Hypothesis Testing. April 21, 2009

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1 Hypothesis Testing April 21, 2009

2 Your Claim is Just a Hypothesis I ve never made a mistake. Once I thought I did, but I was wrong.

3 Your Claim is Just a Hypothesis Confidence intervals quantify how sure we are about a sample statistic. When should a sample statistic provide reason to reject a previously made claim?

4 Examples of Claims Students drink an average of one alcoholic drink when they go out for the night. Senator Obama s tax plan will raise taxes on 56.8% of small business income, Senator Obama s tax plan will raise taxes on only 1.4% of small businesses. Source: smallbusiness/small_biz_taxes_factcheck.smb/

5 Contradictory Hypotheses: The Drinking Claim Claim: Students drink an average of one alcoholic drink when they go out for the night. X = the number of alcoholic drinks a student drinks H 0 : Null Hypthesis: µ 1 H a : Alternative Hypothesis: µ > 1

6 Contradictory Hypotheses: Obama Opponents Claim Claim: Senator Obama s tax plan will raise taxes on 56.8% of small business income. p = proportion of small business income that will pay a higher rate under Obama s plan. H 0 : Null Hypothesis: p.568 H a : Alternative Hypothesis: p <.568

7 Contradictory Hypotheses: Obama Supporters Claim Senator Obama s tax plan will raise taxes on only 1.4% of small businesses. p = the proportion of small businesses that will pay a higher rate under Obama s plan. H 0 : Null Hypthesis: p.014 H a : Alternative Hypothesis: p >.014

8 Types of Results Action H 0 True H 0 False Do not reject H 0 Correct Outcome Type II Error Reject H 0 Type I Error Correct Outcome α = probability of type I error. β = probability of type II error.

9 Procedural Outline 1. Set up two contradictory hypotheses. 2. Collect sample data (in homework problems, the data or summary statistics will be given to you). 3. Determine the correct distribution to perform the hypothesis test. 4. Analyze sample data by performing the calculations that ultimately will support one of the hypotheses. 5. Make a decision and write a meaningful conclusion.

10 Obama s Supporters Claim Revisited Suppose you sample twenty business at random and find that one of these businesses will have their taxes raised under Obama s tax plan. Conduct a hypothesis test to determine if the proportion of business that will have raised taxes in your population is statistically greater at significance α =.05 from the claimed proportion of

11 Obama s Supporters Example Solved via a 1PropZ-Test on a TI-83+ or TI-84 Press STAT and arrow over to TESTS. Press 5:1-PropZTest. Enter for p 0 and press ENTER. Enter 1 for x and press ENTER. Enter 20 for n and press ENTER. Arrow over to > p 0 under the Prop section. Press ENTER. Arrow down to Calculate and press ENTER. The calculator calculates the p value ( p = ) and the test statistic, i.e., the z score (z = 1.371). Prop >.014 is the alternate hypothesis. Do this set of instructions again except arrow to Draw (instead of Calculate). Press ENTER. A shaded graph appears with z = (test statistic) and p = (p value). Make sure when you use Draw that no other equations are highlighted in Y = and the plots are turned off.

12 Conclusion for Obama s Supporters Example Comparing our p value p = to the preconceived significance of α = 0.05 we have p > α, so we can not reject the claim.

13 Example: Known Standard Deviation, Two Tailed The cost of a daily newspaper varies from city to city. However, the variation among prices remains steady with a standard deviation of 6 cents. A study was done to test the claim that the average cost of a daily newspaper is 35 cents. Twelve costs yield an average cost of 30 cents with a standard deviation of 4 cents. Do the data support the claim at the 1% level?

14 Previous example solved via a Z-Test on a TI-83+ or a TI-84 The p-value can easily be calculated using the TI-83+ and the TI-84 calculators: Press STAT and arrow over to TESTS. Press 1:Z-Test. Arrow over to Stats and press ENTER. Arrow down and enter 35 for µ 0 (null hypothesis), 6 for σ, 30 for the sample mean, and 12 for n. Arrow down to µ: (alternate hypothesis) and arrow over to µ 0. Press ENTER. Arrow down to Calculate and press ENTER. The calculator not only calculates the p-value ( p = ) but it also calculates the test statistic, i.e., the z-score (z = 2.89) for the sample mean. µ < 30 is the alternate hypothesis. Do this set of instructions again except arrow to Draw (instead of Calculate). Press ENTER. A shaded graph appears with z = 2.89 (test statistic) and p = (p value). Make sure when you use Draw that no other equations are highlighted in Y = and the plots are turned off.

15 Conclusion for the Newspaper Price Example Comparing our p value p = to the preconceived significance of α = 0.01 we have p < α, so we reject the claim.

16 Example: Unknown Standard Deviation, Single Tailed An article in the San Jose Mercury News stated that students in the California state university system take an average of 4.5 years to finish their undergraduate degrees. Suppose you believe that the average time is longer. You conduct a survey of 49 students and obtain a sample mean of 5.1 with a sample standard deviation of 1.2. Do the data support your claim at the 1% level?

17 Previous example solved via a T-Test on a TI-83+ or a TI-84 Press STAT and arrow over to TESTS. Press 2:T-Test. Arrow over to Stats and press ENTER. Arrow down and enter 4.5 for µ 0. Arrow down and enter 5.1 for x and press ENTER. Arrow down and enter 1.2 for Sx and press ENTER. Arrow down and enter 49 for n and press ENTER. Arrow down to µ and arrow over to > µ 0. Press ENTER. Arrow down to Calculate and press ENTER. The calculator not only calculates the p value ( p = ) but it also calculates the test statistic, i.e., the t score (t = 3.5) for the sample mean, the sample mean, and the sample standard deviation. µ > 4.5 is the alternate hypothesis. Do this set of instructions again except arrow to Draw (instead of Calculate). Press ENTER. A shaded graph appears with t = 3.5 (test statistic) and p = (p value). Make sure when you use Draw that no other equations are highlighted in Y = and the plots are turned off.

18 Conclusion for the Time Till Degree Example Comparing our p value p = to the preconceived significance of α = 0.01 we have p < α, so we reject the claim.

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