Statistical Hypothesis Tests Overview Statistical Hypotheses Hypothesis Tests Determine Ho in addition to H1
Grady, Michael, Features Reporter has reference to this Academic Journal, PHwiki organized this Journal Statistical Hypothesis TestsNotes of STAT6205 by Dr. Fan16205OverviewIntroduction of hypotheses tests (Sections 7.1,7.2)General logicTwo types of errorParametric tests as long as one mean in addition to as long as proportionsWhat is the best test as long as a given situationOrder Statistics (Section 8.3)Wilcoxon tests (Section 8.5)26205Statistical HypothesesA statistical hypothesis is an assumption or statement concerning one or more population parameters.Simple vs. composite hypothesesE.g. A pharmaceutical company wants to be able to claim that as long as its newest medication the proportion of patients who experience side effects is less than 20%.Q. What are the two possible conclusions (hypotheses) here 36205
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Hypothesis TestsA statistical test is to check a statistical hypothesis using data. It involves the five steps:Set up the null (Ho) in addition to alternative (H1) hypothesesFind an appropriate test statistic (T.S.)Find the rejection (critical) region (R.R.)Reject Ho if the observed test statistic falls into R.R. in addition to not reject Ho otherwise Report the result in the context of the situation462055Determine Ho in addition to H1The null hypothesis Ho is the no-change hypothesisThe alternative hypothesis H1 says that Ho is falseThe Logic of Hypothesis Tests:Assume Ho is a possible truth until proven falseAnalogical toPresumed innocent until proven guiltyThe logic of the US judicial systemQ: What are the two possible conclusions6205Determine Ho in addition to H1Golden Rule: Ho must be a simple hypothesis.Practical Rule: If possible, the hypothesis we hope to prove (called research hypothesis) goes to H1.Back to the drug example, setting Ho in addition to H1.66205
7 Good!(Correct!)H0 trueH0 falseType II Error, or ErrorType I Error, or Error Good!(Correct)we accept H0we reject H0Types of ErrorsMore Termsa = Significance level of a test = Type I error ratePower of a test = 1-Type II error rate=1- bWe only control a not b, so we dont say accept Ho.620589Report the ConclusionReject Ho: the data shows strong evidence supporting HaEg. The data shows strong evidence that the proportion of users who will experience side effects is less than 20% at significant level of 10%.Fail to reject Ho: the data does not provide sufficient evidence supporting HaEg. Based on the data, there is not sufficient evidence to support the proportion is less than 20% at significant level of 5%.
Tests as long as One Mean10620511Z TestFor normal populations or large samples (n > 30)And the computed value of Z is denoted by Z.6205620512
13Types of Tests620514Types of Tests620515Types of Tests6205
Example 1 (Conti.)Conduct a test as long as Ho: mu=2500 vs. H1: mu =3000 at 5% significant level.What is the R.R.What is the power of the test Z test is the most powerful test!62051617P-ValuesThe p-value is the smallest level of significance to reject Ho at the observed value, also called the observed significance level. p-value > a fail to reject Ho p-value < a reject Ho (= accept Ha)That is, p-value is the probability of seeing as extreme as (or more extreme) what we observe, given Ho is true.18P-ValueThe level of significance (called a level) is usually 0.05p-value > a fail to reject Ho ()p-value < a reject Ho (= accept Ha) 19Computing the p-Value as long as the Z-Test20Computing the p-Value as long as the Z-Test21Computing the p-Value as long as the Z-TestP-value = P(Z > z )= 2 x P(Z > z)
62052223t TestFor normal populations with unknown s Eg. Revisit Example 124One Population
62052526Testing Hypotheses about a ProportionThree possible Ho in addition to HaWrite them all as p=po in the future27The z-test as long as a ProportionWhen 1) the sample is a r in addition to om sample 2) n(po) in addition to n(1-po) are both at least 10, an appropriate test statistic as long as p is
28Example: New Drug (Conti.)Ho: p > 20% vs. Ha: p < 20%Z-test statistic; a = 0.05Find rejection region or p-valueDecide if reject Ho or notReport the conclusion in the context of the situation29Hypothesis Test as long as the Difference of Two Population ProportionsStep 1. Set up hypothesesHo: p1 = p2 in addition to three possible Has: Ha: p1 = p2 (two-tailed)or Ha: p1 < p2 (lower-tailed)or Ha: p1 > p2 (upper-tailed)30Hypothesis Test as long as the Difference between Two Population ProportionsStep 2. calculate test statistic where
Exercise 8.5-9X = the life time of light bulb of br in addition to AY = the life time of light bulb of br in addition to BData: (in 100 hours)X: 5.6 4.6 6.8 4.9 6.1 5.3 4.5 5.8 5.4 4.7Y: 7.2 8.1 5.1 7.3 6.9 7.8 5.9 6.7 6.5 7.1Conduct the Wilcoxon test at 5 % level to test if br in addition to B has longer life time in general.A: W(Y)=145 > 128 or Z= 3.024 > 1.645; reject HoConstruct in addition to interpret a Q-Q plot of these data.620537R Code as long as Q-Q Plot> x<-c(5.6, 4.6, 6.8, 4.9, 6.1, 5.3, 4.5, 5.8, 5.4, 4.7)> y<-c(7.2, 8.1, 5.1, 7.3, 6.9, 7.8, 5.9, 6.7, 6.5, 7.1)> qqplot(x,y,xlab=”life time of br in addition to A”, ylab=”life time of br in addition to B”, main=”qqplot of Life time of Br in addition to A vs. Br in addition to B”)620538620539
Grady, Michael Features Reporter
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