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.

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.

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 level of significance. Solution: d.f.  14 and   t  2.145

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.

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

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 

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 

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

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 and determining the values of the test statistic that would lead to rejecting the null hypothesis. 3.Gather data and calculate the necessary sample statistics. 4.Draw a conclusion. Hypothesis Testing 10.2 Hypothesis Testing for Means (Small Samples)

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   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

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 $ with a standard deviation of $9.07. Test the store manager’s claim at the 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,   t  Reject if t ≥ t , or if t >  ≤ 100  >

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,  , s  9.07, Finally, draw a conclusion: Since t is greater than t , we will reject the null hypothesis