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

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

HAWKES LEARNING SYSTEMS math courseware specialists Hypothesis Testing 10.2 Hypothesis Testing for Means (Small Samples) Find the critical value: 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 a = 0.025 t0.025 = 2.145 - The higher the confidence level, The smaller the level of significance, The bigger the critical-t score

Determined by two things: HAWKES LEARNING SYSTEMS math courseware specialists Hypothesis Testing 10.2 Hypothesis Testing for Means (Small Samples) Rejection Regions: Determined by two things: The type of hypothesis test. (left, right or two-tailed) The level of significance, a. (e.g. 0.01, 0.05, 0.10) Finding a Rejection Region: Look up the critical value, tc, to determine the cutoff for the rejection region. (e.g. t0.025 = 2.145) If the test statistic you calculate from the sample data falls in the a area, then reject H0. (e.g. If t is greater than 2.145, then reject H0).

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

HAWKES LEARNING SYSTEMS math courseware specialists Hypothesis Testing 10.2 Hypothesis Testing for Means (Small Samples) Rejection Regions for Left-Tailed Tests, Ha contains <: As we become more confident  alpha is smaller (e.g. 0.1 to 0.05), critical-t gets smaller, the less chance to reject the null hypothesis. Reject if t ≤ –t

HAWKES LEARNING SYSTEMS math courseware specialists Hypothesis Testing 10.2 Hypothesis Testing for Means (Small Samples) Rejection Regions for Right-Tailed Tests, Ha contains >: As we become more confident  alpha is smaller (e.g. 0.1 to 0.05), critical-t gets larger, the less chance to reject the null hypothesis. Reject if t ≥ t

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

State the null and alternative hypotheses. HAWKES LEARNING SYSTEMS math courseware specialists Hypothesis Testing 10.2 Hypothesis Testing for Means (Small Samples) Steps for Hypothesis Testing: State the null and alternative hypotheses. 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. Gather data and calculate the necessary sample statistics. Draw a conclusion.

Since t is greater than ta , we will reject the null hypothesis. HAWKES LEARNING SYSTEMS math courseware specialists Hypothesis Testing 10.2 Hypothesis Testing for Means (Small Samples) Draw a conclusion: 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 a = 0.10. Solution: n = 27,  = 9, = 9.8, s = 1.5, d.f. = 26, a = 0.10 t0.10 = Since t is greater than ta , we will reject the null hypothesis. H0: Ha: m > 9 m ≤ 9 Current accepted belief Testing hypothesis 1.315 2.771  2.771 > 1.315

First state the hypotheses: H0: Ha: HAWKES LEARNING SYSTEMS math courseware specialists Hypothesis Testing 10.2 Hypothesis Testing for Means (Small Samples) Draw a conclusion: 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 manager’s claim at the 0.010 level of significance. Solution: First state the hypotheses: H0: Ha: Next, set up the hypothesis test and determine the critical value: d.f. = 23, a = 0.010 t0.010 = Reject if t ≥ t , or if t > 2.500. m ≤ 100 m > 100 2.500

Gather the data and calculate the necessary sample statistics: 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, Reject if t ≥ t , or if t > 2.500. Finally, draw a conclusion: 2.663 > 2.500 Since t is greater than ta , we will reject the null hypothesis. 2.663

H0: Ha: m < 423 m ≥ 423 Current accepted belief Testing hypothesis d.f. = 13 Alpha = 0.01 T-value = -2.6503 (left tail)

H0: Ha: m > 5.8 m ≤ 5.8 Current accepted belief Testing hypothesis d.f. = 11 Alpha = 0.05 T-value = 1.7959 (right tail)

Current accepted belief H0: Ha: m < 447 m ≥ 447 Current accepted belief Testing hypothesis Reject if t < t , or if t < -2.0518 Computed-T is -0.7299 and it is Not < -2.0518. It is > -2.0518. Fail to Reject the Null Hypothesis. Not enough evidence.

Current accepted belief H0: Ha: m > 7.9 m ≤ 7.9 Current accepted belief Testing hypothesis Reject if t > t , or if t > 1.3195 Computed-T is 1.9596 and it is > 1.3195. Reject the Null Hypothesis. Enough evidence that performs above specs.