1. Hypothesis Testing Chapter 10 MDH Ch10 Lecture 1 9th ed. v Spring 2015 EGR 252 2015 Slide 1

2. Hypothesis Testing Basics  A hypothesis is a statistical assertion concerning one or more populations.  Null hypothesis: A hypothesis to be tested. We use the symbol H 0 to represent the null hypothesis  Alternative hypothesis: A hypothesis to be considered as an alternative to the null hypothesis. We use the symbol H 1 to represent the alternative hypothesis.  - The alternative hypothesis is the one believed to be true, or what you are trying to prove is true. MDH Ch10 Lecture 1 9th ed. v Spring 2015 EGR 252 2015 Slide 2

3. Practical Significance vs. Statistical Significance  When no practical difference exist, it may be possible to detect a statistically significant difference  Hypothesis tests are performed to determine if a claim has significant statistical merit  Although a hypothesis claims may be found statistically significant, the effort or expense to implement any changes may not be worth it.  For example if a study showed that a budget helps people save an extra $10 per year, a budget that only saves $10 extra per year does not have any practical significance. MDH Ch10 Lecture 1 9th ed. v Spring 2015 EGR 252 2015 Slide 3

4. MDH Ch10 Lecture 1 9th ed. v Spring 2015 EGR 252 2015 Slide 4 Statistical Hypothesis Testing  In statistics, a hypothesis test is conducted on a set of two mutually exclusive statements: H 0 : null hypothesis H 1 : alternate hypothesis  Example H 0 : μ = 17 H 1 : μ ≠ 17  We sometimes refer to the null hypothesis as the “equals” hypothesis.

5. Hypothesis Testing Basics  Null hypothesis must be accepted (fail to reject) or rejected  Test Statistic: A value which functions as the decision maker. The decision to “reject” or “fail to reject” is based on information contained in a sample drawn from the population of interest.  Rejection region: If test statistic falls in some interval which support alternative hypothesis, we reject the null hypothesis.  Acceptance Region: It test statistic falls in some interval which support null hypothesis, we fail to reject the null hypothesis.  Critical Value: The point which divide the rejection region and acceptance MDH Ch10 Lecture 1 9th ed. v Spring 2015 EGR 252 2015 Slide 5

6. Hypothesis Testing Basics  Test statistic; n is large, standard deviation is known  Test statistic: n is small, standard deviation is unknown MDH Ch10 Lecture 1 9th ed. v Spring 2015 EGR 252 2015 Slide 6 Z-statistic T-statistic

7. MDH Ch10 Lecture 1 9th ed. v Spring 2015 EGR 252 2015 Slide 7 Tests of Hypotheses - Graphics I  We can make a decision about our hypotheses based on our understanding of probability.  We can visualize this probability by defining a rejection region on the probability curve.  The general location of the rejection region is determined by the alternate hypothesis. H0 : μ = _____ H1 : μ < _____ H0 : μ = _____ H1 : μ ≠ _____ H0 : p = _____ H1 : p > _____ One-sided Two-sided One-sided

8. MDH Ch10 Lecture 1 9th ed. v Spring 2015 EGR 252 2015 Slide 8 Choosing the Hypotheses  Your turn … Suppose a coffee vending machine claims it dispenses an 8-oz cup of coffee. You have been using the machine for 6 months, but recently it seems the cup isn’t as full as it used to be. You plan to conduct a statistical hypothesis test. What are your hypotheses? H0 : μ = _____ H1 : μ ≠ _____ H0 : μ = _____ H1 : μ < _____

9. MDH Ch10 Lecture 1 9th ed. v Spring 2015 EGR 252 2015 Slide 9 Potential errors in decision-making  α  Probability of committing a Type I error (incorrect rejection of a true null hypothesis)  Probability of rejecting the null hypothesis given that the null hypothesis is true  P (reject H 0 | H 0 is true)  β  Probability of committing a Type II error. (failure to reject a false null hypothesis)  Power of the test = 1 - β (probability of rejecting the null hypothesis given that the alternate is true.)  Power = P (reject H 0 | H 1 is true)

10. MDH Ch10 Lecture 1 9th ed. v Spring 2015 EGR 252 2015 Slide 10 Hypothesis Testing – Approach 1  Approach 1 - Fixed probability of Type 1 error. 1.State the null and alternative hypotheses. 2.Choose a fixed significance level α. 3.Specify the appropriate test statistic and establish the critical region based on α. Draw a graphic representation. 4.Calculate the value of the test statistic based on the sample data. 5.Make a decision to reject or fail to reject H 0, based on the location of the test statistic. 6.Make an engineering or scientific conclusion.

11. MDH Ch10 Lecture 1 9th ed. v Spring 2015 EGR 252 2015 Slide 11 Hypothesis Testing – Approach 2 p-value is a measure of the significance of your results 1.State the null and alternative hypotheses. 2.Choose an appropriate test statistic. 3.Calculate value of test statistic and determine p- value. Draw a graphic representation. 4.Make a decision to reject or fail to reject H 0, based on the p-value by comparing it to , the level of significance. ***If not given assume  = 0.05 5.Make an engineering or scientific conclusion. p-value <  Reject Null“p-value is low, null must go” p-value >  Fail to Reject Null (Accept) “p-value is high, null must fly”

12. MDH Ch10 Lecture 1 9th ed. v Spring 2015 EGR 252 2015 Slide 12 Hypothesis Testing Tells Us …  Strong conclusion:  If our calculated t-value is “outside” t α,ν (approach 1) or we have a small p-value (approach 2), then we reject H 0 : μ = μ 0 in favor of the alternate hypothesis.  Weak conclusion:  If our calculated t-value is “inside” t α,ν (approach 1) or we have a “large” p-value (approach 2), then we cannot reject H 0 : μ = μ 0.  Failure to reject H 0 does not imply that μ is equal to the stated value (μ 0 ), only that we do not have sufficient evidence to support H 1.

13. Types of Tests Non-directional, two-tail test:  H 0 : parameter = value  H 1 : parameter ≠ value Directional, right-tail test:  H 0 : parameter  value  H 1 : parameter > value Directional, left-tail test:  H 0 : parameter  value  H 1 : parameter < value MDH Ch10 Lecture 1 9th ed. v Spring 2015 EGR 252 2015 Slide 13

14. Non-directional - Two-tail Test Reject Region MDH Ch10 Lecture 1 9th ed. v Spring 2015 EGR 252 2015 Slide 14

15. Directional-Right-tail Test Reject Region MDH Ch10 Lecture 1 9th ed. v Spring 2015 EGR 252 2015 Slide 15

16. Directional-Left-tail Test Reject Region MDH Ch10 Lecture 1 9th ed. v Spring 2015 EGR 252 2015 Slide 16

17. MDH Ch10 Lecture 1 9th ed. v Spring 2015 EGR 252 2015 Slide 17 Example: Single Sample Test of the Mean A sample of 20 cars driven under varying highway conditions achieved fuel efficiencies as follows: Sample mean x = 34.271 mpg Sample std dev s = 2.915 mpg Test the hypothesis that the population mean equals 35.0 mpg vs. μ < 35. Step 1: State the hypotheses. H 0 : μ = 35 H 1 : μ < 35 Step 2: Determine the appropriate test statistic. σ unknown, n = 20 Therefore, use t distribution

18. MDH Ch10 Lecture 1 9th ed. v Spring 2015 EGR 252 2015 Slide 18 Example (concl.) Approach 1: significance level (alpha) Step 1: State hypotheses. Step 2: Let’s set alpha at 0.05. Step 3: Determine the critical value of t that separates the reject H 0 region from the do not reject H 0 region. t , n-1 = t 0.05,19 = 1.729 Since H 1 format is “μ< μ 0,” t crit = -1.729 Step 4: t calc = -1.11842 Step 5: Decision Fail to reject H 0 Step 6: Conclusion: The population mean is not significantly less than 35 mpg. ****Do not conclude that the population mean equals 35 mpg.****

19. MDH Ch10 Lecture 1 9th ed. v Spring 2015 EGR 252 2015 Slide 19 Single Sample Example (cont.) Approach 2 p-value approach: = -1.11842 Find probability from chart or use Excel’s tdist function. P(x ≤ -1.118) = TDIST (1.118, 19, 1) = 0.139665 0______________________________1 Decision: Fail to reject null hypothesis (Accept) Conclusion: The mean is not significantly less than 35 mpg. P-value = 0.14  = 0.05  < p-value