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### Lecture24

Course: ECON 220, Spring 2011
School: University of Toronto
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Word Count: 775

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Lecture ECO220Y 24 Introduction to Hypothesis Testing Migiwa Tanaka Reading:11.1 1 Outline Outline Introduction to Hypothesis Testing Testing population mean when the population variance is known Rejection Region Method p-value Method One-tail vs. two-tail Test Calculating the probability of a type II error 2 Hypothesis Hypothesis Testing Step1: Identify hypotheses about a population parameter...

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Lecture ECO220Y 24 Introduction to Hypothesis Testing Migiwa Tanaka Reading:11.1 1 Outline Outline Introduction to Hypothesis Testing Testing population mean when the population variance is known Rejection Region Method p-value Method One-tail vs. two-tail Test Calculating the probability of a type II error 2 Hypothesis Hypothesis Testing Step1: Identify hypotheses about a population parameter Step2: Collect a sample Step3: Conduct hypothesis testing: Conclude whether the hypotheses set up in step 1 is true or not, using evidence found in the sample. A. B. Calculate a test statistic Compare A. with the decision criteria 3 Analogy with Criminal Trial (1) Hypothesis Testing Criminal Trial Unknown: Population Parameter Construct hypotheses about population population parameter. Collect a sample from the population Calculate test statistics. Based on the evidence in a sample, test hypothesis. 4 Unknown: Defendants Guilt Construct hypotheses about defendants guilt Collect evidence relevant to the crime. Based on the evidence, test hypothesis Hypotheses Hypotheses 1. All hypotheses are constructed in a pair. Null Hypothesis (H0): Initial presumption about the population, which is not based on the data. Status quo. Alternative Hypothesis (H1): The statement that can be proven by the data. New regime. It is also called research hypothesis hypothesis. 2. Example: H0: Defendant is innocent H1: Defendant is guilty H0: Mean daily demand for gasoline is 1000 gallon 1000 H1: Mean daily demand for gasoline is not 1000 gallon 5 Analogy Analogy with Criminal Trial (2) Juries consider following hypotheses: H0: Defendant is innocent H1: Defendant is guilty Two choices: 1. 2. Defendant is not guilty Not enough evidence to conclude H0 is true. Do not reject H0 Defendant is guilty Enough evidence beyond reasonable doubt. Reject H0 in favor of H1 Does 1. implies the defendant is proven innocent? 6 Construction of Hypotheses: Example 1: Criminal Trial Current Legal Standard: Alternative Legal Standard: Innocent until proven guilty beyond a reasonable doubt Guilty until proven innocent innocent beyond a reasonable doubt H0: Defendant is innocent H1: Defendant is guilty What will be the decision if there is no evidence? H0 : H1 : What will be the decision if there is no evidence? 7 What we can do in hypothesis testing: Reject null hypothesis in favor of alternative(research) hypothesis. There is enough evidence beyond a reasonable doubt. There is not enough evidence beyond reasonable a doubt to prove the guilty of the defendant. Do not reject null What we cannot do: Accept null hypothesis. (i.e. the defendant is innocent.) (i Prove null hypothesis. (i.e. Prove the innocence of defendant defendant.) 8 Type Type I Error and Type II Error Errors in Decision H0 is true (innocent) Reject H0 (Convict) Do not reject H0 (Acquit) (Acquit) H1 is true (guilty) Type Type I Error Correct Decision Correct Decision Type II Error Type I error is occurs when rejecting true null. Convict an innocent person. Acquit a guilty person. Type II error occurs when not rejecting false null. 9 Type Type I Error and Type II Error H0 is true (innocent) Reject H0 (Convict) Do not reject H0 (Acquit) H1 is true (guilty) Type I Error Correct Decision Correct Decision Type II Error Probability of making type I error= =P(Reject H0 | H0 is true) This is called level of significance in hypothesis testing. Probability of making type II error = =P(Do not reject H0 | H1 is true) How and are related? 10 Relationship Relationship between and The probability of type I error( ) and the probability of type II error () are inversely related. Decrease in Increase in burden of proof Increase the chance of acquittal for both innocent and guilty Increase in (chance of letting guilty person go free) Decrease in , Decrease in burden of proof. Decrease the chance of acquittal for both innocent and guilty defendant. Increase in . 11 Construction Construction of Statistical Hypotheses: Hypothesis: a claim about a population parameter Null hypothesis H0: =some value (population mean is 0) Alternative hypothesis (three ways.) A. B. C. H1: some value (population mean is not 0) H1: <some value (population mean is less than 0) H1: >some value (population mean is greater than 0) A. leads to two-tailed test: When researcher wants to know if the parameter is different from null. B. and C. lead to one-tailed test. : When researcher wants to know if the parameter is greater/less than null. 12 Construction of Hypothesis: Example 2: Inventory Management The retail stores manage their inventory depending on demand for the product. For example, your store stocks canned tomato soup such that it can accommodate 100 cans each week. You want to know if there is an increase in the demand and if you should increase the inventor y level. H0 : H1 : You want to know if there is any change in the demand and if you should maintain the current inventory level. H0 : H1 : 13
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University of Toronto - ECON - 220
ECO220Y Lecture25 Hypothesis Testing (2)Migiwa Tanaka Reading: 11.2 (pp 349-352)1Outline OutlineTesting population mean when the population variance is knownTwo Approaches to Hypothesis Testing Rejection Region Method p-value MethodOne-tail vs. tw
University of Toronto - ECON - 220
ECO220Y: Homework, Lectures 1 &amp; 2 Readings: Sections 1.1, 1.3, 2.1, 2.3 Exercises: 1.2, 1.4, 1.6, 1.7, 2.6, 2.34, 2.36 (by hand) Problems: (1) Give original examples of data that are: cross-sectional, time series, and longitudinal. For each, describe the
University of Toronto - ECON - 220
ECO220Y: Homework, Lectures 1 &amp; 2 SOLUTIONS (1) Many possible solutions. Cross-sectional: percent of regular faculty that are women in a sample of 34 economics departments in universities around the world in 2004. Time series: percent of regular faculty t
University of Toronto - ECON - 220
ECO220Y: Homework, Lectures 3 &amp; 4 Readings: Sections 2.2 2.6 Exercises: 2.86 (by hand) Problems: (1) Using the descriptive terms for histograms given in lecture, how would you describe the graph below? Roughly how many observations are negative?n: 228 .0
University of Toronto - ECON - 220
ECO220Y: Homework, Lectures 3 &amp; 4 SOLUTIONS (1) Symmetric, bell shaped, unimodal. Resist the temptation to call this bimodal and positively skewed: it is not. When we are describing the shape we see in a histogram we are trying to make inferences about th
University of Toronto - ECON - 220
ECO220Y: Homework, Lectures 5 &amp; 6 Readings: Sections 4.1 4.3 Exercises: 4.2, 4.10, 4.22, 4.25 4.30, 4.40, 4.42, 4.46 (but these data not this data) Problems: (1) Considering the tabulation, find the mean, median, and mode of x. x| Freq. Percent Cum. -+-0|
University of Toronto - ECON - 220
ECO220Y: Homework, Lectures 5 &amp; 6 SOLUTIONS (1) Mean is 0.6572, median is 0.7, and mode is 1. (You should show your work.) (2) Many possible solutions. One example: 0 0 0 0 0 2 2 2 2 2 (3) The sample mean is about 150. It is reasonable to infer that this
University of Toronto - ECON - 220
ECO220Y: Homework, Lectures 7 &amp; 8 Readings: Sections 4.4, 4.7, 4.8 Exercises: 4.55, 4.56, 4.58: include complete interpretation in paragraph-form for answer to Part e. of 4.58 Applets (CD-ROM): Applet 1 (page 128), Applet 2 (page 128) Problems: (1) Suppos
University of Toronto - ECON - 220
ECO220Y: Homework, Lectures 9 &amp; 10 Readings: Sections 5.1 5.4, Handout Chapter 1: Economic Questions and Data from Introduction to Econometrics, Second Edition, by James H. Stock and Mark W. Watson, 2007, Sections 6.1 6.3 Exercises: 5.1 5.3, 5.6, 5.7, 5.9
University of Toronto - ECON - 220
ECO220Y: Homework, Lectures 11 &amp; 12 Readings: Sections 7.1 7.2 Exercises: 7.2, 7.4, 7.5, 7.8, 7.16 7.18, 7.32 7.34, 7.36, 7.40 7.42, 7.47 7.50, 7.52, 7.54, 7.56, 7.58, 7.59 Problems: (1) Suppose you calculate x = \$36, y = \$60, sX = \$12, sY = \$15, and sXY
University of Toronto - ECON - 220
ECO220Y: Homework, Lectures 11 &amp; 12 SOLUTIONS (1) In 2007 \$1 CAN \$0.9 U.S., which means that we must multiply X and Y by 0.9. [You may use the most recent exchange rate.] Use the Laws of Expectation and Variance to get reported statistics in U.S. dollars.
University of Toronto - ECON - 220
ECO220Y: Homework, Lectures 13 &amp; 14 Readings: Section 7.4 Exercises: 7.92, 7.94, 7.96, 7.97, 7.98 (Note: Solve these Exercises without a computer and without using any tables in the appendices) Problems: (1) What two factors affect the probability of any
University of Toronto - ECON - 220
ECO220Y: Homework, Lectures 13 &amp; 14 SOLUTIONS (1) The number of trials and the probability of success. Go over how this relates to the example and provide the intuition. (2) The first example would be Binomial but the second case would not. Drawing cards
University of Toronto - ECON - 220
ECO220Y: Homework, Lectures 15 Readings: Sections 8.1 Exercises: 8.4, 8.6, 8.9 8.14 Problems: (1) What is the mean and variance of Z if Z = X1 + X2 and X1 is binomially distributed with p = 0.1 and n = 20, X2 is binomially distributed with p = 0.9 and n =
University of Toronto - ECON - 220
ECO220Y: Homework, Lecture 15 SOLUTIONS (1)E [ X 1 ] 20 0.1 2, E [ X 1 X 2 ] 2 18 20,E[ X 2 ] 20 0.9 18, V [ X 2 ] 20 0.1 0.9 1.8,V [ X 1 ] 20 0.1 0.9 1.8, V [ X 1 X 2 ] 1.8 1.8 3.6(2) 16(3) mean = 0, var = 0.67; sd = 0.82
University of Toronto - ECON - 220
ECO220Y: Homework, Lecture 16 Readings: Section 8.2, Handout lecture16 (posted on the portal. Go to Content Handout lecture16), and Section 8.4 (excluding 2) Exercises: 8.15 8.33, 8.38, 8.46, 8.48, 8.52, 8.58, 8.64, 8.69, 8.70 (Note: Use table in The Stan
University of Toronto - ECON - 220
ECO220Y: Homework, Lecture 20 &amp; 21Readings: Section 9.1 (pages 301 303), Sections 10.1 Exercises: 10.1, 10.3, 10.4, 10.6, 10.8 Applets (CD-ROM): Applets 9 12 Problems: (1) In this problem, you are asked to use 3 Loonies (one dollar coins) to simulate a s
University of Toronto - ECON - 220
ECO220Y: Homework, Lecture 17 Readings: Sections 9.1 Exercises: 9.1 9.4 Applets (CD-ROM): Applet 9 (page 302) Problems: (1) Recall the telework example in Lecture 17. Here is the relevant information about the population: Number of Permits 0 1 2 Fraction
University of Toronto - ECON - 220
ECO220Y: Homework, Lecture 18 Readings: Sections 9.1, 9.3, 9.4 Exercises: 9.5, 9.6, 9.8, 9.10, 9.16, 9.22, 9.28, 9.48, 9.50, 9.54 Applets (CD-ROM): Applet 10 (page 302), Applet 11 (page 303), Applet 12 (page 303), Applet 14 (page 317)Problems: (1) It is
University of Toronto - ECON - 220
ECO220Y: Homework, Lecture 19 Readings: Sections 9.2 Exercises: 9.30, 9.32, 9.34, 9.36, 9.40, Applets (CD-ROM): Applet 13 (page 312) Problems: Consider the box shown below, which contains 30 balls. Consider sampling 50 balls with replacement from this box
University of Toronto - ECON - 220
ECO220Y: Homework, Lecture 19 SOLUTIONS (1) The rule of thumb requires that the entire interval defined by p 3(p(1-p)/n)0.5 has to be contained between 0 and 1. Since p=8/30=0.27, n=50, the interval is [0.08,0.46], which falls within [0,1]. Thus, sampling
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