Res-Econ TBL 12 Notes - Covers the basics on Hypothesis Testing and the steps associated with it including PDF

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Covers the basics on Hypothesis Testing and the steps associated with it including formulas and step-by-step procedures
Professor: Wayne Roy Gayle...

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Resource Economics 212 December 1 2016 Textbook Notes TBL 12: Hypothesis Testing Hypothesis Testing  Confidence in a characteristic of a population, such as the mean, variance, or proportion, is often pivotal for decision making in business, science, and everyday life.  Statistical Hypothesis o A hypothesis is a claim about a characteristic of a population. o A hypothesis test is a procedure for testing a characteristic of a population. o The method of hypothesis testing is based on the Rare Event Rule for Inferential Statistics:  If, under a given claim about the value of the population parameter, the probability of an estimate of the population parameter is very mall, we conclude that the claim is probably not correct. o The method of hypothesis testing formalizes how you draw these conclusions, taking into account the uncertainty that is inherent in your estimator due to sampling variation.  Hypothesis Testing – 6 Steps o Identify the null hypothesis o Identify the alternative hypothesis o Specify Decision Rule: Construct the Rejection Region  Determine if a z or t table should be used – is  known?  Pick , the significance level  Find the critical value, z-value or t-value o Estimate your parameter of interest o Make a decision: compare your sample estimate to your decision rule o Conclusion: state your conclusion in words  Null Hypothesis, H 0 , is the statement about the population (mean or proportion) that is assumed to be true unless there is convincing evidence to the contrary.  Alternative Hypothesis, H 1 or H a is a statement about the population that is contradictory to the null hypothesis. It is accepted as true only if there is convincing evidence in favor of it.  Null vs. Alternative Hypothesis o The null and alternative hypotheses should be constructed so that they are mutually exclusive and collectively exhaustive.  Mutually Exclusive: both hypothees cannot be “true” at the same time  Collectively Exhaustive: one of the hypotheses must be “true”.  Hypothesis Testing – Types of Errors o The null hypothesis is either true or false o We have only limited information about the population distribution which is contained in the sample.

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o With this information, we have to decide to reject or not reject our null hypothesis. o With limited info, we are likely to make mistakes:  Type 1 Error: There is a chance that we reject the null hypothesis when it is true.  Type 2 Error: There is a chance that you fail to reject (FTR) the null hypothesis when it is false. o We can never tell whether we make an error in our decision about whether to reject the null hypothesis. However, we can compute the probability of making a type 1 and type 2 error:  Pr(Type I error) = P(Reject H0|H0 is true) =   Pr(Type II error) = P(FTR H0|H0 is false) =  Decision Rule: criterion for rejecting the null hypothesis. The decision rule is set before the experiment is performed or before the sample is drawn. o Depends on the significance level,  which defines the rejection region of the test. o Here we admit that we do not know the true population parameter, but we claim or hypothesize that it is at least, at most, or equal to a given value.  Left-Tailed Test:  �0: � ≥ �0  �1: � < �0  Right-Tailed Test:  �0: � ≤ �0  �1: � > �0  Two-Tailed Test:  �0: � = �0  �1: � ≠ �0 o The critical values of a hypothesis test are the points that partition the confidence region and the rejection region.  If  is known then for a given significance level,  and sample size n.  Left-Tailed Test H1: � < �0  - �0 < 0 K α =μ0 −z α ❑ o √n  If  is unknown then for a given significance level,  and sample size n.  Left-Tailed Test H1: � < �0  - �0 < 0 S K ❑=μ0 −t ,n−1 o √n  For a significance level,  and sample size n.  Left-Tailed Test �1: � < �0 ⇔ � − �0 < 0. o K ❑= π 0− z α √ π 0(1−π 0 )/ n o The same procedure applies for the Right-Tailed Test when:  H1: � > μ0  � - μ0 > 0  Except, you will add instead of substracting.

K α =μ0 + z α ❑ √n o The Two-Tailed Test adds and subtracts. The Confidence Region: the region around our hypothesized population parameter where we do not reject the null hypothesis if our sample estimate falls within it. The Rejection or Critical Region: the region outside the confidence region, where we reject the null hypothesis if our sample estimator falls with it. Type 1 vs. Type II Error o A good testing procedure is one that minimizes both the type I and type II errors o However, for any fixed sample size n, both are inversely related. If you try to reduce the probability of one type of error, you increase it for the other. o The only way to reduce both is to increase the sample size. o EG:

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