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Copyright © 2011 Pearson Education, Inc. Statistical Tests Chapter 16
16.1 Concepts of Statistical Tests A manager is evaluating software to filter SPAM s (cost $15,000). To make it profitable, the software must reduce SPAM to less than 20%. Should the manager buy the software? Use a statistical test to answer this question Consider the plausibility of a specific claim (claims are called hypotheses) Copyright © 2011 Pearson Education, Inc. 3 of 40
16.1 Concepts of Statistical Tests Null and Alternative Hypotheses Statistical hypothesis: claim about a parameter of a population. Null hypothesis (H 0 ): specifies a default course of action, preserves the status quo. Alternative hypothesis (H a ): contradicts the assertion of the null hypothesis. Copyright © 2011 Pearson Education, Inc. 4 of 40
16.1 Concepts of Statistical Tests SPAM Software Example Let p = that slips past the filter H 0 : p ≥ 0.20 H a : p < 0.20 These hypotheses lead to a one-sided test. Copyright © 2011 Pearson Education, Inc. 5 of 40
16.1 Concepts of Statistical Tests One- and Two-Sided Tests One-sided test: the null hypothesis allows any value of a parameter larger (or smaller) than a specified value. Two-sided test: the null hypothesis asserts a specific value for the population parameter. Copyright © 2011 Pearson Education, Inc. 6 of 40
16.1 Concepts of Statistical Tests Type I and II Errors Reject H 0 incorrectly (buying software that will not be cost effective) Retain H 0 incorrectly (not buying software that would have been cost effective) Copyright © 2011 Pearson Education, Inc. 7 of 40
16.1 Concepts of Statistical Tests Type I and II Errors indicates a correct decision Copyright © 2011 Pearson Education, Inc. 8 of 40
16.1 Concepts of Statistical Tests Other Tests Visual inspection for association, normal quantile plots and control charts all use tests of hypotheses. For example, the null hypothesis in a visual test for association is that there is no association between two variables shown in the scatterplot. Copyright © 2011 Pearson Education, Inc. 9 of 40
16.1 Concepts of Statistical Tests Sampling Distribution Statistical tests rely on the sampling distribution of the statistic that estimates the parameter specified in the null and alternative hypotheses. Key question: What is the chance of getting a sample that differs from H 0 by as much as this one if H 0 is true? Copyright © 2011 Pearson Education, Inc. 10 of 40
16.2 Testing the Proportion SPAM Software Example Based on n = 100, = Assuming H 0 is true, the sampling distribution of is approximately normal with mean p = 0.20 and SE( ) = 0.04 (note that the hypothesized value p 0 = 0.20 is used to calculate SE). Copyright © 2011 Pearson Education, Inc. 11 of 40
16.2 Testing the Proportion SPAM Software Example What is the chance of making a Type I error? Possible sampling distributions for. Chance of a Type I error shown in shaded area. Copyright © 2011 Pearson Education, Inc. 12 of 40
16.2 Testing the Proportion z–Test and p-Value p-Value: the largest chance of a Type I error if H 0 is rejected based on the observed test statistic. z-Test: test of H 0 based on a count of the standard errors separating H 0 from the test statistic. Copyright © 2011 Pearson Education, Inc. 13 of 40
16.2 Testing the Proportion z–Test for SPAM Software Example = Copyright © 2011 Pearson Education, Inc. 14 of 40
16.2 Testing the Proportion p–Value for SPAM Software Example Interpret the p-value as a weight of evidence against H 0 ; small values mean that H 0 is not plausible. Copyright © 2011 Pearson Education, Inc. 15 of 40
16.2 Testing the Proportion α-Value α-Value: threshold that sets the maximum tolerance for a Type I error. Statistically significant: data contradict the null hypothesis and lead us to reject H 0 (p-value < α). The p-value in the SPAM example is less than the typical α of 0.05; should buy the software. Copyright © 2011 Pearson Education, Inc. 16 of 40
16.2 Testing the Proportion Type II Error Power: probability that a test rejects H 0. If a test has little power when H 0 is false, it is likely to miss meaningful deviations from the null hypothesis and produce a Type II error. Copyright © 2011 Pearson Education, Inc. 17 of 40
16.2 Testing the Proportion Summary Copyright © 2011 Pearson Education, Inc. 18 of 40
16.2 Testing the Proportion Checklist SRS condition: the sample is a simple random sample from the relevant population. Sample size condition (for proportion): both np 0 and n(1 - p 0 ) are larger than 10. Copyright © 2011 Pearson Education, Inc. 19 of 40
4M Example 16.1: DO ENOUGH HOUSEHOLDS WATCH? Motivation The Burger King ad featuring Coq Roq won critical acclaim. In a sample of 2,500 homes, MediaCheck found that only 6% saw the ad. An ad must be viewed by 5% or more of households to be effective. Based on these sample results, should the local sponsor run this ad? Copyright © 2011 Pearson Education, Inc. 20 of 40
4M Example 16.1: DO ENOUGH HOUSEHOLDS WATCH? Mehod Set up the null and alternative hypotheses. H 0 : p ≤ 0.05 H a : p > 0.05 Use α = Note that p is the population proportion who watch this ad. Both SRS and sample size conditions are met. Copyright © 2011 Pearson Education, Inc. 21 of 40
4M Example 16.1: DO ENOUGH HOUSEHOLDS WATCH? Mechanics Perform a one-sided z-test for a proportion. z = 2.3 with p-value of Reject H 0. Copyright © 2011 Pearson Education, Inc. 22 of 40
4M Example 16.1: DO ENOUGH HOUSEHOLDS WATCH? Message The results are statistically significant. We can conclude that more than 5% of households watch this ad. The Burger King Coq Roq ad is cost effective and should be run. Copyright © 2011 Pearson Education, Inc. 23 of 40
16.3 Testing the Mean Similar to Tests of Proportions The hypothesis test of µ replaces with. Unlike the test of proportions, σ is not specified. Use s from the sample as an estimate of σ to calculate the estimated standard error of. Copyright © 2011 Pearson Education, Inc. 24 of 40
16.3 Testing the Mean Example: Denver Rental Properties A firm is considering expanding into the Denver area. In order to cover costs, the firm needs rents in this area to average more than $500 per month. Are Denver rents high enough to justify the expansion? Copyright © 2011 Pearson Education, Inc. 25 of 40
16.3 Testing the Mean Null and Alternative Hypotheses Let µ = mean monthly rent for all rental properties in the Denver area Set up hypotheses as: H 0 : µ ≤ µ 0 = $500 H a : µ > µ 0 = $500 Copyright © 2011 Pearson Education, Inc. 26 of 40
16.3 Testing the Mean t - Statistic Used in the t-test for µ (since s estimates σ) The t-statistic, with n-1 df, is Copyright © 2011 Pearson Education, Inc. 27 of 40
16.3 Testing the Mean Example: Denver Rental Properties The firm obtained rents for a sample of size n=45; the average rent was $647 with s = $299. t = with 44 df; p-value = Reject H 0 ; mean rent exceeds break-even value. Copyright © 2011 Pearson Education, Inc. 28 of 40
16.3 Testing the Mean Finding the p-Value in the t-Table t = is larger than any value in the row Copyright © 2011 Pearson Education, Inc. 29 of 40
16.3 Testing the Mean Summary Copyright © 2011 Pearson Education, Inc. 30 of 40
16.3 Testing the Mean Checklist SRS condition: the sample is a simple random sample from the relevant population. Sample size condition. Unless the population is normally distributed, a normal model can be used to approximate the sampling distribution of if n is larger than 10 times both the squared skewness and absolute value of kurtosis. Copyright © 2011 Pearson Education, Inc. 31 of 40
4M Example 16.2: COMPARING RETURNS ON INVESTMENTS Motivation Does stock in IBM return more, on average, than T-Bills? From 1980 through 2005, T- Bills returned 5% each month. Copyright © 2011 Pearson Education, Inc. 32 of 40
4M Example 16.2: COMPARING RETURNS ON INVESTMENTS Method Let µ = mean of all future monthly returns for IBM stock. Set up the hypotheses as H 0 : µ ≤ H a : µ > Sample consists of monthly returns on IBM for 312 months (January 1980 – December 2005) Copyright © 2011 Pearson Education, Inc. 33 of 40
4M Example 16.2: COMPARING RETURNS ON INVESTMENTS Mechanics Sample yields = with s = t = 1.22 with 311 df; p-value = Copyright © 2011 Pearson Education, Inc. 34 of 40
4M Example 16.2: COMPARING RETURNS ON INVESTMENTS Message Monthly IBM returns from 1980 through 2005 do not bring statistically significantly higher earnings than comparable investments in US Treasury Bills during this period. Copyright © 2011 Pearson Education, Inc. 35 of 40
16.4 Other Properties of Tests Significance versus Importance Statistical significance does not mean that you have made an important or meaningful discovery. The size of the sample affects the p-value of a test. With enough data, a trivial difference from H 0 leads to a statistically significant outcome. Copyright © 2011 Pearson Education, Inc. 36 of 40
16.4 Other Properties of Tests Confidence Interval or Test? A confidence interval provides a range of parameter values that are compatible with the observed data. A test provides a precise analysis of a specific hypothesized value for a parameter. Copyright © 2011 Pearson Education, Inc. 37 of 40
Best Practices Pick the hypotheses before looking at the data. Choose the null hypothesis on the basis of profitability. Pick the α level first, taking into account both types of error. Think about whether α = 0.05 is appropriate for each test. Copyright © 2011 Pearson Education, Inc. 38 of 40
Best Practices (Continued) Make sure to have an SRS from the right population. Use a one-sided test. Report a p–value to summarize the outcome of a test. Copyright © 2011 Pearson Education, Inc. 39 of 40
Pitfalls Do not confuse statistical significance with substantive importance. Do not think that the p–value is the probability that the null hypothesis is true. Avoid cluttering a test summary with jargon. Copyright © 2011 Pearson Education, Inc. 40 of 40
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