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Session 8b

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Decision Models -- Prof. Juran2 Overview Hypothesis Testing Review of the Basics –Single Parameter –Differences Between Two Parameters Independent Samples Matched Pairs –Goodness of Fit Simulation Methods

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Decision Models -- Prof. Juran3 Basic Hypothesis Testing Method 1.Formulate Two Hypotheses 2.Select a Test Statistic 3.Derive a Decision Rule 4.Calculate the Value of the Test Statistic; Invoke the Decision Rule in light of the Test Statistic

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Applied Regression -- Prof. Juran 5 Hypothesis Testing: Gardening Analogy Rocks Dirt

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Applied Regression -- Prof. Juran 6 Hypothesis Testing: Gardening Analogy

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Applied Regression -- Prof. Juran 7 Hypothesis Testing: Gardening Analogy

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Applied Regression -- Prof. Juran 8 Hypothesis Testing: Gardening Analogy

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Applied Regression -- Prof. Juran 9 Hypothesis Testing: Gardening Analogy Screened out stuff: Correct decision or Type I Error? Stuff that fell through: Correct decision or Type II Error?

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Decision Models -- Prof. Juran10 The p -value of a test is the probability of observing a sample at least as “unlikely” as ours. In other words, it is the “minimum level” of significance that would allow us to reject H 0. Small p -value = unlikely H 0

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Decision Models -- Prof. Juran11 Example: Buying a Laundromat A potential entrepreneur is considering the purchase of a coin-operated laundry. The present owner claims that over the past 5 years the average daily revenue has been $675. The buyer would like to find out if the true average daily revenue is different from $675. A sample of 30 selected days reveals a daily average revenue of $625 with a standard deviation of $75.

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Decision Models -- Prof. Juran13 Test Statistic: Decision Rule, based on alpha of 1%: Reject H 0 if the test statistic is greater than or less than

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Decision Models -- Prof. Juran15 We reject H 0. There is sufficiently strong evidence against H 0 to reject it at the 0.01 level. We conclude that the true mean is different from $675.

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Decision Models -- Prof. Juran16 Example: Reliability Analysis

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Decision Models -- Prof. Juran17 The brand manager wants to begin advertising for this product, and would like to claim a mean time between failures (MTBF) of 45 hours. The product is only in the prototype phase, so the design engineer uses Crystal Ball simulation to estimate the product’s reliability characteristics. Extracted data:

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Decision Models -- Prof. Juran18 H 0 here is that the product lasts 45 hours (on the average). There is sufficiently strong evidence against H 0 to reject it at any reasonable significance level. We conclude that the true MTBF is less than 45 hours.

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Decision Models -- Prof. Juran19 Example: Effects of Sales Campaigns In order to measure the effect of a storewide sales campaign on nonsale items, the research director of a national supermarket chain took a random sample of 13 pairs of stores that were matched according to average weekly sales volume. One store of each pair (the experimental group) was exposed to the sales campaign, and the other member of the pair (the control group) was not.

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Decision Models -- Prof. Juran20 The following data indicate the results over a weekly period:

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Decision Models -- Prof. Juran21 Is the campaign effective? Basically this is asking: Is there a difference between the average sales from these two populations (with and without the campaign)?

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Decision Models -- Prof. Juran22 Two Methods Independent Samples –General Method Matched Pairs –Useful Only in Specific Circumstances –More Powerful Statistically –Requires Logical One-to-One Correspondence between Pairs

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Decision Models -- Prof. Juran23 Independent Samples Method

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Decision Models -- Prof. Juran24 We do not reject the null hypothesis. The campaign made no significant difference in sales.

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Decision Models -- Prof. Juran26 Matched-Pairs Method

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Decision Models -- Prof. Juran27 This time we do reject the null hypothesis, and conclude that the campaign actually did have a significant positive effect on sales.

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Decision Models -- Prof. Juran29 TSB Problem Revisited

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Decision Models -- Prof. Juran30 Independent Samples

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Decision Models -- Prof. Juran33 Matched Pairs

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Decision Models -- Prof. Juran36 Goodness-of-Fit Tests Determine whether a set of sample data have been drawn from a hypothetical population Same four basic steps as other hypothesis tests we have learned An important tool for simulation modeling; used in defining random variable inputs

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37 Example: Barkevious Mingo Financial analyst Barkevious Mingo wants to run a simulation model that includes the assumption that the daily volume of a specific type of futures contract traded at U.S. commodities exchanges (represented by the random variable X ) is normally distributed with a mean of 152 million contracts and a standard deviation of 32 million contracts. (This assumption is based on the conclusion of a study conducted in 2013.) Barkevious wants to determine whether this assumption is still valid. Decision Models -- Prof. Juran

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38 He studies the trading volume of these contracts for 50 days, and observes the following results (in millions of contracts traded): Decision Models -- Prof. Juran

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39Decision Models -- Prof. Juran

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40 Here is a histogram showing the theoretical distribution of 50 observations drawn from a normal distribution with μ = 152 and σ = 32, together with a histogram of Mingo’s sample data: Decision Models -- Prof. Juran

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41 The Chi-Square Statistic Decision Models -- Prof. Juran

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42 Essentially, this statistic allows us to compare the distribution of a sample with some expected distribution, in standardized terms. It is a measure of how much a sample differs from some proposed distribution. A large value of chi-square suggests that the two distributions are not very similar; a small value suggests that they “fit” each other quite well. Decision Models -- Prof. Juran

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43 Like Student’s t, the distribution of chi- square depends on degrees of freedom. In the case of chi-square, the number of degrees of freedom is equal to the number of classes (a.k.a. “bins” into which the data have been grouped) minus one, minus the number of estimated parameters. Decision Models -- Prof. Juran

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46 Note: It is necessary to have a sufficiently large sample so that each class has an expected frequency of at least 5. We need to make sure that the expected frequency in each bin is at least 5, so we “collapse” some of the bins, as shown here. Decision Models -- Prof. Juran

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47 The number of degrees of freedom is equal to the number of bins minus one, minus the number of estimated parameters. We have not estimated any parameters, so we have d.f. = 4 – 1 – 0 = 3. The critical chi-square value can be found either by using a chi-square table or by using the Excel function: =CHIINV(alpha, d.f.) = CHIINV(0.05, 3) = We will reject the null hypothesis if the test statistic is greater than Decision Models -- Prof. Juran

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48 Our test statistic is not greater than the critical value; we cannot reject the null hypothesis at the 0.05 level of significance. It would appear that Barkevious is justified in using the normal distribution with μ = 152 and σ = 32 to model futures contract trading volume in his simulation. Decision Models -- Prof. Juran

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49 The p -value of this test has the same interpretation as in any other hypothesis test, namely that it is the smallest level of alpha at which H 0 could be rejected. In this case, we calculate the p -value using the Excel function: = CHIDIST(test stat, d.f.) = CHIDIST(7.439,3) = Decision Models -- Prof. Juran

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50 Example: Catalog Company If we want to simulate the queueing system at this company, what distributions should we use for the arrival and service processes? Decision Models -- Prof. Juran

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51 Arrivals Decision Models -- Prof. Juran

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56 Services Decision Models -- Prof. Juran

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63 Other uses for the Chi-Square statistic Tests of the independence of two qualitative population variables. Tests of the equality or inequality of more than two population proportions. Inferences about a population variance, including the estimation of a confidence interval for a population variance from sample data. The chi-square technique can often be employed for purposes of estimation or hypothesis testing when the z or t statistics are not appropriate. In addition to the goodness-of-fit application described above, there are at least three other important uses for chi-square: Decision Models -- Prof. Juran

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64 Summary Hypothesis Testing Review of the Basics –Single Parameter –Differences Between Two Parameters Independent Samples Matched Pairs –Goodness of Fit Simulation Methods

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