Schaum s Outline Probability and Statistics Chapter 7 HYPOTHESIS TESTING exhibited by Professor Carol Dahl Exam

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Schaum's Layout Likelihood and Insights Section 7 Theory TESTING introduced by Educator Tune Dahl Cases by Alfred Aird Kira Jeffery Catherine Keske Hermann Logsend Yris Olaya. Blueprint of Points. Themes Secured. Factual Choices Measurable Speculations

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Schaum's Outline Probability and Statistics Chapter 7 HYPOTHESIS TESTING introduced by Professor Carol Dahl Examples by Alfred Aird Kira Jeffery Catherine Keske Hermann Logsend Yris Olaya

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Outline of Topics Covered Statistical Decisions Statistical Hypotheses Null Hypotheses Tests of Hypotheses Type I and Type II Errors Level of Significance Tests Involving the Normal Distribution One and Two – Tailed Tests P – Value

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Outline of Topics (Continued ) Special Tests of Significance Large Samples Small Samples Estimation Theory/Hypotheses Testing Relationship Operating Characteristic Curves and Power of a Test Fitting Theoretical Distributions to Sample Frequency Distributions Chi-Square Test for Goodness of Fit

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"The Truth Is Out There" The Importance of Hypothesis Testing Hypothesis testing assesses models based upon genuine information empowers one to assemble a factual model improves your validity as analyst market analyst

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Statistical Decisions Innocent until demonstrated blameworthy rule Want to demonstrate somebody is liable Assume the inverse or business as usual - honest H o : Innocent H 1 : Guilty Take subsample of conceivable data If prove not predictable with pure - dismiss Person not articulated honest but rather not liable

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Statistical Decisions Status quo guiltlessness = invalid speculation Evidence = test result Reasonable uncertainty = certainty level

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Statistical Decisions Eg. Tantalum metal store achievable if quality > 0.0600g/kg with 99% certainty 100 specimens gathered from huge store aimlessly. Test dissemination mean of 0.071g/kg standard deviation 0.0025g/kg.

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Statistical Decisions Should the store be produced? Confirm = 0.071 (example mean) Reasonable uncertainty = 99% Status quo = don't build up the store H o :  < 0.0600 H 1 :  > 0.0600

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Statistical Hypothesis General Principles Inferences about populace utilizing test measurement Prove An is valid by accepting it isn't genuine Results of (test) contrasted and display If consequences of model improbable, dismiss demonstrate If comes about clarified by model, don't dismiss

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Statistical Hypothesis Area B z 0 A Event A genuinely likely, model would be held Event B far-fetched, model would be rejected

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Statistical Decisions Should the store be produced? Prove = 0.071 (example mean) Reasonable uncertainty = 99% Status quo = don't build up the store H o :  = 0.0600 H 1 :  > 0.0600 How likely H o given = 0.071

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Need Sampling Statistic Need measurement with population parameter estimate for populace parameter its circulation

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Need Sampling Statistic Population Normal - Two Choices Small Sample <30 Known Variance Unknown Variance N(0,1) t n-1

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Need Sampling Statistic Population Not-Normal Large Sample Known Variance Unknown Variance N(0,1) N(0,1) Doesn't make any difference if know change of not If populace is limited testing no substitution require modification

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Normal Distribution 27 X~N(0,1)  =0 SD=1 (68%) SD=2 (95%) SD=3 (99.7%)

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Statistical Decisions Should the store be produced? Prove: 0.071 (example mean) 0.0025g/kg (test difference) 0.05 (specimen standard deviation) Reasonable uncertainty = 99% Status quo = don't build up the store H o :  = 0.0600 H 1 :  > 0.0600 One followed test How likely H o given = 0.071

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Hypothesis test Evidence: 0.071 (specimen mean) 0.05g/kg (test standard deviation) Reasonable uncertainty = 99% Status quo = don't build up the store H o :  = 0.0600 H 1 :  > 0.0600

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Statistical Hypothesis Eg. Z = (0.071 – 0.0600)/(0.05/ 100) = 2.2 Conclusion: Don't dismiss H o , don't create store Z c =2.33 2.2

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Null Hypothesis Hypotheses can't be demonstrated reject or neglect to dismiss in view of probability of occasion happening invalid theory is not acknowledged

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Test of Hypotheses Maple Creek Mine and Potaro Diamond field in Guyana Mine potential for delivering huge precious stones Experts need to know genuine mean carat measure created True mean said to be 4 carats Experts need to know whether valid with 95% certainty Random example taken Sample mean observed to be 3.6 carats Based on test, is 4 carats genuine mean for mine?

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Tests of Hypotheses Tests alluded to as: "Trial of Hypotheses" "Trial of Significance" "Principles of Decision"

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Types of Errors H o : µ = 4 (Suppose this is valid) H 1 : µ  4 Two followed test Choose  = 0.05 Sample n = 100 (expect X is ordinary),  = 1

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Type I mistake ( ) –reject genuine H o : µ = 4 assume genuine /2 /2

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Type II Error (ß) - Accept False μ = 4 μ = 6 0 2 H o : µ = 4 not genuine µ = 6 genuine X-µ not mean 0 but rather mean 2 ß

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Lower Type I What happens to Type II μ = 4 μ = 6 0 2 Ho: µ = 4 not genuine µ = 6 genuine ß

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Higher µ What happens to Type II? μ = 4 μ = 7 0 3 Ho: µ = 4 not genuine µ = 7 genuine X-µ not mean 0 but rather mean 3 ß

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Type I and Type II Errors Two sorts of mistakes can happen in theory testing To diminish blunders, increment test measure when conceivable

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To Reduce Errors Increase test estimate when conceivable Population, n = 5, 10, 20

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Error Examples Type I Error – dismissing a genuine invalid speculation Convicting a pure individual Rejecting genuine mean carat size is 4 when it is Type II Error – not dismissing a false invalid speculation Setting a liable individual free Not dismissing mean carat size is 4 when it's not

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Level of Significance ( ) α = max likelihood we're willing to hazard Type I Error = tail territory of likelihood thickness work If Type I Error's "cost" high, pick α low α characterized before theory test directed α regularly characterized as 0.10, 0.05 or 0.01 α = 0.10 for 90% certainty of right test choice α = 0.05 for 95% certainty of right test choice α = 0.01 for 99% certainty of right test choice

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Diamond Hypothesis Test Example H o : µ = 4 H 1 : µ  4 Choose α = 0.01 for 99% certainty Sample n = 100,  = 1 X = 3.6, - Z c = - 2.575, Z c = 2.575 - 2.575 .005

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21 Example Continued 1 Observed not "altogether" not the same as anticipated that Fail would dismiss invalid speculation We're 99% sure genuine mean is 4 carats

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Tests Involving the t Distribution Billy Ray has acquired vast, 25,000 section of land residence Located on edges of Murfreesboro, Arkansas, close: Crater of Diamonds State Park Prairie Creek Volcanic Pipe Land now utilized for agricultural recreational No official mining has occurred

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Case Study in Statistical Analysis Billy Ray's Inheritance Billy Ray should now settle ashore use Options: Exploration for precious stones Conservation Land biodiversity and diversion Agriculture and amusement Land advancement

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Consider Costs and Benefits of Mining Cost and Benefits of Mining Opportunity cost Excessive jewel investigation harms land's esteem Exploration and Mining Costs Benefit Value of mineral created

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Consider Costs and Benefits of Mining Cost and Benefits of Mining Sample for geologic pointers for precious stones kimberlite or lamporite larger test more prone to speak to "genuine populace" larger test will cost more

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How to choose one followed or two followed One followed test Do we change business as usual just if its greater than invalid Do we change existing conditions just if its littler than invalid Two followed test Change the present state of affairs if its greater of on the off chance that it littler

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Tests of Mean Normal or t population ordinary known variance small test Normal populace typical obscure variance little specimen t substantial populace Normal

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Difference Normal and t "fatter" tail than ordinary ringer bend

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Hypothesis and Sample Need no less than 30 g/m 3 mine Null theory H o : µ = 20 Alternative speculation H 1 : ? Test information: n=16 (gaps penetrated) X near typical X =31 g/m³ change ( ŝ 2/n) =0.286 g/m³

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Normal or t? One followed Null theory H o : µ = 30 Alternative speculation H 1 : µ > 30 Sample information: n = 16 (openings bored) X = 31 g/m³ difference ( ŝ 2 ) = 4.29 g/m³ = 4.29 standard deviation ŝ = 2.07 little specimen, evaluated fluctuation, X near ordinary not precisely t but rather close if X near typical

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Tests Involving the t Distribution t n-1 = X - µ ŝ/n t 16-1  =0 Reject 5% t c =1.75

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Tests Involving the t Distribution t n-1 = X - µ = (31 - 30) = 1.93 ŝ/n 2.07/16 t 16-1  =0 Reject 5% t c =1.75

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Wells produces oil X= API Gravity approximate typical with mean 37  periodically test to check whether the mean has changed too substantial or too light reexamine contract H o : H 1 : Sample of 9 wells, X= 38, ŝ 2 = 2 What is test measurement? Ordinary or t?

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Two followed t test on mean t n-1 = X - µ ŝ/n  =0 Reject /2 % Reject /2 % t c t c

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Two followed t test on mean H o : µ= 37 H 1 : µ 37 Sample of 9 wells, X= 38, ŝ 2 = 2,  = 10% t n-1 = X - µ = (38 – 37) = 1.5 ŝ/n 2/ 9

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P-values - one followed

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