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Session 8bDecision Models -- Prof. Juran1Decision Models -- Prof. Juran2OverviewHypothesis Testing Review of the BasicsSingle ParameterDifferences Between Two ParametersIndependent SamplesMatched PairsGoodness of FitSimulation MethodsDecision Models -- Prof. Juran2Decision Models -- Prof. Juran3Basic Hypothesis Testing MethodFormulate Two HypothesesSelect a Test StatisticDerive a Decision RuleCalculate the Value of the Test Statistic; Invoke the Decision Rule in light of the Test StatisticDecision Models -- Prof. Juran3Decision Models -- Prof. Juran4

Decision Models -- Prof. Juran4Applied Regression -- Prof. Juran5Hypothesis Testing: Gardening Analogy

RocksDirtApplied Regression -- Prof. Juran5Applied Regression -- Prof. Juran6

Hypothesis Testing: Gardening AnalogyApplied Regression -- Prof. Juran6Applied Regression -- Prof. Juran7Hypothesis Testing: Gardening Analogy

Applied Regression -- Prof. Juran7Applied Regression -- Prof. Juran8Hypothesis Testing: Gardening Analogy

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Screened out stuff:Correct decision or Type I Error?Stuff that fell through:Correct decision or Type II Error?Applied Regression -- Prof. Juran9Decision Models -- Prof. Juran10The 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 H0.

Small p-value = unlikely H0Decision Models -- Prof. Juran10Decision Models -- Prof. Juran11Example: Buying a LaundromatA 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. Decision Models -- Prof. Juran11Decision Models -- Prof. Juran12

Decision Models -- Prof. Juran12Decision Models -- Prof. Juran13Test Statistic:

Decision Rule, based on alpha of 1%: Reject H0 if the test statistic is greater than 2.575 or less than -2.575.

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Decision Models -- Prof. Juran14Decision Models -- Prof. Juran15We reject H0. There is sufficiently strong evidence against H0 to reject it at the 0.01 level. We conclude that the true mean is different from $675.

Decision Models -- Prof. Juran15Decision Models -- Prof. Juran16Example: Reliability Analysis

Decision Models -- Prof. Juran16Decision Models -- Prof. Juran17The 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 products reliability characteristics. Extracted data:

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H0 here is that the product lasts 45 hours (on the average).There is sufficiently strong evidence against H0 to reject it at any reasonable significance level. We conclude that the true MTBF is less than 45 hours. Decision Models -- Prof. Juran18Decision Models -- Prof. Juran19Example: Effects of Sales CampaignsIn 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. Decision Models -- Prof. Juran19Decision Models -- Prof. Juran20The following data indicate the results over a weekly period:

Decision Models -- Prof. Juran20Decision Models -- Prof. Juran21Is the campaign effective?Basically this is asking: Is there a difference between the average sales from these two populations (with and without the campaign)?Decision Models -- Prof. Juran21Decision Models -- Prof. Juran22Two MethodsIndependent SamplesGeneral MethodMatched PairsUseful Only in Specific CircumstancesMore Powerful StatisticallyRequires Logical One-to-One Correspondence between PairsDecision Models -- Prof. Juran22Decision Models -- Prof. Juran23Independent Samples Method

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We do not reject the null hypothesis. The campaign made no significant difference in sales. Decision Models -- Prof. Juran24Decision Models -- Prof. Juran25

Decision Models -- Prof. Juran25Decision Models -- Prof. Juran26Matched-Pairs Method

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This time we do reject the null hypothesis, and conclude that the campaign actually did have a significant positive effect on sales. Decision Models -- Prof. Juran27Decision Models -- Prof. Juran28

Decision Models -- Prof. Juran28Decision Models -- Prof. Juran29TSB Problem Revisited

Decision Models -- Prof. Juran29Decision Models -- Prof. Juran30Independent Samples

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Decision Models -- Prof. Juran32Decision Models -- Prof. Juran33Matched Pairs

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Decision Models -- Prof. Juran35Decision Models -- Prof. Juran36Goodness-of-Fit TestsDetermine whether a set of sample data have been drawn from a hypothetical population Same four basic steps as other hypothesis tests we have learnedAn important tool for simulation modeling; used in defining random variable inputs Decision Models -- Prof. Juran3637Example: Barkevious MingoFinancial 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. JuranApplied Regression -- Prof. Juran3738He studies the trading volume of these contracts for 50 days, and observes the following results (in millions of contracts traded):

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Decision Models -- Prof. JuranApplied Regression -- Prof. Juran3940Here is a histogram showing the theoretical distribution of 50 observations drawn from a normal distribution with = 152 and = 32, together with a histogram of Mingos sample data:

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The Chi-Square StatisticDecision Models -- Prof. JuranApplied Regression -- Prof. Juran4142Essentially, 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. JuranApplied Regression -- Prof. Juran4243Like Students 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. JuranApplied Regression -- Prof. Juran4344

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Decision Models -- Prof. JuranApplied Regression -- Prof. Juran4546Note: 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. JuranApplied Regression -- Prof. Juran4647The 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) = 7.815

We will reject the null hypothesis if the test statistic is greater than 7.815.Decision Models -- Prof. JuranApplied Regression -- Prof. Juran4748

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 = 152 and = 32 to model futures contract trading volume in his simulation. Decision Models -- Prof. JuranApplied Regression -- Prof. Juran4849

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 H0 could be rejected. In this case, we calculate the p-value using the Excel function:= CHIDIST(test stat, d.f.) = CHIDIST(7.439,3) = 0.0591Decision Models -- Prof. JuranApplied Regression -- Prof. Juran4950Example: Catalog CompanyIf we want to simulate the queueing system at this company, what distributions should we use for the arrival and service processes?Decision Models -- Prof. JuranApplied Regression -- Prof. Juran5051Arrivals

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63Other uses for the Chi-Square statisticTests 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. JuranApplied Regression -- Prof. Juran63Decision Models -- Prof. Juran64SummaryHypothesis Testing Review of the BasicsSingle ParameterDifferences Between Two ParametersIndependent SamplesMatched PairsGoodness of FitSimulation Methods

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