In any hypothesis testing problem, there are two contradictory hypotheses under consideration.

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1 8.1 Hypotheses and Test Procedures: A hypothesis One example of a hypothesis is p =.5, if we are testing if a new formula for a soda is preferred to the old formula (p=.5 assumes that they are preferred equally) In any hypothesis testing problem, there are two contradictory hypotheses under consideration. The objective of hypothesis testing is to decide based on sample evidence which of the two hypotheses is correct. The judicial system is a perfect example of this: We are deciding guilt or innocence. We initially assume innocence and then if strong enough evidence is shown to contradict this assumption, we find the defendant guilty.

2 In Mathematics, the two contradictory hypotheses have names: 0 If sample evidence is strong enough, we say If the sample evidence is NOT strong enough

3 A test of hypotheses is The alternative hypothesis is often called the researchers hypothesis because it is the claim that you are attempting to prove. The alternative hypothesis bears the burden of proof. For simplifications sake, The null hypothesis normally has the form: The alternative hypotheses will have one of the following forms:

4 Test Procedures: A test procedure is specified by the following: 1. 2.

5 For example Old coke vs. new coke, the null hypoth: people prefer old coke (why switch if they don t like new coke better?) Let p = the proportion of people polled who prefer the new coke. Null: p =.5 (they are same) vs. alter: p >.5. We would not reject for p = Let s say you plan to poll 100 people independently: You would expect to see a split if there is no difference in preference. So how many people do you think you would like to see choose the new formula before you are willing to stop producing old coke and start producing only new coke? 1. Test Statistic: 2. Rejection region: Then you sample the 100 people. And

6 Errors in Hypothesis Testing: There are two types of errors one can encounter when doing a hypothesis test. These errors will help us determine a good vs. a bad rejection region: Type I error: Type II error: These errors happen because there is variability within each sample. It is only possible to avoid these if you sample the entire population, which is impractical or impossible!

7 We cannot find procedures for which these errors are impossible, but we can find procedures for which they are unlikely. The rejection region creates boundaries for the probability of these errors. The probability of a type I error is denoted by : The probability of a type II error is denoted by: Type I error depends on the null value so there is only one value for α (recall the null is always θ = θ ). Type II error 0 depends on the alternative value, which may take on a number of values, so there are many values for β (recall H is either θ > θ, θ < θ, θ θ so the point is that θ θ because θ really equals some 0 different number (but it may be anything!) a

8 Back to our coke example: Old coke vs. new coke, the null hypoth: people prefer old coke (why switch if they don t like new coke better?) Let p = the proportion of people polled who prefer the new coke. Null: p =.5 (they are same) vs. alter: p >.5. Let s say you plan to poll 100 people independently: You would expect to see a split if there is no difference in preference. 1. Test Statistic: 2. Rejection region: When H 0 is true, X is binomial, with n = 100, p =.5. The probability of a type I error: (since n is so large, we can approximate with the normal)

9 For type II errors, things are not quite as straight forward: You don't reject the null but the null is false; the nullis false, because p is not.5, but in fact something larger (not that we actually know what p IS). The type two error works like this: What is the probability of not rejecting the null, when the null is false because it actually equals...um, say,.75. or maybe.65, or even.9. You could find the β value for each of these situations, and each is valid becaue they are all larger than.5. let's do it for p =.75 P(not rejecting H when H is false, because p =.75) 0 0

10 Let s say we changed our rejection region to be X> 90. Now compute the type I and type II errors:

11 This example shows that if the sample size of an experiment is fixed and a test statistic is chosen then decreasing the size of the rejection region to obtaina smaller value of α will result in a larger value for β. The chosen value of α is called the significance level of the test. The test done is called a level α test.

12 A mixture of pulverized fuel ash and Portland cement to be used for grouting should have a compressive strength of more than The mixture will not be used unless experimental evidence indicates conclusively that the strength specification has been met. Suppose compressive strength for the specimens of this mixture is normally distributed with. σ = 60. Let μ denote the true average compressive strength. a. What are appropriate null and alternative hypotheses? b. Let X denote the sample average compressivestrength for n = 20 randomly selected specimens. Consider the test procedure with test statistic X and rejection region x What is the probability of the test statistic when H is true? What is theprobability of a type I error? 0

13 c. How would you change the test procedure of part b to obtain a test with significance level.05?