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Part 13: Statistical Tests – Part 1 13-1/37 Statistics and Data Analysis Professor William Greene Stern School of Business IOMS Department Department of Economics

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Part 13: Statistical Tests – Part 1 13-2/37 Statistics and Data Analysis Part 13 – Statistical Tests: 1

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Part 13: Statistical Tests – Part 1 13-3/37 Statistical Testing Methodology: The scientific method and statistical testing Classical hypothesis testing Setting up the test Test of a hypothesis about a mean Other kinds of statistical tests Mechanics of hypothesis testing Applications

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Part 13: Statistical Tests – Part 1 13-4/37 Classical Hypothesis Testing The scientific method applied to statistical hypothesis testing Hypothesis: The world works according to my hypothesis Testing or supporting the hypothesis Data gathering Rejection of the hypothesis if the data are inconsistent with it Retention and exposure to further investigation if the data are consistent with the hypothesis Failure to reject is not equivalent to acceptance.

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Part 13: Statistical Tests – Part 1 13-5/37 http://query.nytimes.com/gst/fullpage.html?res=9C00E4DF113BF935A3575BC0A9649C8B63

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Part 13: Statistical Tests – Part 1 13-6/37 Methodology The standard approach would be to hypothesize that there is no link and seek data (evidence) that are (is) inconsistent with the hypothesis. That is the way the NCI usually carries out an investigation. This one was different.

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Part 13: Statistical Tests – Part 1 13-7/37 Errors in Testing Correct Decision Type II Error Type I Error Correct Decision Hypothesis is Hypothesis is True False I Do Not Reject the Hypothesis I Reject the Hypothesis

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Part 13: Statistical Tests – Part 1 13-8/37 A Legal Analogy: The Null Hypothesis is INNOCENT Correct Decision Type II Error Guilty defendant goes free T ype I Error Innocent defendant is convicted Correct Decision Null Hypothesis Alternative Hypothesis Not Guilty Guilty Finding: Verdict Not Guilty Finding: Verdict Guilty The errors are not symmetric. Most thinkers consider Type I errors to be more serious than Type II in this setting.

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Part 13: Statistical Tests – Part 1 13-9/37 (Worldwide) Standard Methodology “Statistical” testing Methodology Formulate the “null” hypothesis Decide (in advance) what kinds of “evidence” (data) will lead to rejection of the null hypothesis. I.e., define the rejection region) Gather the data Carry out the test.

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Part 13: Statistical Tests – Part 1 13-10/37 Formulating the Hypothesis Stating the hypothesis: A belief about the “state of nature” A parameter takes a particular value There is a relationship between variables And so on… The null vs. the alternative By induction: If we wish to find evidence of something, first assume it is not true. Look for evidence that leads to rejection of the assumed hypothesis.

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Part 13: Statistical Tests – Part 1 13-11/37 Terms of Art Null Hypothesis: The proposed state of nature Alternative hypothesis: The state of nature that is believed to prevail if the null is rejected.

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Part 13: Statistical Tests – Part 1 13-12/37 Example: Credit Rule Investigation: I believe that Fair Isaacs relies on home ownership in deciding whether to “accept” an application. Null hypothesis: There is no relationship Alternative hypothesis: They do use homeownership data. What decision rule should I use?

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Part 13: Statistical Tests – Part 1 13-13/37 Some Evidence = Homeowners 48% of cardholders are homeowners. 38% of nonholders are homeowners.

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Part 13: Statistical Tests – Part 1 13-14/37 The Rejection Region What is the “rejection region?” Data (evidence) that are inconsistent with my hypothesis Evidence is divided into two types: Data that are inconsistent with my hypothesis (the rejection region) Everything else

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Part 13: Statistical Tests – Part 1 13-15/37 Application: Breast Cancer On Long Island Null Hypothesis: There is no link between the high cancer rate on LI and the use of pesticides and toxic chemicals in dry cleaning, farming, etc. Neyman-Pearson Procedure Examine the physical and statistical evidence If there is convincing covariation, reject the null hypothesis What is the rejection region? The NCI study: Working hypothesis: There is a link: We will find the evidence. How do you reject this hypothesis?

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Part 13: Statistical Tests – Part 1 13-16/37 Formulating the Testing Procedure Usually: What kind of data will lead me to reject the hypothesis? Thinking scientifically: If you want to “prove” a hypothesis is true (or you want to support one) begin by assuming your hypothesis is not true, and look for plausible evidence that contradicts the assumption.

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Part 13: Statistical Tests – Part 1 13-17/37 Hypothesis Testing Strategy Formulate the null hypothesis Gather the evidence Question: If my null hypothesis were true, how likely is it that I would have observed this evidence? Very unlikely: Reject the hypothesis Not unlikely: Do not reject. (Retain the hypothesis for continued scrutiny.)

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Part 13: Statistical Tests – Part 1 13-18/37 Hypothesis About a Mean I believe that the average income of individuals in a population is (about) $30,000. H 0 : μ = $30,000 (The null) H 1 : μ ≠ $30,000 (The alternative) I will draw the sample and examine the data. The rejection region is data for which the sample mean is far from $30,000. How far is far????? That is the test.

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Part 13: Statistical Tests – Part 1 13-19/37 Application The mean of a population takes a specific value: Null hypothesis: H 0 : μ = $30,000 H 1 : μ ≠ $30,000 Test: Sample mean close to hypothesized population mean? Rejection region: Sample means that are far from $30,000

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Part 13: Statistical Tests – Part 1 13-20/37 Deciding on the Rejection Region If the sample mean is far from $30,000, I will reject the hypothesis. I choose, the region, for example, 30,500 The probability that the mean falls in the rejection region even though the hypothesis is true (should not be rejected) is the probability of a type 1 error. Even if the true mean really is $30,000, the sample mean could fall in the rejection region. 29,500 30,000 30,500 Rejection

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Part 13: Statistical Tests – Part 1 13-21/37 Reduce the Probability of a Type I Error by Making the Rejection Region Smaller 28,500 29,500 30,000 30,500 31,500 Reduce the probability of a type I error by moving the boundaries of the rejection region farther out. You can make a type I error impossible by making the rejection region very far from the null. Then you would never make a type I error because you would never reject H 0. Probability outside this interval is large. Probability outside this interval is much smaller.

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Part 13: Statistical Tests – Part 1 13-22/37 Setting the α Level “α” is the probability of a type I error Choose the width of the interval by choosing the desired probability of a type I error, based on the t or normal distribution. (How confident do I want to be?) Multiply the corresponding z or t value by the standard error of the mean.

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Part 13: Statistical Tests – Part 1 13-23/37 Testing Procedure The rejection region will be the range of values greater than μ 0 + zσ/√N or less than μ 0 - zσ/√N Use z = 1.96 for 1 - α = 95% Use z = 2.576 for 1 - α = 99% Use the t table if small sample and sampling from a normal distribution.

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Part 13: Statistical Tests – Part 1 13-24/37 Deciding on the Rejection Region If the sample mean is far from $30,000, reject the hypothesis. Choose, the region, say, Rejection I am 95% certain that I will not commit a type I error (reject the hypothesis in error). (I cannot be 100% certain.)

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Part 13: Statistical Tests – Part 1 13-25/37 The Testing Procedure (For a Mean)

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Part 13: Statistical Tests – Part 1 13-26/37 The Test Procedure Choosing z = 1.96 makes the probability of a Type I error 0.05. Choosing z = 2.576 would reduce the probability of a Type I error to 0.01.

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Part 13: Statistical Tests – Part 1 13-27/37 What to use for σ? The known value if there is one The sample estimate if random sampling.

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Part 13: Statistical Tests – Part 1 13-28/37 Application

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Part 13: Statistical Tests – Part 1 13-29/37

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Part 13: Statistical Tests – Part 1 13-30/37 If you choose 1-Sample Z… to use the normal distribution, Minitab assumes you know σ and asks for the value.

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Part 13: Statistical Tests – Part 1 13-31/37 Specify the Hypothesis Test Minitab assumes 95%. You can choose some other value.

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Part 13: Statistical Tests – Part 1 13-32/37 The Test Results (Are In)

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Part 13: Statistical Tests – Part 1 13-33/37 An Intuitive Approach Using the confidence interval The confidence interval gives the range of plausible values. If this range does not include the null hypothesis, reject the hypothesis. If the confidence interval contains the hypothesized value, retain the hypothesis. Includes $30,000.

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Part 13: Statistical Tests – Part 1 13-34/37 The P value The “P value” is the probability that you would have observed the evidence that you did observe if the null hypothesis were true. If the P value is less than the Type I error probability (usually 0.05) you have chosen, you will reject the hypothesis.

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Part 13: Statistical Tests – Part 1 13-35/37 Insignificant Results The test results are “significant” if the P value is less than α. These test results are “insignificant” at the 5% level. This is 1 – α.

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Part 13: Statistical Tests – Part 1 13-36/37 Application: One sided test of a mean Hypothesis: The mean is greater than some value Business application: Does a new machine that we might buy produce grommets faster than the one we have now? H 0 : μ ≤ M (where M is the mean for the old machine.) H 1 : μ > M Rejection region: Mean of a sample of production rates from the new machine is far above M. Buy the new machine, Academic Application: Do SAT Test Courses work? Null hypothesis: The mean grade on the do-overs is less than the mean on the original test. Reject means the do- over appears to be better.

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Part 13: Statistical Tests – Part 1 13-37/37 Summary Methodological issues: Science and hypothesis tests Standard methods: Formulating a testing procedure Determining the “rejection region” Many different kinds of applications

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