Elementary Statistics Triola, Elementary Statistics 11/e Unit 17 The Basics of Hypotheses Testing

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1 (Section 8-2) Hypotheses testing is not all that different from confidence intervals, so let s do a quick review of the theory behind the latter. If it s our goal to estimate the mean of a population, we re going to start with the mean of our sample. We can think of this mean is one of many comprising the sampling distribution, whose mean is and whose standard deviation is (Review Unit 13, Central Limit Theorem.) Since the sampling distribution is normally shaped, 95% of the are going to fall within standard deviation units from (The 95% centered area under the Standard Normal Distribution lies between ) A standard deviation unit for the sampling distribution is and because, (from the Central Limit Theorem) we have that 95% of the averages, are going to lie between Now, if 95% of the are within 1.96 of then will be within 1.96 of 95% of all the from the sampling distribution. (Think about that). Hence there s a 95% chance the our is one of the from above, so there s a 95% chance that is within 1.96 of our Now there is one more adjustment we need to make. Since we don t know we have to replace 1.96 with is Therefore, our margin of error E is and our 95% confidence interval Read the above paragraph over again to really appreciate the beauty of all this. Recall that we were able to use confidence intervals to verify claims. Let s suppose that we take a sample from a population where the claim is that the mean of the population, is 25.0, and that the 95% confidence interval based on our sample is ( ) Notice that 25.0 is not in the interval. What does that mean? It means one of two things, either we somehow managed to select a very unusual sample having less than a 5% chance of being selected, or that the claim is wrong. Which do you think is more likely, that we picked a funky sample, which we only had a 5% chance of doing so, or the claim is wrong? Most statisticians would agree that most likely the claim is wrong. This is essentially what we do with hypotheses tests. We test claims. However, with hypotheses tests we are able to test a greater variety of claims. For example we can test, a. The claim is less than some number. b. The claim is greater than some number. c. The claim is at most some number. d. The claim is at least some number e. The claim is equal to some number. These are the same five relations that gave you so much trouble back when you were working with the Binomial and Poisson distributions. These five relations result in different types of hypotheses testing 60

2 Elementary Statistics involving the tails of the Normal or Student t distribution. So first, we have to learn something about the tails. Consider the following figure, The reddish areas are the left and right tails of the distribution. Don t be concerned with the numbers at this point. They change with respect to the confidence level. Pictured above is a 95% confidence level because the area between the two tails is We will be taking raw scores, like and converting these scores to t-scores using the following formula, So for example, if ( ) then we have, x s n t Locate approximately this value for t on the graph above. Does it fall under one of the red areas? 1. Now calculate the value of t but instead of using use the value of the claim. What did you get? 2. 61

3 Locate this value on the graph above. Notice that corresponds to Now, comes a very important concept. is a distance of units from adjusted for the sample standard deviation and size, and that distance does not put under one of the red areas. If the distance between and is too great then we cannot accept the claim as being true. What is too great? It s too great when t ends up being under the red zone. Please reread the above paragraph several times until it makes sense. It is the central idea of hypotheses testing. There are three basic types of of hypotheses testing and they involve the tails, Left Tail (LT), Right Tail(RT) and Two Tail (2T). To understand the difference between these tests and why we would use one over the other, let s work with an example. Suppose we get in a shipment of 50,000 washers. (These are circular discs with a hole in the center). We might be concerned that the hole is too small, in which case we would use a left-tail test. Or we might be concerned that the outside diameter of the washer is too big, in which case we would use a right-tail test. Finally, we might be concerned that the hole is too small or too big, in which case we would use the two-tail test. Let s take a deeper look at why we would use the left-tail test if we were concerned about the hole being two small. Again, suppose the manufacturer is claiming that on average, the hole size is mm. We are either going to reject this claim because we have very convincing evidence that it is wrong, or we will not reject the claim based on our evidence. So let s consider two cases, case 1 where we cannot reject the claim, and case 2, where we do reject the claim. Case 1. Let s say the our sample yields the following results, the claim). The t value is (this is x s n t Now, look at the graph above. Does t fall under the red area? (The border if the left red area is because the graph is symmetrical). The answer is no, and it means that for a sample of size 30 with a standard deviation of 3.45, just isn t far enough way from to provide convincing proof that the holes of the batch of 50,000 washers does not have a mean diameter of mm. Remember we took just a simple random sample of only 30 washers. If we were to take another simple random sample of 30 washer, we could very easily end up with a sample average hole diameter of Case 2. In this case, let s say we have the following results, value is The t x s n t

4 This time we see that t is in the red zone. Given a standard deviation of 1.74 and a sample size of 30, is just too far from forcing us to reject the claim. Given the evidence of our sample, it is very unlikely (less than 5%) that the claim is correct and our sample was just a weird one. The five mathematical relationship above result in three different type tail hypotheses tests as shown in the following table, Relation Less than Greater than At most At least Equals Tail Type LT RT RT LT 2T All you have to do is determine which of the relational tests you want to run and then according to the table, select the appropriate tail test. The key is to find the words in the problem statement that match (or mean the same thing) as the words in the table. The homework will give you plenty of practice doing this. Finally, there is one more step to setting an hypotheses test. We need to formally state the hypotheses. This involves defining the null hypothesis, and the alternative hypothesis,. The null hypothesis is always stated one way, that No matter how the problem is worded, i.e., is less than, greater than, at most, at least or equals some number, the null hypothesis is always state as, The alternative hypothesis reflects the wording of the problem, but can only use the following relations, To get the right relation, you need a table, Relation Tail Type Alt Hyp Less than LT Greater than RT At most RT At least LT Equals 2T It may seem that the relation for at most and at least are contradictory, but they result from the fact that we cannot use in hypothesis statement. 63

5 Elementary Statistics Worked Example In this example, we are going to make only one claim about the weight of elephants. We are going to claim that the average weight of elephants is less than 5000 lbs, and we are going to test this hypothesis at the 95% confidence level. We collect a simple random sample of 30 elephants and find that Hypotheses Statement Calculate t If you locate this value on the graph above, you will see that t is greater than and hence does not ends up in the red zone. The conclusion is that we cannot reject the null hypothesis (which is effectively saying that is greater than or equal to 5000) and therefore, we would reject the claim that on average elephants weigh less than 5000 lbs. In all likelihood, you found the above paragraph to be confusing. Look at this way, 4900 is just not far enough away from 5000 for us to assume that elephants weight less than 5000 lbs. When running a hypotheses test, the burden of proof is on the claim. In the next unit, we will explore the other relations as well as dig a little deeper into hypotheses testing. This is the end of Unit 16. In class, you will get more practice with these concepts by working exercises in MyMathLab. 64

6 Answers 1. No