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Section 10.2 Hypothesis Testing for Means (Small Samples) HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2008 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved.

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What this lesson is about Learn to perform a hypothesis test The previous lesson was only about how to set up a hypothesis test. – Reading and understanding the real-life scenario. – Getting the right letter, μ or p. – Getting the right relational operators in the right places: = and, and >, and <. – Getting the right value of μ or p (and setting aside the noise numbers in the problem statement.) (Added content by D.R.S.)

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Choice: Do a t Test or a z Test? Small Samples: t Test Small means sample size is n < 30. Theres an assumption that the population is normally distributed. If the population is not normally distributed, this method we use is NOT valid. Easy for today: everything we do is a t Test. Large Samples: z Test Large means sample size is n 30. To be discussed in a later lesson. The Bluman book has slightly different rules from the way this Hawkes book does it. Just be aware of that. (Added content by D.R.S.)

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HAWKES LEARNING SYSTEMS math courseware specialists Test Statistic for Small Samples, n < 30: Hypothesis Testing 10.2 Hypothesis Testing for Means (Small Samples) with d.f. n – 1 To determine if the test statistic calculated from the sample is statistically significant we will need to look at the critical value. The critical values for n < 30 are found from the t-distribution.

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HAWKES LEARNING SYSTEMS math courseware specialists Find the critical value: Hypothesis Testing 10.2 Hypothesis Testing for Means (Small Samples) Find the critical t-score for a right-tailed test that has 14 degrees of freedom at the 0.025 level of significance. Solution: d.f. 14 and 0.025 t 0.025 2.145 (Added info) Its in Table C, Critical Values of t Inputs: Column for α (alpha) Choose the right column for one- or two-tailed Row for d.f., degrees of freedom (= sample size n, minus 1)

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HAWKES LEARNING SYSTEMS math courseware specialists Rejection Regions: Hypothesis Testing 10.2 Hypothesis Testing for Means (Small Samples) Determined by two things: 1.The type of hypothesis test. 2.The level of significance,. Finding a Rejection Region: 1.Look up the critical value, t c, to determine the cutoff for the rejection region. 2.If the test statistic you calculate from the sample data falls in the area, then reject H 0.

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HAWKES LEARNING SYSTEMS math courseware specialists Types of Hypothesis Tests: Hypothesis Testing 10.2 Hypothesis Testing for Means (Small Samples) Alternative Hypothesis < Value > Value Value Type of Test Left-tailed test Right-tailed test Two-tailed test

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HAWKES LEARNING SYSTEMS math courseware specialists Rejection Regions for Left-Tailed Tests, H a contains <: Hypothesis Testing 10.2 Hypothesis Testing for Means (Small Samples) Reject if t –t

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HAWKES LEARNING SYSTEMS math courseware specialists Rejection Regions for Right-Tailed Tests, H a contains >: Hypothesis Testing 10.2 Hypothesis Testing for Means (Small Samples) Reject if t t

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HAWKES LEARNING SYSTEMS math courseware specialists Rejection Regions for Two-Tailed Tests, H a contains : Hypothesis Testing 10.2 Hypothesis Testing for Means (Small Samples) Reject if | t | t /2

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HAWKES LEARNING SYSTEMS math courseware specialists Steps for Hypothesis Testing: 1.State the null and alternative hypotheses. 2.Set up the hypothesis test by choosing the test statistic [that is, make a decision on whether its a t or z problem] and determining the values of the test statistic that would lead to rejecting the null hypothesis [the critical value(s)]. 3.Gather data and calculate the necessary sample statistics [t = or z = ]. 4.Draw a conclusion [Stating it two ways: reject/fail to reject, and also in plain English]. Hypothesis Testing 10.2 Hypothesis Testing for Means (Small Samples) IMPORTANT !!!!

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HAWKES LEARNING SYSTEMS math courseware specialists Draw a conclusion: Hypothesis Testing 10.2 Hypothesis Testing for Means (Small Samples) 27 college students are asked how many parking tickets they get a semester to test the claim that the average student gets more than 9 tickets a semester. The sample mean for the number of tickets was found to be 9.8 with a standard deviation of 1.5. Use 0.10. Solution: n 27, 9, 9.8, s 1.5, d.f. 26, 0.10 t 0.10 Since t is greater than t, we will reject the null hypothesis. 1.315 2.771 H 0 : μ 9 tickets H a : μ > 9 tickets. This is the CRITICAL VALUE. Either use table or invT(0.10,26). Draw a PICTURE, too. Mark 1.315 and highlight the critical region. This is the TEST STATISTIC. Mark 2.771 on your picture.

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Remarks about the parking ticket example There was a choice made to do a t Test because the sample size was < 30. There was an implicit assumption that the distribution of the count of parking tickets fits a normal distribution. It was a RIGHT-TAILED TEST because of the > in the alternative hypothesis. (Added content by D.R.S.)

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Remarks about the parking ticket example, continued Hypothesis tests are really essay questions. The outline for the essay is the four-step procedure described in the earlier slide. Each of the four steps needs to be explained plainly with a lot of words: Complete thoughts and complete sentences. The final statement is in plain English, suitable for the general public to understand. (Added content by D.R.S.)

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The Parking Ticket problem done as an essay question 1. State the hypotheses We investigate the claim that the average student receives more than nine parking tickets in a semester. Our hypotheses are: Null hypothesis, H 0 : μ 9 Alternative hypothesis: Ha: μ > 9, more than nine tickets per semester. 2. Find the critical value This is a t Test, right tailed. The sample size is n = 27. The degrees of freedom is d.f. = n – 1 = 26. The level of significance chosen is α = 0.10 The critical value is t α=0.10,d.f.=26 = 1.315 (Added content by D.R.S.)

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The Parking Ticket problem done as an essay question 3. Compute the test statistic (As shown on the earlier slide – formula & details) 4. Conclusions Since the test value 2.771 is greater than the critical value 1.315, we reject the null hypothesis. There is sufficient evidence to support the claim that the average student gets more than 9 parking tickets per semester. (Added content by D.R.S.) 2.771

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HAWKES LEARNING SYSTEMS math courseware specialists Draw a conclusion: Hypothesis Testing 10.2 Hypothesis Testing for Means (Small Samples) A hometown department store has chosen its marketing strategies for many years under the assumption that the average shopper spends no more than $100 in their store. A newly hired store manager claims that the current average is higher, and wants to change their marketing scheme accordingly. A group of 24 shoppers is chosen at random and found to have spent on average $104.93 with a standard deviation of $9.07. Test the store managers claim at the 0.010 level of significance. Solution: First state the hypotheses: H0:H0: Ha:Ha: Next, set up the hypothesis test and determine the critical value: d.f. 23, 0.010 t 0.010 Reject if t t, or if t > 2.500. 100 > 100 2.500

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HAWKES LEARNING SYSTEMS math courseware specialists Hypothesis Testing 10.2 Hypothesis Testing for Means (Small Samples) Solution (continued): Gather the data and calculate the necessary sample statistics: n 24, 100, 104.93, s 9.07, Finally, draw a conclusion: Since t is greater than t, we will reject the null hypothesis. 2.663

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Added content Repeating several of the slides with extra comments about TI-84 Also an important reminder: using this method for small sample sizes requires that the population being studied is NORMALLY DISTRIBUTED. Not uniform, not skewed, but a bell curve distribution is assumed. (This book somewhat glosses over this point.

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HAWKES LEARNING SYSTEMS math courseware specialists Find the critical value: Hypothesis Testing 10.2 Hypothesis Testing for Means (Small Samples) Find the critical t-score for a right-tailed test that has 14 degrees of freedom at the 0.025 level of significance. Solution: d.f. 14 and 0.025 t 0.025 2.145 The critical values for n < 30 are found from the t-distribution. invT(area to left, d.f.) = t value Plus or Minus Sign? Either by symmetry or by adjusting the area value for a right-tailed test. You still have to understand whether its left-tailed, right- tailed, or two-tailed. The calculator wont do that for you !

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HAWKES LEARNING SYSTEMS math courseware specialists Draw a conclusion: Hypothesis Testing 10.2 Hypothesis Testing for Means (Small Samples) 27 college students are asked how many parking tickets they get a semester to test the claim that the average student gets more than 9 tickets a semester. The sample mean for the number of tickets was found to be 9.8 with a standard deviation of 1.5. Use 0.10. Solution: n 27, 9, 9.8, s 1.5, d.f. 26, 0.10 t 0.10 1.315 Again, fix up the sign by knowing that its a right-tailed test, therefore positive critical value. The calculator will not do this thinking for you.

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HAWKES LEARNING SYSTEMS math courseware specialists Hypothesis Testing 10.2 Hypothesis Testing for Means (Small Samples) (continued from previous slide) Solution: n 27, 9, 9.8, s 1.5, d.f. 26, 0.10 t 0.10 Since t is greater than t, we will reject the null hypothesis. 1.315 2.771 EXTRA ( ) around complicated numerators and denominators !!!

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HAWKES LEARNING SYSTEMS math courseware specialists Draw a conclusion: Hypothesis Testing 10.2 Hypothesis Testing for Means (Small Samples) A hometown department store has chosen its marketing strategies for many years under the assumption that the average shopper spends no more than $100 in their store. A newly hired store manager claims that the current average is higher, and wants to change their marketing scheme accordingly. A group of 24 shoppers is chosen at random and found to have spent on average $104.93 with a standard deviation of $9.07. Test the store managers claim at the 0.010 level of significance. Solution: First state the hypotheses: H0:H0: Ha:Ha: Next, set up the hypothesis test and determine the critical value: d.f. 23, 0.010 t 0.010 Reject if t t, or if t > 2.500. 100 > 100 2.500

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HAWKES LEARNING SYSTEMS math courseware specialists Hypothesis Testing 10.2 Hypothesis Testing for Means (Small Samples) Solution (continued): Gather the data and calculate the necessary sample statistics: n 24, 100, 104.93, s 9.07, Finally, draw a conclusion: Since t is greater than t, we will reject the null hypothesis. 2.663

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TI-84 T-Test The TI-84 has a built in Hypothesis Testing tool STAT menu, TESTS submenu, 2:T-Test You must understand how to do hypothesis testing with charts and formulas, however. The calculator is not a substitute for that. Mere button smashing will lead you to failure.

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HAWKES LEARNING SYSTEMS math courseware specialists Example: Hypothesis Testing 10.2 Hypothesis Testing for Means (Small Samples) 27 college students are asked how many parking tickets they get a semester to test the claim that the average student gets more than 9 tickets a semester. The sample mean for the number of tickets was found to be 9.8 with a standard deviation of 1.5. Use 0.10. Solution: Choose Data if the 27 data values were in TI-84 Lists, Stats if we have summary statistics already calculated Null hypothesiss mean Samples Mean, Standard deviation, and Size Direction of the Alternative Hypothesis Highlight Calculate and press ENTER

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HAWKES LEARNING SYSTEMS math courseware specialists Example, continued: Hypothesis Testing 10.2 Hypothesis Testing for Means (Small Samples).... Use 0.10. Verify that it did the Test you wanted and that it has the correct Alternative Hypothesis. Verify that the sample data is correct. The t= is the Test Statistic. It comes from the same formula as the one weve been using. The p = is the p-value. It is the area to the right of that t value (in the case of this right- tailed test.) It is the probability of getting a t value as extreme as the t value we got. When using the calculators T-Test, we use the p-value method. You dont need a t critical value. Instead, you compare your p-value to the α (alpha) level of significance. If your p < α(alpha), then the decision is Reject H 0.

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HAWKES LEARNING SYSTEMS math courseware specialists The other example, done with TI-84 T-Test and the p-value method: Hypothesis Testing 10.2 Hypothesis Testing for Means (Small Samples) A hometown department store has chosen its marketing strategies for many years under the assumption that the average shopper spends no more than $100 in their store. A newly hired store manager claims that the current average is higher, and wants to change their marketing scheme accordingly. A group of 24 shoppers is chosen at random and found to have spent on average $104.93 with a standard deviation of $9.07. Test the store managers claim at the 0.010 level of significance. H0:H0: Ha:Ha: 100 > 100 Compare your p-value p=.0069501788 to alpha: α=0.010 and make the decision: Should we reject H 0 ?

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