Chapter ElevenIntroduction to Hypothesis Testing618Hypothesis Testing• Along with Estimation, Hypothesis Testing constitutes the second type of statistical inference.• The goal of hypothesis testing is to determine whether there is enough “statistical evidence” to conclude that a conjecture or hypothesisabout a population parameter is supported by the data.619Null and Alternative Hypothesis• When we do hypothesis testing, we test between two hypothesis or conjectures. • The first such conjecture is called the null hypothesis and is denoted as H0.• The second conjecture is called the alternative hypothesis and is denoted as H1.• The goal is to design a statistical procedure that allows us to determine whether the data supports the null or the alternative hypothesis. 620• How do we determine which one is the null (H0) and which one is the alternative (H1) hypothesis?• We consider the null hypothesis (H0) as the one that is more costly (socially, economically, etc.) to reject when it is true. • The statistical testing procedures are designed in a wa y that keeps the probability of wrongly rejecting H0low.621• The basic illustrative example of hypothesis testing and the distinction between H0 and H1 is the issue of establishing culpability of a person in a trial.• The worst possible social outcome in this case is to con vict an innocent person. Thus, the null hypothesis becomes:H0 : The defendant is innocent• And the alternative hypothesis is:H1 : The defendant is guilty• Again, H0 is the hypothesis (conjecture) considered more costly (socially, economically, etc.) to wrongly reject.622Type I and Type II Errors• There are two types of errors (mistakes) we can make in a hypothesis test:I. Type I Error : Rejecting a true null hypothesis H0II. Type II Error: Failing to reject a false null hypothesis H0 • Significance Level: Is the probability of committing a Type I error. It is typically denoted by α• The probability of committing a Type II error is typically denoted by β• As we discussed above, the Type I error is deemed more costly than Type II. • The goal is to design testing procedures where the researcher can control the significance level and fix it to be low. For example, fix α=5%. 623• Ideally, one would want procedures where both error probabilities α and β Are low. Unfortunately, these error probabilities are inversely related. Therefore, we settle on fixing α to be low.• In our previous example, we have:• α= Probability of convicting an innocent defendant.• β = Probability of letting a guilty defendant free. 624Logical Steps of a Hypothesis Test• We begin with a null H0 and alternative hypothesis H1.• The testing procedure begins with the assumption that H0is true. • The goal is to determine whether there is enough evidence in the data to support H1. • There are only two possible decisions:• Conclude that there is enough evidence to support H1• Conclude that there is not enough evidence to support H1• The decision rule revolves around Pr(Type I error) = α625Hypothesis Tests of a Population Mean µ when the Standard Deviation is Known• Here, the goal is to compare µ against a pre‐specified value, µ*.• There are two types of alternative hypothesis:• Two‐tail test.‐ Here, we haveH0: µ = µ* vs. H1: µ ≠ µ*• One‐tail tests.‐ Here, we could have eitherH0: µ ≤ µ* vs. H1: µ > µ*or…H0: µ ≥ µ* vs. H1: µ < µ*626Re‐expressing One‐Tail Tests• Suppose we have H0: µ ≤ µ* vs. H1: µ > µ*• Notice that if we have enough evidence in favor of H1when we start with the assumption that µ = µ*, the evidence in favor of H1would be even stronger if we assumed that µ < µ*.• For this reason, it would suffice to useH0: µ = µ* vs. H1: µ > µ*• If we reject µ = µ* in favor of µ > µ*, we would automatically also reject µ ≤ µ* in favor of µ > µ*. 627• Analogous reasoning leads us to conclude that, in order to testH0: µ ≥ µ* vs. H1: µ < µ*it suffices to testH0: µ = µ* vs. H1: µ < µ*• This is because, if we reject µ = µ* in favor of µ < µ* we would necessarily also reject µ ≥ µ* in favor of µ < µ*. 628• Therefore, we can re‐express our tests as:• Two‐tail test.‐H0: µ = µ* vs. H1: µ ≠ µ*• One‐tail tests.‐H0: µ = µ* vs. H1: µ > µ*or…H0: µ = µ* vs. H1: µ < µ*629• Example 10.1: This is the example about the manager wanting to estimate the demand for computers.• Back then, we determined that a 95% confidence interval for the mean demand µwas [340.76 , 399.56] • Suppose now that the computer firm wants to determine whet her the mean demand µ is different from 350. • In this case, we would have H0: µ=350 vs. H1: µ≠350 630• Next, suppose a new inventory str
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