equal to in Utrecht (p 2 = p 1, or equally ok is p 2 p 1 ); alternative H A : percentage of people in Amsterdam is lower than in Utrecht (p 2 < p 1 ). Under the null hypothesis, the proportions should be the same, so we can use the pooled estimation to compute the SE: pooledˆp = = SE = pooledˆp (1 pooledˆp) pooledˆp (1 pooledˆp) + = n 1 n = The Z statistic is Z = (p 2 p 1 ) 0 = , so from the Normal table we find pvalue= (1 SE ) = , and we would reject the null hypothesis. Note that the calculation using the SE from the non-pooled version would lead to the same conclusion (very small difference in the numbers). The pooled SE is preferrable for this hypothesis testing, but the difference is very often negligible.
3 Question 5. An independent random sample is selected from an approximately normal population with unknown standard deviation. Find the degrees of freedom and the critical t-value (t ) for the given sample size and confidence level. (a) n = 6, CL = 98% (b) n = 12, CL = 95% (c) n = 29, CL = 99% Results obtained by inspecting the table for two-tails since it is a confidence interval: (a) df = 6 1 = 5, t 5;0.02 = (b) df = 12 1 = 11, t 11;0.05 = (c) df = 29 1 = 28, t 28;0.99 = Question 6. Air quality measurements were collected in a random sample of 40 country capitals in 2013, and then again in the same cities in We would like to use these data to decide if the average air quality has decreased in 2014 with respect to (a) Should we use a one-sided or a two-sided test? Explain your reasoning. (b) Should we use a paired or non-paired test? Explain your reasoning. (c) Should we use a t-test or a z-test? Explain your reasoning. (a) One-sided, we are evaluating whether the air quality has decreased. (b) Paired, data are recorded in the same cities at two different time points. (c) t-test, population standard deviation is unknown. Question 7. Suppose you tossed a coin 10 times independently and obtained heads exactly twice. Using the ideas of hypothesis testing, is there strong evidence to support that the coin is not uniformly distributed? Compute an appropriate p-value and explain your reasoning. The null hypothesis is H 0 : p = 0.5, where p is the chance of heads. The alternative is H A : p 0.5. Pvalue represents the chance of observing something as extreme or more against the null hypothesis, considering that the null hypothesis is true. In this case, we observed 2 in 10, which using the binomial has probability: P r(x = 2) = ( n 2 ) p 2 (1 p) n 2 = ( ) = = 45/ More extreme cases against H 0 would be k = 1 and k = 0: ( ) ( ) n 10 P r(x = 1) = p 1 (1 p) n 1 = = = 10/1024, 1 2 ( ) ( ) n 10 P r(x = 0) = p 0 (1 p) n 0 = = = 1/ Values X=8,9,10 are also as extreme or more than the observed. By symmetry, we have P r(x = 0) = P r(x = 10), P r(x = 1) = P r(x = 9), P r(x = 2) = P r(x = 8). So the pvalue is 2 ( ) = 108/1024 =
4 Question 8. Sally gets a cup of coffee and a muffin every day for breakfast from one of the many coffee shops in her neighbourhood. She picks a coffee shop each morning at random and independently of previous days. The average price of a cup of coffee is 1.40 with a standard deviation of 0.30, the average price of a muffin is 2.50 with a standard deviation of 0.15, and the two prices are independent of each other. (a) What is the mean and standard deviation of the amount she spends on breakfast daily? (b) What is the mean and standard deviation of the amount she spends on breakfast weekly (7 days)? (a) Let X be the coffee price with µ X = 1.4 and σ X = 0.3 and Y the muffin price with µ Y = 2.5 and σ Y = Since they are independent, we have E(X +Y ) = = 3.9, var(x +Y ) = var(x)+var(y ) = = and so the standard deviation is = (b) We can write X 1,..., X 7 and Y 1,..., Y 7 to represent the days of the week. Since they are all independent, the same reasoning holds: E((X 1 + Y 1 ) (X 7 + Y 7 )) = = 27.3, var((x 1 + Y 1 ) (X 7 + Y 7 )) = var(x 1 + Y 1 ) var(x 7 + Y 7 ) = = , and so the standard deviation will be = Question 9. In each part below, there is a value of interest and two scenarios (I and II). For each part, report if the value of interest is larger under scenario I, scenario II, or whether the value is equal under the scenarios. (a) The standard error of sample mean when sample standard deviation is 120 and (I) n=25 or (II) n=125. (b) The margin of error of a confidence interval when the confidence level is (I) 90% or (II) 80%. (c) The p-value for a Z-statistic of 2.5 when (I) n = 500 or (II) n = (a) SE = s n, hence larger under scenario (I). (b) To achieve greater confidence, we need a wider interval, so margin of error is larger under scenario (I). (c) Z = PointEstimate NullValue = 2.5, so n is already taken into account when we compute the Z SE statistic and does not matter afterwards (the distribution is Normal and does not depend on n). Thus, there is no difference in pvalues between the scenarios. Question 10. Rock-paper-scissors is a hand game played by two or more people where players choose to sign either rock, paper, or scissors with their hands. You want to evaluate whether players choose between these three options randomly, or if certain options are favoured above others. You ask two friends to play rock-paper-scissors and count the times each option is played. The data are: Rock 43 times, Paper 21 times, Scissors 35 times. Compute the Chi-Square statistics and its p-value to evaluate whether players choose between these three options randomly, or if certain options are favoured above others.
5 The total number of counts is = 99. The expected counts for each situation are the same: (1/3) * 99 = 33 for each option. We can use a Chi-Square with df = 3 1 = 2. The statistic is χ 2 stat for df=2 = ((43 33) 2 )/33 + ((21 33) 2 )/33 + ((35 33) 2 )/33 = 7.52, and so 0.02 < pvalue < 0.05 (by inspecting the table for 7.52 with df = 2). Since a significance level was not defined, we cannot decide if we can reject the null hypothesis. Nevertheless, the pvalue is reasonably small. For those who want to discuss further the topic: First of all, 99 samples can be considered strange, since there are two players each time. We could think that once one of the players did not show a hand, which happens (and he/she lost that game). Second, one may argue about the independence if the same players keep playing the game. We can assume that you ask two friends to play, then other two friends, and so on (you need many friends though). Question 11. Let X be a random variable that takes the value 0 with probability 1/2 (that is, P r(x = 0) = 1/2), and takes the value 1 with probability 1/2. Let Z be a random variable, independent of X, that takes the value -1 with probability 1/2, and takes the value 1 with probability 1/2. Let U be a random variable constructed as U = X Z. (a) Prove that X and U are not independent (use the definition of independence). (b) Compute the expected value E(U) and the variance var(u). (a) We can show that the rule of independence (that is, P r(u X) = P r(u)) does not hold: P r(u = 0 X = 0) = 1, while P r(u = 0 X = 1) = 0, and therefore P r(u = 0) = 1/2 and P r(u = 0 X = 0) P r(u = 0). (b) VERSION1: First, let us compute E(Z). E(Z) = x Z x P r(z = x) = = 0. Because X and Z are independent, we have cov(x, Z) = 0 and hence E(XZ) = E(X)E(Z), so E(U) = E(XZ) = E(X)E(Z) = 0. As for the variance: var(u) = E(U 2 ) (E(U)) 2 = E(U 2 ) = E(X 2 Z 2 ) = E(X 2 )E(Z 2 ), E(X 2 ) = x X x 2 P r(x = x) = = 1 2, E(Z 2 ) = x 2 P r(z = x) = ( 1) = 1. Therefore var(u) = 1 2. x Z (b) VERSION2: Another possible approach is to write down all possible values of U and their probabilities, and go from there. We know that U = XZ with X independent of Z, hence P r(u = 0) = 1/2, P r(u = 1) = 1/4 and P r(u = 1) = 1/4. Therefore, E(U) = x P r(u = x) = ( 1) = 0, x U E(U 2 ) = x U x 2 P r(u = x) = ( 1) = 1 2, var(u) = E(U 2 ) (E(U)) 2 = E(U 2 ) = 1 2.
6 Formulas (assumptions/conditions to use these formulas are not explicit here but must be considered) Sample x 1,..., x n. Sample mean: x = 1 n n i=1 x i. Sample variance: s 2 = 1 n 1 n i=1 (x i x) 2. ˆx µ T-statistic, Z-statistic:, where ˆx depends on what is being tested. For instance, it can SE be x, or a proportion ˆp, or a difference of means, etc. Degrees of freedom of T are n 1, or min(n 1, n 2 ) 1 for two samples. s Standard error: SE = 2, where the sample variance n s2 depends on what is being estimated (if population variance σ 2 is known, then we use it instead of s 2 ). For instance, for proportions we have s 2 = p (1 p), where p may come from the null hypothesis, or from the sample(s), or pooled from 2 groups. For a binomial distribution over n trials and success chance of p, a normal approximation would have µ = n p and σ 2 = n p (1 p). For difference of estimators from two samples, we use s 2 1 SE = + s2 2, n 1 n 2 where s 1, n 1 come from group 1, and s 2, n 2 from group 2. Margin of error: ME = v SE, where v comes from the appropriate table (Normal, t-table with appropriate degrees of freedom). Confidence interval: (point-estimate M E; point-estimate +M E), where point-estimate depends on what is your target (mean, proportion, a difference). Chisquare statistics: k i=1 (O i E i ) 2 E i, where k is the number of types/categories/possibilities, O i the number of observations of type i, E i the number of expected cases of type i. Degrees of freedom are k 1. Expected value (E) of a discrete variable X: E(X) = x X x P r(x = x), E(f(X)) = x X f(x) P r(x = x). Variance (var) and covariance (cov) of random variables X, Y in terms of their expected values: var(x) = E(X 2 ) (E(X)) 2, var(x + Y ) = var(x) + var(y ) + 2 cov(x, Y ), cov(x, Y ) = E(X Y ) E(X) E(Y ). If X,Y are independent of each other, then var(x + Y ) = var(x) + var(y ). Binomial distribution (n trials, success p): P r(x = k) = ( n k) p k (1 p) n k. Geometric distribution (success p): P r(x = k) = p(1 p) k 1.
7 Normal Probability Table Y positive Z Second decimal place of Z Z For Z 3.50, the probability is greater than
8 t-probability Table One tail One tail Two tails Figure 1. Tails for the t-distribution. one tail two tails df
9 Chi-Square Probability Table Figure 2. Areas in the chi-square table always refer to the right tail. Upper tail df In the case of df > 50, one may use the fact that χ 2 df df 2 df Normal(µ = 0, σ = 1) and so the number from the table of the standard Normal distribution can be employed after such transformation.
1 Basic continuous random variable problems
Name M362K Final Here are problems concerning material from Chapters 5 and 6. To review the other chapters, look over previous practice sheets for the two exams, previous quizzes, previous homeworks and
More information1 Basic continuous random variable problems
Name M362K Final Here are problems concerning material from Chapters 5 and 6. To review the other chapters, look over previous practice sheets for the two exams, previous quizzes, previous homeworks and
More informationThis does not cover everything on the final. Look at the posted practice problems for other topics.
Class 7: Review Problems for Final Exam 8.5 Spring 7 This does not cover everything on the final. Look at the posted practice problems for other topics. To save time in class: set up, but do not carry
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 informationProbability Methods in Civil Engineering Prof. Dr. Rajib Maity Department of Civil Engineering Indian Institution of Technology, Kharagpur
Probability Methods in Civil Engineering Prof. Dr. Rajib Maity Department of Civil Engineering Indian Institution of Technology, Kharagpur Lecture No. # 36 Sampling Distribution and Parameter Estimation
More informationExam Empirical Methods VU University Amsterdam, Faculty of Exact Sciences h, February 12, 2015
Exam Empirical Methods VU University Amsterdam, Faculty of Exact Sciences 18.30 21.15h, February 12, 2015 Question 1 is on this page. Always motivate your answers. Write your answers in English. Only the
More informationWISE MA/PhD Programs Econometrics Instructor: Brett Graham Spring Semester, Academic Year Exam Version: A
WISE MA/PhD Programs Econometrics Instructor: Brett Graham Spring Semester, 2016-17 Academic Year Exam Version: A INSTRUCTIONS TO STUDENTS 1 The time allowed for this examination paper is 2 hours. 2 This
More informationWISE MA/PhD Programs Econometrics Instructor: Brett Graham Spring Semester, Academic Year Exam Version: A
WISE MA/PhD Programs Econometrics Instructor: Brett Graham Spring Semester, 2016-17 Academic Year Exam Version: A INSTRUCTIONS TO STUDENTS 1 The time allowed for this examination paper is 2 hours. 2 This
More informationGEOMETRIC -discrete A discrete random variable R counts number of times needed before an event occurs
STATISTICS 4 Summary Notes. Geometric and Exponential Distributions GEOMETRIC -discrete A discrete random variable R counts number of times needed before an event occurs P(X = x) = ( p) x p x =,, 3,...
More informationPHP2510: Principles of Biostatistics & Data Analysis. Lecture X: Hypothesis testing. PHP 2510 Lec 10: Hypothesis testing 1
PHP2510: Principles of Biostatistics & Data Analysis Lecture X: Hypothesis testing PHP 2510 Lec 10: Hypothesis testing 1 In previous lectures we have encountered problems of estimating an unknown population
More informationChapter 9 Inferences from Two Samples
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 Review
More informationSTA 2201/442 Assignment 2
STA 2201/442 Assignment 2 1. This is about how to simulate from a continuous univariate distribution. Let the random variable X have a continuous distribution with density f X (x) and cumulative distribution
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 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 informationContinuous Improvement Toolkit. Probability Distributions. Continuous Improvement Toolkit.
Continuous Improvement Toolkit Probability Distributions The Continuous Improvement Map Managing Risk FMEA Understanding Performance** Check Sheets Data Collection PDPC RAID Log* Risk Analysis* Benchmarking***
More informationECE 313: Conflict Final Exam Tuesday, May 13, 2014, 7:00 p.m. 10:00 p.m. Room 241 Everitt Lab
University of Illinois Spring 1 ECE 313: Conflict Final Exam Tuesday, May 13, 1, 7: p.m. 1: p.m. Room 1 Everitt Lab 1. [18 points] Consider an experiment in which a fair coin is repeatedly tossed every
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 informationProbability & Statistics - FALL 2008 FINAL EXAM
550.3 Probability & Statistics - FALL 008 FINAL EXAM NAME. An urn contains white marbles and 8 red marbles. A marble is drawn at random from the urn 00 times with replacement. Which of the following is
More informationMidterm Exam 1 Solution
EECS 126 Probability and Random Processes University of California, Berkeley: Fall 2015 Kannan Ramchandran September 22, 2015 Midterm Exam 1 Solution Last name First name SID Name of student on your left:
More informationPage Max. Possible Points Total 100
Math 3215 Exam 2 Summer 2014 Instructor: Sal Barone Name: GT username: 1. No books or notes are allowed. 2. You may use ONLY NON-GRAPHING and NON-PROGRAMABLE scientific calculators. All other electronic
More informationPrecept 4: Hypothesis Testing
Precept 4: Hypothesis Testing Soc 500: Applied Social Statistics Ian Lundberg Princeton University October 6, 2016 Learning Objectives 1 Introduce vectorized R code 2 Review homework and talk about RMarkdown
More informationCourse: ESO-209 Home Work: 1 Instructor: Debasis Kundu
Home Work: 1 1. Describe the sample space when a coin is tossed (a) once, (b) three times, (c) n times, (d) an infinite number of times. 2. A coin is tossed until for the first time the same result appear
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 information18.05 Final Exam. Good luck! Name. No calculators. Number of problems 16 concept questions, 16 problems, 21 pages
Name No calculators. 18.05 Final Exam Number of problems 16 concept questions, 16 problems, 21 pages Extra paper If you need more space we will provide some blank paper. Indicate clearly that your solution
More informationUC Berkeley Department of Electrical Engineering and Computer Science. EE 126: Probablity and Random Processes. Solutions 5 Spring 2006
Review problems UC Berkeley Department of Electrical Engineering and Computer Science EE 6: Probablity and Random Processes Solutions 5 Spring 006 Problem 5. On any given day your golf score is any integer
More informationExam 2 Practice Questions, 18.05, Spring 2014
Exam 2 Practice Questions, 18.05, Spring 2014 Note: This is a set of practice problems for exam 2. The actual exam will be much shorter. Within each section we ve arranged the problems roughly in order
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 informationexp{ (x i) 2 i=1 n i=1 (x i a) 2 (x i ) 2 = exp{ i=1 n i=1 n 2ax i a 2 i=1
4 Hypothesis testing 4. Simple hypotheses A computer tries to distinguish between two sources of signals. Both sources emit independent signals with normally distributed intensity, the signals of the first
More informationMAS223 Statistical Inference and Modelling Exercises
MAS223 Statistical Inference and Modelling Exercises The exercises are grouped into sections, corresponding to chapters of the lecture notes Within each section exercises are divided into warm-up questions,
More informationClass 8 Review Problems 18.05, Spring 2014
1 Counting and Probability Class 8 Review Problems 18.05, Spring 2014 1. (a) How many ways can you arrange the letters in the word STATISTICS? (e.g. SSSTTTIIAC counts as one arrangement.) (b) If all arrangements
More informationSTAT 516 Midterm Exam 3 Friday, April 18, 2008
STAT 56 Midterm Exam 3 Friday, April 8, 2008 Name Purdue student ID (0 digits). The testing booklet contains 8 questions. 2. Permitted Texas Instruments calculators: BA-35 BA II Plus BA II Plus Professional
More information18.05 Exam 1. Table of normal probabilities: The last page of the exam contains a table of standard normal cdf values.
Name 18.05 Exam 1 No books or calculators. You may have one 4 6 notecard with any information you like on it. 6 problems, 8 pages Use the back side of each page if you need more space. Simplifying expressions:
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 informationMidterm Exam 1 (Solutions)
EECS 6 Probability and Random Processes University of California, Berkeley: Spring 07 Kannan Ramchandran February 3, 07 Midterm Exam (Solutions) Last name First name SID Name of student on your left: Name
More information6.4 Type I and Type II Errors
6.4 Type I and Type II Errors Ulrich Hoensch Friday, March 22, 2013 Null and Alternative Hypothesis Neyman-Pearson Approach to Statistical Inference: A statistical test (also known as a hypothesis test)
More informationSTAT 4385 Topic 01: Introduction & Review
STAT 4385 Topic 01: Introduction & Review Xiaogang Su, Ph.D. Department of Mathematical Science University of Texas at El Paso xsu@utep.edu Spring, 2016 Outline Welcome What is Regression Analysis? Basics
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 informationFinal Exam. Math Su10. by Prof. Michael Cap Khoury
Final Exam Math 45-0 Su0 by Prof. Michael Cap Khoury Name: Directions: Please print your name legibly in the box above. You have 0 minutes to complete this exam. You may use any type of conventional calculator,
More informationStatistics 224 Solution key to EXAM 2 FALL 2007 Friday 11/2/07 Professor Michael Iltis (Lecture 2)
NOTE : For the purpose of review, I have added some additional parts not found on the original exam. These parts are indicated with a ** beside them Statistics 224 Solution key to EXAM 2 FALL 2007 Friday
More informationChapter 26: Comparing Counts (Chi Square)
Chapter 6: Comparing Counts (Chi Square) We ve seen that you can turn a qualitative variable into a quantitative one (by counting the number of successes and failures), but that s a compromise it forces
More information18.05 Practice Final Exam
No calculators. 18.05 Practice Final Exam Number of problems 16 concept questions, 16 problems. Simplifying expressions Unless asked to explicitly, you don t need to simplify complicated expressions. For
More informationCOSC 341 Human Computer Interaction. Dr. Bowen Hui University of British Columbia Okanagan
COSC 341 Human Computer Interaction Dr. Bowen Hui University of British Columbia Okanagan 1 Last Topic Distribution of means When it is needed How to build one (from scratch) Determining the characteristics
More informationDo not copy, post, or distribute
14 CORRELATION ANALYSIS AND LINEAR REGRESSION Assessing the Covariability of Two Quantitative Properties 14.0 LEARNING OBJECTIVES In this chapter, we discuss two related techniques for assessing a possible
More informationThe Chi-Square Distributions
MATH 183 The Chi-Square Distributions Dr. Neal, WKU The chi-square distributions can be used in statistics to analyze the standard deviation σ of a normally distributed measurement and to test the goodness
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 information79 Wyner Math Academy I Spring 2016
79 Wyner Math Academy I Spring 2016 CHAPTER NINE: HYPOTHESIS TESTING Review May 11 Test May 17 Research requires an understanding of underlying mathematical distributions as well as of the research methods
More informationSTAT 285 Fall Assignment 1 Solutions
STAT 285 Fall 2014 Assignment 1 Solutions 1. An environmental agency sets a standard of 200 ppb for the concentration of cadmium in a lake. The concentration of cadmium in one lake is measured 17 times.
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 information, 0 x < 2. a. Find the probability that the text is checked out for more than half an hour but less than an hour. = (1/2)2
Math 205 Spring 206 Dr. Lily Yen Midterm 2 Show all your work Name: 8 Problem : The library at Capilano University has a copy of Math 205 text on two-hour reserve. Let X denote the amount of time the text
More informationIB Mathematics HL Year 2 Unit 7 (Core Topic 6: Probability and Statistics) Valuable Practice
IB Mathematics HL Year 2 Unit 7 (Core Topic 6: Probability and Statistics) Valuable Practice 1. We have seen that the TI-83 calculator random number generator X = rand defines a uniformly-distributed random
More information16.400/453J Human Factors Engineering. Design of Experiments II
J Human Factors Engineering Design of Experiments II Review Experiment Design and Descriptive Statistics Research question, independent and dependent variables, histograms, box plots, etc. Inferential
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 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 information1 Probability theory. 2 Random variables and probability theory.
Probability theory Here we summarize some of the probability theory we need. If this is totally unfamiliar to you, you should look at one of the sources given in the readings. In essence, for the major
More informationProbability Year 10. Terminology
Probability Year 10 Terminology Probability measures the chance something happens. Formally, we say it measures how likely is the outcome of an event. We write P(result) as a shorthand. An event is some
More informationCONTINUOUS RANDOM VARIABLES
the Further Mathematics network www.fmnetwork.org.uk V 07 REVISION SHEET STATISTICS (AQA) CONTINUOUS RANDOM VARIABLES The main ideas are: Properties of Continuous Random Variables Mean, Median and Mode
More informationSTAT 414: Introduction to Probability Theory
STAT 414: Introduction to Probability Theory Spring 2016; Homework Assignments Latest updated on April 29, 2016 HW1 (Due on Jan. 21) Chapter 1 Problems 1, 8, 9, 10, 11, 18, 19, 26, 28, 30 Theoretical Exercises
More informationCSE 103 Homework 8: Solutions November 30, var(x) = np(1 p) = P r( X ) 0.95 P r( X ) 0.
() () a. X is a binomial distribution with n = 000, p = /6 b. The expected value, variance, and standard deviation of X is: E(X) = np = 000 = 000 6 var(x) = np( p) = 000 5 6 666 stdev(x) = np( p) = 000
More informationSTAT 135 Lab 11 Tests for Categorical Data (Fisher s Exact test, χ 2 tests for Homogeneity and Independence) and Linear Regression
STAT 135 Lab 11 Tests for Categorical Data (Fisher s Exact test, χ 2 tests for Homogeneity and Independence) and Linear Regression Rebecca Barter April 20, 2015 Fisher s Exact Test Fisher s Exact Test
More informationWISE International Masters
WISE International Masters ECONOMETRICS Instructor: Brett Graham INSTRUCTIONS TO STUDENTS 1 The time allowed for this examination paper is 2 hours. 2 This examination paper contains 32 questions. You are
More informationSTAT 418: Probability and Stochastic Processes
STAT 418: Probability and Stochastic Processes Spring 2016; Homework Assignments Latest updated on April 29, 2016 HW1 (Due on Jan. 21) Chapter 1 Problems 1, 8, 9, 10, 11, 18, 19, 26, 28, 30 Theoretical
More information. Find E(V ) and var(v ).
Math 6382/6383: Probability Models and Mathematical Statistics Sample Preliminary Exam Questions 1. A person tosses a fair coin until she obtains 2 heads in a row. She then tosses a fair die the same number
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 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 informationLinear Regression with 1 Regressor. Introduction to Econometrics Spring 2012 Ken Simons
Linear Regression with 1 Regressor Introduction to Econometrics Spring 2012 Ken Simons Linear Regression with 1 Regressor 1. The regression equation 2. Estimating the equation 3. Assumptions required for
More informationLast few slides from last time
Last few slides from last time Example 3: What is the probability that p will fall in a certain range, given p? Flip a coin 50 times. If the coin is fair (p=0.5), what is the probability of getting an
More information2. Suppose (X, Y ) is a pair of random variables uniformly distributed over the triangle with vertices (0, 0), (2, 0), (2, 1).
Name M362K Final Exam Instructions: Show all of your work. You do not have to simplify your answers. No calculators allowed. There is a table of formulae on the last page. 1. Suppose X 1,..., X 1 are independent
More informationDiscrete Distributions
A simplest example of random experiment is a coin-tossing, formally called Bernoulli trial. It happens to be the case that many useful distributions are built upon this simplest form of experiment, whose
More informationMTH 302 long questions solved by Pisces girl My Lord! Increase me in knowledge.
MTH 302 long questions solved by Pisces girl My Lord! Increase me in knowledge. Question No: 5 ( Marks: 3 ) In a school, 50% students study science subjects and 30% of them study biology. What is the probability
More informationThis exam is closed book and closed notes. (You will have access to a copy of the Table of Common Distributions given in the back of the text.
TEST #3 STA 5326 December 4, 214 Name: Please read the following directions. DO NOT TURN THE PAGE UNTIL INSTRUCTED TO DO SO Directions This exam is closed book and closed notes. (You will have access to
More informationFinal Exam - Solutions
Ecn 102 - Analysis of Economic Data University of California - Davis March 17, 2010 Instructor: John Parman Final Exam - Solutions You have until 12:30pm to complete this exam. Please remember to put your
More informationLecture 2: Repetition of probability theory and statistics
Algorithms for Uncertainty Quantification SS8, IN2345 Tobias Neckel Scientific Computing in Computer Science TUM Lecture 2: Repetition of probability theory and statistics Concept of Building Block: Prerequisites:
More informationAdvanced Herd Management Probabilities and distributions
Advanced Herd Management Probabilities and distributions Anders Ringgaard Kristensen Slide 1 Outline Probabilities Conditional probabilities Bayes theorem Distributions Discrete Continuous Distribution
More informationSolutions to Homework Set #6 (Prepared by Lele Wang)
Solutions to Homework Set #6 (Prepared by Lele Wang) Gaussian random vector Given a Gaussian random vector X N (µ, Σ), where µ ( 5 ) T and 0 Σ 4 0 0 0 9 (a) Find the pdfs of i X, ii X + X 3, iii X + X
More informationQuestion. Hypothesis testing. Example. Answer: hypothesis. Test: true or not? Question. Average is not the mean! μ average. Random deviation or not?
Hypothesis testing Question Very frequently: what is the possible value of μ? Sample: we know only the average! μ average. Random deviation or not? Standard error: the measure of the random deviation.
More informationLecture 2: Review of Probability
Lecture 2: Review of Probability Zheng Tian Contents 1 Random Variables and Probability Distributions 2 1.1 Defining probabilities and random variables..................... 2 1.2 Probability distributions................................
More informationHW5 Solutions. (a) (8 pts.) Show that if two random variables X and Y are independent, then E[XY ] = E[X]E[Y ] xy p X,Y (x, y)
HW5 Solutions 1. (50 pts.) Random homeworks again (a) (8 pts.) Show that if two random variables X and Y are independent, then E[XY ] = E[X]E[Y ] Answer: Applying the definition of expectation we have
More informationz and t tests for the mean of a normal distribution Confidence intervals for the mean Binomial tests
z and t tests for the mean of a normal distribution Confidence intervals for the mean Binomial tests Chapters 3.5.1 3.5.2, 3.3.2 Prof. Tesler Math 283 Fall 2018 Prof. Tesler z and t tests for mean Math
More informationLECTURE 1. Introduction to Econometrics
LECTURE 1 Introduction to Econometrics Ján Palguta September 20, 2016 1 / 29 WHAT IS ECONOMETRICS? To beginning students, it may seem as if econometrics is an overly complex obstacle to an otherwise useful
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 informationStatistics 427: Sample Final Exam
Statistics 427: Sample Final Exam Instructions: The following sample exam was given several quarters ago in Stat 427. The same topics were covered in the class that year. This sample exam is meant to be
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 informationProbability Year 9. Terminology
Probability Year 9 Terminology Probability measures the chance something happens. Formally, we say it measures how likely is the outcome of an event. We write P(result) as a shorthand. An event is some
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 informationData Mining. Chapter 5. Credibility: Evaluating What s Been Learned
Data Mining Chapter 5. Credibility: Evaluating What s Been Learned 1 Evaluating how different methods work Evaluation Large training set: no problem Quality data is scarce. Oil slicks: a skilled & labor-intensive
More informationAP Statistics Final Examination Free-Response Questions
AP Statistics Final Examination Free-Response Questions Name Date Period Section II Part A Questions 1 4 Spend about 50 minutes on this part of the exam (70 points) Directions: You must show all work and
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 informationMAT 271E Probability and Statistics
MAT 271E Probability and Statistics Spring 2011 Instructor : Class Meets : Office Hours : Textbook : Supp. Text : İlker Bayram EEB 1103 ibayram@itu.edu.tr 13.30 16.30, Wednesday EEB? 10.00 12.00, Wednesday
More informationtheir contents. If the sample mean is 15.2 oz. and the sample standard deviation is 0.50 oz., find the 95% confidence interval of the true mean.
Math 1342 Exam 3-Review Chapters 7-9 HCCS **************************************************************************************** Name Date **********************************************************************************************
More informationSTA 2101/442 Assignment 2 1
STA 2101/442 Assignment 2 1 These questions are practice for the midterm and final exam, and are not to be handed in. 1. A polling firm plans to ask a random sample of registered voters in Quebec whether
More informationSTA301- Statistics and Probability Solved Subjective From Final term Papers. STA301- Statistics and Probability Final Term Examination - Spring 2012
STA30- Statistics and Probability Solved Subjective From Final term Papers Feb 6,03 MC004085 Moaaz.pk@gmail.com Mc004085@gmail.com PSMD0 STA30- Statistics and Probability Final Term Examination - Spring
More informationMath 50: Final. 1. [13 points] It was found that 35 out of 300 famous people have the star sign Sagittarius.
Math 50: Final 180 minutes, 140 points. No algebra-capable calculators. Try to use your calculator only at the end of your calculation, and show working/reasoning. Please do look up z, t, χ 2 values for
More informationDiscrete distribution. Fitting probability models to frequency data. Hypotheses for! 2 test. ! 2 Goodness-of-fit test
Discrete distribution Fitting probability models to frequency data A probability distribution describing a discrete numerical random variable For example,! Number of heads from 10 flips of a coin! Number
More informationHypothesis testing: Steps
Review for Exam 2 Hypothesis testing: Steps Repeated-Measures ANOVA 1. Determine appropriate test and hypotheses 2. Use distribution table to find critical statistic value(s) representing rejection region
More informationInstitute of Actuaries of India
Institute of Actuaries of India Subject CT3 Probability & Mathematical Statistics May 2011 Examinations INDICATIVE SOLUTION Introduction The indicative solution has been written by the Examiners with the
More informationHYPOTHESIS TESTING: THE CHI-SQUARE STATISTIC
1 HYPOTHESIS TESTING: THE CHI-SQUARE STATISTIC 7 steps of Hypothesis Testing 1. State the hypotheses 2. Identify level of significant 3. Identify the critical values 4. Calculate test statistics 5. Compare
More informationEECS 126 Probability and Random Processes University of California, Berkeley: Spring 2015 Abhay Parekh February 17, 2015.
EECS 126 Probability and Random Processes University of California, Berkeley: Spring 2015 Abhay Parekh February 17, 2015 Midterm Exam Last name First name SID Rules. You have 80 mins (5:10pm - 6:30pm)
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 information1. Let X be a random variable with probability density function. 1 x < f(x) = 0 otherwise
Name M36K Final. Let X be a random variable with probability density function { /x x < f(x = 0 otherwise Compute the following. You can leave your answers in integral form. (a ( points Find F X (t = P
More informationStat 135 Fall 2013 FINAL EXAM December 18, 2013
Stat 135 Fall 2013 FINAL EXAM December 18, 2013 Name: Person on right SID: Person on left There will be one, double sided, handwritten, 8.5in x 11in page of notes allowed during the exam. The exam is closed
More information