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Elementary Statistics Triola, Elementary Statistics 11/e Unit 22 The Basics of Hypotheses Testing Hypotheses testing is not all that different from confidence intervals, so letβs do a quick review of the theory behind the latter. If itβs our goal to estimate the mean of a population, weβre 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 ππΜ = π and whose standard deviation is ππΜ = π . βπ (Review Unit 18, Central Limit Theorem.) Since the sampling distribution is normally shaped, 95% of the π₯Μ β² π are going to fall within ±1.96 standard deviation units from ππΜ . (The 95% centered area under the Standard Normal Distribution lies between π§ = β1.96 πππ π§ = +1.96.) A standard deviation unit for the sampling distribution is ππΜ = π βπ and because ππΜ = π, (from the Central Limit Theorem) we have that 95% of the averages, π₯, Μ are going to lie π π βπ between π ±1.96 π. Now, if 95% of the π₯Μ β² π are within ±1.96 β π βπ of π then π will be within ±1.96 of 95% of all the π₯Μ β² π from the sampling distribution. (Think about that). Hence thereβs a 95% chance that π βπ π is within ±1.96 of the average, πΜ of our sample. Now there is one more adjustment we need to π βπ make. Since we donβt know π we have to replace ±1.96 with π‘πΌβ2 π ββπ. . Therefore, our margin of error E is πΈ = π‘πΌβ2 π ββπ and our 95% confidence interval is π₯Μ ± π‘πΌβ2 π ββπ. 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. Letβs suppose that we take a sample from a population where is claimed that π = 25.0., and the 95% confidence interval based on our sample is (23.75,24.87). 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. This is an example of the Rare Event Rule. Which do you think is more likely, that we picked a βfunkyβ 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. b. c. d. e. The claim is less than some number. The claim is greater than some number. The claim is at most some number. The claim is at least some number 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 involving the βtailsβ of the Normal or Student t distribution. So first, we have to learn something about the tails. Consider the following figure, 55 Unit 22 The Basics of Hypotheses Testing The reddish areas are the left and right tails of the distribution. Donβt 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 0.95. We will be taking raw scores, like 25.25 and converting these scores to t-scores using the following formula, π‘= π₯Μ β π π₯Μ β π π = π βπ βπ So for example, if π₯Μ = 25.25, π = 1.32, π = 20, π = 25.00 (this is the claim), then we have, π‘= x 25.25 25.25 β 25.00 β20 = 0.8470 1.32 s n 1.32 20 t 0.8470 You can double click the above table, because it is a spreadsheet calculator. Question #1 Locate approximately this value for t on the graph above. Does it fall under one of the red areas? Question #2 Now calculate the value of t but instead of using π₯Μ = 25.25, use π₯Μ = 25.00, the value of the claim. What did you get? Locate this value on the graph above. Notice that π = 25.00 corresponds to π‘ = 0. Now, comes a very important concept. π₯Μ = 25.25 is a distance of 0.8470 units from π = 25.00 adjusted for the sample standard deviation and size, and that distance does not put 25.25 under one of the red areas. If the Μ and π is βtoo greatβ then we cannot accept the claim as being true. What is too distance between π great? Itβs 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. This is the end of Unit 22. these concepts. Now turn to MyMathLab to get more practice with 56