#### Transcript Chapter 9

Chapter 9 Hypothesis Testing McGraw-Hill/Irwin Copyright © 2009 by The McGraw-Hill Companies, Inc. All Rights Reserved. Hypothesis • Hypothesis is a statement about a certain population parameter. • It is usually about a parameter equal to (not equal to), less than or equal to ( greater than), greater than or equal to(or less than) some given number. • For example: the supporting rate of a candidate among all voters is greater than 50%. • A hypothesis testing is a statistical technique for evaluating if there is enough evidence to support a hypothesis. The strength of evidence is evaluated by the probability of certain statistic, called test statistic. 9-2 Rule of Rare Events • If under current assumption, the probability to observe the sample (statistic) we have is extremely small, then we conclude the assumption is not right. • How small is small? if the probability to observe the sample is less than the Level of significance . =0.1, .05, .01 It is determined by the risk requirement of the problem. The risk is the probability to make a mistake. 9-3 Rule of Events, final version • If under current assumption (H-null), the probability to observe the test statistic is , then we conclude the assumption is not right. • If under H-null, the p-value is , then we conclude H-null assumption is not right. 9-4 How to set up the symbolic form of the hypothesisexpress relationship with math expressions • Find out the hypothesis of interest about the population parameter (mean), and write down its symbolic expression. • Write down the symbolic expression of the complement. • The expression with equal sign (=, ≤, ≥) is called H-null hypothesis (denoted by H0 ), and the remaining expression not containing equal sign is called the H-A hypothesis (denoted by Ha). 9-5 Types of Hypotheses • Right-Sided tailed, “Greater Than” Alternative H0: 0 vs. Ha: > 0 • Left-Sided tailed, “Less Than” Alternative H0 : 0 vs. Ha : < 0 • Two-Sided tailed, “Not Equal To” Alternative H0 : = 0 vs. Ha : 0 where 0 is a given constant value (with the appropriate units) that is a comparative value 9-6 Types of Hypotheses-the red part is the conversion followed by some textbooks • Right-Sided (Right-Tailed), “Greater Than” Alternative H0: 0 vs. Ha: > 0 H0: = 0 vs. Ha: > 0 • Left-Sided (Left-Tailed), “Less Than” Alternative H0 : 0 vs. Ha : < 0 H0 : = 0 vs. Ha : < 0 • Two-Sided (Two-Tailed) , “Not Equal To” Alternative H0 : = 0 vs. Ha : 0 where 0 is a given constant value (with the appropriate units) that is a comparative value 9-7 Z Tests about a Population Mean: σ known • The population standard deviation σ is known. • Suppose the population being sampled is normally distributed, or sample size n is at least 30. Under these two conditions, use the Z distribution to calculate the p-value and then use the rule of rare events to perform the hypothesis testing. 9-8 Z Test Statistic-σ is known • Use the “test statistic” x 0 t.s. n • If the population is normal or n is large*, the test statistic t.s. follows a normal distribution * n ≥ 30, by the Central Limit Theorem 9-9 Z Tests about a Population Mean: σ known Alternative Type of test p-value Ha: µ > µ0 P(Z>t.s.) Ha: µ < µ0 P(Z<t.s.) Ha: µ µ0 2*P(Z>t.s.) for t.s.>0 2*P(Z<t.s.) for t.s.<0 Reject H0 if: p-value≤ t.s. z x 0 x x 0 n 9-10 Right-tailed Test • Right-tailed test: P(Z>t.s.) •H0: μ ≤ k •P is the area to the right of the test statistic. •Ha: μ > k •-3 •-2 •-1 •0 •1 •2 •3 •z •Test statistic 9-11 Left-tailed Test • Left-tailed test: P(Z<t.s.) •H0: μ k •Ha: μ < k •P is the area to the left of the test statistic. •3 •2 •1 •0 •1 •2 •3 •z •Test statistic 9-12 Two-tailed Test Two-tailed test: 2*P(Z>t.s.) for t.s.>0 2*P(Z<t.s.) for t.s.<0 •H0: μ = k •Ha: μ k •P is twice the area to the right of the positive test statistic. •P is twice the area to the left of the negative test statistic. •-3 •-2 •-1 •Test statistic •0 •1 •2 •Test statistic •3 •z 9-13 Hypothesis Testing Conclusion • If p-value≤, then we say we reject Hnull and accept Ha. • If p-value>, we say we fail to reject Hnull and do not accept Ha. Never say we accept H-null. 9-14 Z Tests about a Population Mean: σ known, rejection region method • To use the rule of rare events we only need to know the relationship between p-value and the given significance level. See slide 3. • The rejection region method explores the property and sets up rejection regions in which any value corresponds to a p-value less than the given significance level . That means if the t.s. is on the rejection region then we reject H0. 9-15 Za and Right Hand Tail Areas •The definition of the critical value Zα •The area to the right if 1-α •Zα •Zα is the percentile such that •P(Z< Zα) =1-α 9-16 Right-tailed Test, rejection region • Right-tailed test, for any given significance level •H0: μ ≤ k •The area to the left of z is α. •Ha: μ > k •-3 •-2 •-1 z •0 •1 •2 •3 •Test statistic •z 9-17 Left-tailed Test, rejection region • Left-tailed test •H0: μ k •Ha: μ < k The area to the left of -z is α. •3 •Test statistic •2 •1 •0 •1 •2 •3 •z -z 9-18 Two-tailed Test, rejection region • Two-tailed test •H0: μ = k •Ha: μ k The area to the left of -z/2 is α/2. •-3 •-2 •-1 -z/2 •0 •1 •2 The area to the right of z/2 is α/2. •3 •z z/2 9-19 Z Tests about a Population Mean: σ known, rejection region method Alternative Reject H0 if: Rejection region Ha: μ> µ0 t.s. ≥ z [z, Ha: μ < µ0 t.s. ≤ –z (-∞, -z] Ha: μ µ0 either (-∞, -z/2] ∞) [z/2, ∞) t.s. ≥ z/2 or t.s. ≤ –z/2 Where the test statistics is x 0 t.s. n 9-20 t Tests about a Population Mean: σ Unknown • The population standard deviation σ is unknown, as is the usual situation, but the sample standard deviation s is given. • The population being sampled is normally distributed or sample size is n≥30. • Under these two conditions, we can use the t distribution to test hypotheses 9-21 Defining the t Statistic: σ Unknown • Let x be the mean of a sample of size n with standard deviation s • Also, µ0 is the claimed value of the population mean • Define a new test statistic x 0 t.s. s n • If the population being sampled is normal or sample size is big enough, and s is given… • The sampling distribution of the t.s. is a t distribution with n – 1 degrees of freedom 9-22 t Tests about a Population Mean: σ Unknown Continued Alternative Reject H0 if: Rejection region Ha: µ > µ0 t.s. ≥ t [t, Ha: µ < µ0 t.s. ≤ –t (-∞, -t] Ha: µ µ0 t.s. ≥ t/2 (-∞, -t/2] ∞) [t/2, ∞) or t.s. ≤ –t/2 t, t/2, and p-values are based on n – 1 degrees of freedom (for a sample of size n) 9-23 Definition of the critical value tα 9-24 Type I and Type II errors • If we reject H0 , then it is possible to make type I error • If we fail to reject H0 and do not accept Ha (or equivalently: fail to reject H0 and reject Ha ), then it is possible to make type II error. 9-25 Selecting an Appropriate Test Statistic for a Test about a Population Mean 9-26