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VCU STAT 210 - Lecture35(2) (1)

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Slide 1Practice ProblemsAdditional Reading and ExamplesTest 6Slide 5Statistical InferenceGeneral Significance Testing ProcedureTests of Significance for m: s KnownTests of Significance for m: s KnownSignificance Test for mTests of Significance: s UnknownSlide 12Tests of Significance for mTests of Significance for mTests of Significance for mP-value using the t-distributionSlide 17TI-83/84 CalculatorExample 93/80Example 93/80Example 93/80Example 93/80Example 93/80Example 93/80Example 93/80Example 93/80Example 93/80Example 93/80Example 93/80Example 93/80Example 93/80Example 93/80Slide 33Example 93/80Example 93/80Example 93/80Example 93/80Example 93/80Example 94/81Example 94/81Example 94/81Example 94/81Example 94/81Example 94/81Example 94/81Example 94/81Example 94/81Example 94/81Example 94/81Example 94/81Example 94/81Example 94/81Example 94/81Example 94/81Example 94/81Example 94/81Example 94/81Example 94/81Example 95/82Example 95/82Example 95/82Example 95/82Example 95/82Example 95/82Example 95/82Example 95/82Motivating ExampleMotivating ExampleMotivating ExampleMotivating ExampleMotivating ExampleMotivating Example SolutionSlide 73STAT 210Lecture 35Tests of Significance on m When s is UnknownNovember 17, 2016Practice ProblemsSailboat: pages 220 through 230Relevant problems: VIII.18, VIII.19 (a) and (c), VIII.20, VIII.21, VIII.22, VIII.23, VIII.24 (a) and (d), VIII.26Recommended problems: VIII.20, VIII.21, VIII.23Hummingbird: pages 250 through 260Relevant problems: IX.18, IX.19 (a) and (c), IX.20, IX.21, IX.22, IX.23, IX.24 (a) and (d), IX.26Recommended problems: IX.20, IX.21, IX.23Additional Reading and ExamplesSailboat: Read pages 213 through 218Pay particular attention to page 218Hummingbird: Read pages 243 through 248Pay particular attention to page 248Test 6Monday, November 21Questions for the first 10 minutes, then test – papers due promptly at the end of class!Covers Chapter 9 in Hummingbird (pages 219 – 260) or Chapter 8 in Sailboat (pages 189 – 230)Combination of multiple choice questions and written/short answer questions and problems.Formulas provided; Bring a calculator!Practice Tests and Formula Sheet on Blackboard.ClickerStatistical InferenceStatistical inference involves using statistics computed from data collected in a sample to make statements about unknown population parameters.In this chapter we are making statistical inferences (confidence intervals and tests of significance) about the population mean m.General Significance Testing Procedure1. State the null and alternative hypotheses, and state the significance level.2. Carry out the experiment, collect the data, verify the assumptions, and compute the value of the test statistic.3. Calculate the p-value.4. Make a decision on the significance of the test (reject or fail to reject H0).5. Make a conclusion statement in the words of the original problem. This is the statistical inference.Tests of Significance for m: s KnownGoal:We hypothesize that the population mean equals some value m0, and state the alternative hypothesis that we wish to prove is true.H0: m = m0Ha: m > m0 or Ha: m < m0 or Ha: m = m0Tests of Significance for m: s KnownAssumptions:1. Simple random sample2. The population is normal or the sample size is large enough for the Central Limit Theorem to apply.3. Population standard deviation s is known.ClickerSignificance Test for mSo the test statistic is:Z = X - m0 s/ nTests of Significance: s UnknownNow suppose that the population standard deviation s is unknown.We still assume that (1) we have a simple random sample and that (2) the population is normal or the sample size is large enough for the Central Limit Theorem to apply.When s is unknown the use of the Z procedures are not valid, and instead we must use the t-distribution.ClickerTests of Significance for mSituation: The population standard deviation s is unknown (we only know the sample standard deviation s)Since s is unknown the Z test statistic is not valid, and insteadwe must use a t statistic.Assumptions: 1. Simple random sample 2. The population is normal or large sampleTest Statistic: t = X - m0 which has df = n - 1 degrees of S/ n freedomTests of Significance for mp-value: the probability, assuming the null hypothesis is true, of observing a test statistic value as extreme or more extreme than the value observed.These are calculated as in the last lecture, except instead of using the Z-distribution we use the t-distribution with n – 1 degrees of freedom.Tests of Significance for m1. Upper one-sided test: Ha: m > m0 p-value = P ( tdf > tobs )2. Lower one-sided test: Ha: m < m0 p-value = P ( tdf < tobs )3. Two-sided test: Ha: m = m0 p-value = 2 P ( tdf > |tobs |)P-value using the t-distributionCalculating the p-value using the t-distribution is very similar to calculating the p-value using the Z-distribution, with the exception being the way the t-table is read.With the Z-distribution we were able to get one number for our p-value. With the t-distribution, using the table in the book we can only find two values that the p-value would be between.p-value = P(t8 > 2.000).025 < p-value < .05TI-83/84 CalculatorSee the end of the chapter for instructions for using a calculator to perform a significance test fora population mean based on the t-distributions.1. Hit STAT2. Scroll over to TESTS3. Choose 2:T-Test4. Next to Inpt choose Stats5. Enter required components and hit CalculateExample 93/80Population of interest: ???Parameter of interest: ???Example 93/80Population of interest: all calls for an ambulance in the large cityParameter of interest = m = mean time before ambulance arrivesExample 93/80Population of interest: all calls for an ambulance in the large cityParameter of interest = m = mean time before ambulance arrives(1) H0: Clicker2 Ha:Example 93/80Population of interest: all calls for an ambulance in the large cityParameter of interest = m = mean time before ambulance arrives(1) H0: m = 4.8 minutes Ha:Example 93/80Population of interest: all calls for an ambulance in the large cityParameter of interest = m = mean time before ambulance arrives(1) H0: m = 4.8 minutes Ha: m < 4.8 (“improved” means less time)Example 93/80Population of interest: all calls for an ambulance in the large cityParameter of interest = m = mean time before ambulance arrives(1) H0: m = 4.8


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VCU STAT 210 - Lecture35(2) (1)

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