Statistical Hypothesis Tests Notes of STAT 6205 by
Statistical Hypothesis Tests Notes of STAT 6205 by Dr. Fan 6205 1
Overview • Introduction of hypotheses tests (Sections 7. 1, 7. 2) o o General logic Two types of error Parametric tests for one mean and for proportions What is the best test for a given situation? • Order Statistics (Section 8. 3) • Wilcoxon tests (Section 8. 5) 6205 2
Statistical Hypotheses • A statistical hypothesis is an assumption or statement concerning one or more population parameters. • Simple vs. composite hypotheses E. g. A pharmaceutical company wants to be able to claim that for its newest medication the proportion of patients who experience side effects is less than 20%. Q. What are the two possible conclusions (hypotheses) here? 6205 3
Hypothesis Tests • A statistical test is to check a statistical hypothesis using data. It involves the five steps: 1. 2. 3. 4. Set up the null (Ho) and alternative (H 1) hypotheses Find an appropriate test statistic (T. S. ) Find the rejection (critical) region (R. R. ) Reject Ho if the observed test statistic falls into R. R. and not reject Ho otherwise 5. Report the result in the context of the situation 6205 4
Determine Ho and H 1 • The null hypothesis Ho is the no-change hypothesis • The alternative hypothesis H 1 says that Ho is false The Logic of Hypothesis Tests: “Assume Ho is a possible truth until proven false” Analogical to “Presumed innocent until proven guilty” The logic of the US judicial system Q: What are the two possible conclusions? 6205 5
Determine Ho and H 1 Golden Rule: Ho must be a simple hypothesis. Practical Rule: If possible, the hypothesis we hope to prove (called research hypothesis) goes to H 1. Back to the drug example, setting Ho and H 1. 6205 6
Types of Errors H 0 true we accept H 0 Good! (Correct!) we reject H 0 Type I Error, or “ Error” H 0 false Type II Error, or “ Error” Good! (Correct) 7
More Terms • a = Significance level of a test = Type I error rate • Power of a test = 1 -Type II error rate=1 - b • We only control a not b, so we don’t say “accept Ho”. 6205 8
Report the Conclusion • Reject Ho: the data shows strong evidence supporting Ha Eg. 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 Ha Eg. Based on the data, there is not sufficient evidence to support the proportion is less than 20% at significant level of 5%. 9
Tests for One Mean 6205 10
Z Test For normal populations or large samples (n > 30) And the computed value of Z is denoted by Z*. 6205 11
6205 12
Types of Tests 6205 13
Types of Tests 6205 14
Types of Tests 6205 15
Example 1 (Conti. ) Conduct a test for Ho: mu=2500 vs. H 1: mu =3000 at 5% significant level. 1) What is the R. R. ? 2) What is the power of the test? Z test is the most powerful test! 6205 16
P-Values • The p-value is the smallest level of significance to reject Ho at the observed value, also called the observed significance level. p-value > a p-value < a fail to 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. 17
P-Value • The level of significance (called a level) is usually 0. 05 • p-value > a • p-value < a fail to reject Ho (? ? ) reject Ho (= accept Ha) 18
Computing the p-Value for the Z-Test 19
Computing the p-Value for the Z-Test 20
Computing the p-Value for the Z-Test P-value = P(|Z| > |z*| )= 2 x P(Z > |z*|) 21
6205 22
t Test • For normal populations with unknown s Eg. Revisit Example 1 23
One Population 24
6205 25
Testing Hypotheses about a Proportion • Three possible Ho and Ha Ho Ha Type p = po Two-sided p > po p < po One-sided (lower-tailed) p < po p > po One-sided (upper-tailed) Write them all as p=po in the future 26
The z-test for a Proportion • When 1) the sample is a random sample 2) n(po) and n(1 -po) are both at least 10, an appropriate test statistic for p is 27
Example: New Drug (Conti. ) 1. 2. 3. 4. 5. Ho: p > 20% vs. Ha: p < 20% Z-test statistic; a = 0. 05 Find rejection region or p-value Decide if reject Ho or not Report the conclusion in the context of the situation 28
Hypothesis Test for the Difference of Two Population Proportions • Step 1. Set up hypotheses Ho: p 1 = p 2 and three possible Ha’s: Ha: p 1 = p 2 (two-tailed) or Ha: p 1 < p 2 (lower-tailed) or Ha: p 1 > p 2 (upper-tailed) 29
Hypothesis Test for the Difference between Two Population Proportions • Step 2. calculate test statistic where 30
Hypothesis Test for the Difference between Two Population Proportions • Step 3: Find p value 1. Must be two independent random samples; both are large samples: And 2. When the above conditions are met, use Z-Table to find p-value. • Steps 4 and 5 are the same as before 31
Example: Bike to School For 80 randomly selected men, 30 regularly bicycled to campus; while for 100 randomly selected women, 20 regularly bicycled to campus. • Find the p-value for testing: Ho: p 1 = p 2 vs. Ha: p 1 > p 2 Answer: z=2. 60, p=0. 0047 1: men; 2: women 32
Order Statistics • Min & Max • Joint and other orders 6205 33
Order Statistics Problem 1: Suppose X 1, X 2, …, X 5 are a random sample from U[0, 1]. Find the pdf of X(2). Problem 2: Suppose X 1, X 2, …, Xn are a random sample from U[0, 1]. Show that X(k) ~ beta(k, n-k+1). 6205 34
Order Statistics The CDF of X(k) and example 6205 35
Wilcoxon Tests Ho: median of X = median of Y vs. H 1: Ho is false Wilcoxon tests (p. 448 - 450) • assume the two distributions are of similar shapes but do not need to be normal • See the supplementary material 6205 36
Exercise 8. 5 -9 X = the life time of light bulb of brand A Y = the life time of light bulb of brand B Data: (in 100 hours) X: 5. 6 4. 6 6. 8 4. 9 6. 1 5. 3 4. 5 5. 8 5. 4 4. 7 Y: 7. 2 8. 1 5. 1 7. 3 6. 9 7. 8 5. 9 6. 7 6. 5 7. 1 (a) Conduct the Wilcoxon test at 5 % level to test if brand B has longer life time in general. A: W(Y)=145 > 128 or Z= 3. 024 > 1. 645; reject Ho (a) Construct and interpret a Q-Q plot of these data. 6205 37
R Code for 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 brand A", ylab="life time of brand B", main="qqplot of Life time of Brand A vs. Brand B") 6205 38
6205 39
- Slides: 39