1. Section 10.3 Hypothesis Testing for Means (Large Samples) HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2008 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved.

2. HAWKES LEARNING SYSTEMS math courseware specialists Test Statistic for Large Samples, n ≥ 30: Hypothesis Testing 10.3 Hypothesis Testing for Means (Large Samples) To determine if the test statistic calculated from the sample is statistically significant we will need to look at the critical value. cOne-Tailed TestTwo-Tailed Test 0.901.28±1.645 0.951.645±1.96 0.982.05±2.33 0.992.33±2.575

3. HAWKES LEARNING SYSTEMS math courseware specialists Draw a conclusion: A research company reports that in 2006 the average American woman is 25 years of age at her first marriage. A researcher claims that for women in California, this estimate is too low. A survey of 213 newlywed women in California gave a mean of 25.4 years with a standard deviation of 2.3 years. Using a 95% level of confidence, determine if the data supports the researcher’s claim. Solution: First state the hypotheses: H0:H0: Ha:Ha: Next, set up the hypothesis test and determine the critical value: c  0.95 zc zc  Reject if z ≥ z c, or if z ≥ 1.645.  ≤ 25  > 25 1.645 Hypothesis Testing 10.3 Hypothesis Testing for Means (Large Samples)

4. HAWKES LEARNING SYSTEMS math courseware specialists Solution (continued): Gather the data and calculate the necessary sample statistics: n  213,   25,  25.4, s  2.3, Finally, draw a conclusion: Since z is greater than z c, we will reject the null hypothesis. 2.54 Hypothesis Testing 10.3 Hypothesis Testing for Means (Large Samples)

5. HAWKES LEARNING SYSTEMS math courseware specialists p-Values: A p-value is the probability of obtaining a sample more extreme than the one observed on your data, when H 0 is assumed to be true. To find the p-value, first calculate the z-score from the sample data and then find the corresponding probability for that z-score. Hypothesis Testing 10.3 Hypothesis Testing for Means (Large Samples)

6. HAWKES LEARNING SYSTEMS math courseware specialists Calculate the p-value: Calculate the p-value for a hypothesis test with the following hypotheses. Assume that data has been collected and the test statistic was calculated to be z  –1.34. H 0 :  ≥ 0.15 H a :  < 0.15 Hypothesis Testing 10.3 Hypothesis Testing for Means (Large Samples) Solution: The alternative hypothesis tells us that this is a left-tailed test. Therefore, the p-value for this situation is the probability that z is less than –1.34. p  0.0901

7. HAWKES LEARNING SYSTEMS math courseware specialists Calculate the p-value: Calculate the p-value for a hypothesis test with the following hypotheses. Assume that data have been collected and the test statistic was calculated to be z  2.78. H 0 :  ≤ 0.43 H a :  > 0.43 Hypothesis Testing 10.3 Hypothesis Testing for Means (Large Samples) Solution: The alternative hypothesis tells us that this is a right-tailed test. Therefore, the p-value for this situation is the probability that z is greater than 2.78. p  0.0027

8. HAWKES LEARNING SYSTEMS math courseware specialists Calculate the p-value: Calculate the p-value for a hypothesis test with the following hypotheses. Assume that data have been collected and the test statistic was calculated to be z  –2.15. H 0 :  = 0.78 H a :  ≠ 0.78 Hypothesis Testing 10.3 Hypothesis Testing for Means (Large Samples) Solution: The alternative hypothesis tells us that this is a two-tailed test. Therefore, the p-value for this situation is the probability that z is either less than –2.15 or greater than 2.15. p  0.0158 (2)  0.0316

9. HAWKES LEARNING SYSTEMS math courseware specialists Conclusions for a Hypothesis Testing Using p-Values: 1.If p ≤ , then reject the null hypothesis. 2.If p > , then fail to reject the null hypothesis. Hypothesis Testing 10.3 Hypothesis Testing for Means (Large Samples)

10. HAWKES LEARNING SYSTEMS math courseware specialists Draw a conclusion: Calculate the p-value for the following data and draw a conclusion based on the given value of alpha. a.Left-tailed test with z  –1.34 and  0.05. p  0.0901 which is greater than 0.05 so we fail to reject the null hypothesis. b.Two-tailed test with z  –2.15 and  0.10. p  0.0158(2)  0.0316 which is less than 0.10 so we reject the null hypothesis. Hypothesis Testing 10.3 Hypothesis Testing for Means (Large Samples)

11. HAWKES LEARNING SYSTEMS math courseware specialists Steps for Using p-Values in Hypothesis Testing: 1.State the null and alternative hypotheses. 2.Set up the hypothesis test by choosing the test statistic and stating the level of significance. 3.Gather data and calculate the necessary sample statistics. 4.Draw a conclusion by comparing the p-value to the level of significance. Hypothesis Testing 10.3 Hypothesis Testing for Means (Large Samples)

12. HAWKES LEARNING SYSTEMS math courseware specialists Draw a conclusion: A recent study showed that the average number of children for women in Europe is 1.48. A global watch group claims that German women have an average fertility rate which is different from the rest of Europe. To test their claim, they surveyed 128 German women and found that had an average fertility rate of 1.39 children with a standard deviation of 0.84. Does this data support the claim made by the global watch group at the 90% level of confidence? Solution: First state the hypotheses: H0:H0: Ha:Ha: Next, set up the hypothesis test and state the level of significance: c  0.90   0.10 Reject if p < , or if p < 0.10.  = 1.48  ≠ 1.48 Hypothesis Testing 10.3 Hypothesis Testing for Means (Large Samples)

13. HAWKES LEARNING SYSTEMS math courseware specialists Solution (continued): Gather the data and calculate the necessary sample statistics: n  128,   1.48,  1.39, s  0.84, Since this is a two-tailed test, p  0.1131(2)  0.2262. Finally, draw a conclusion: Since p is greater than , we will fail to reject the null hypothesis. The evidence does not support the watch group’s claim at the 90% level of confidence. –1.21 Hypothesis Testing 10.3 Hypothesis Testing for Means (Large Samples)