EGR Ch. 101 Tests of Hypotheses (Statistical) Hypothesis: an assertion concerning one or more populations. In statistics, there are only two states of the world: H 0 : “equals” (null hypothesis) H 1 : _______(alternate hypothesis) Examples: H 0 : μ = 17 H 1 : μ ≠ 17 H 0 : μ = 8 H 1 : μ > 8 H 0 : p = 0.5 H 1 : p < 0.5

EGR Ch. 102 Choosing a Hypothesis 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 test of your hypothesis. What are your hypotheses?

EGR Ch. 103 Hypothesis testing Level of significance, α –Probability of committing a Type I error = P (rejecting H 0 | H 0 is true) β –Probability of committing a Type II error –Power of the test = ________ (probability of rejecting the null hypothesis given that the alternate is true.)

EGR Ch. 104 Determining α & β Example: Proportion of adults in a small town who are college graduates is estimated to be p = 0.6. A random sample of 15 adults is selected to test this hypothesis. If we find that between 6 and 12 adults are college graduates, we will accept H 0 : p = 0.6; otherwise we will reject the hypothesis and conclude the proportion is something different (for this example, use H 1 : p = 0.5). α = ________________ β = ________________

EGR Ch. 105 Hypothesis Testing Approach 1 - Fixed probability of Type I 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.Compute 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.Draw an engineering or scientific conclusion.

EGR Ch. 106 Hypothesis Testing Approach 2 - Significance testing (P-value approach) 1.State the null and alternative hypotheses. 2.Choose an appropriate test statistic. 3.Compute value of test statistic and determine P-value. 4.Draw conclusion based on P- value. P = 0P =

EGR Ch. 107 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. In other words: –Failure to reject H 0 does not imply that μ is equal to the stated value, only that we do not have sufficient evidence to favor H 1.

EGR Ch. 108 Single Sample Test of the Mean A sample of 20 cars driven under varying highway conditions achieved fuel efficiencies as follows: Sample mean x = mpg Sample std dev s = mpg Test the hypothesis that the population mean equals 35.0 mpg vs. μ < 35. H 0 : ________n = ________ H 1 : ________ σ unknown use ___ distribution.

EGR Ch. 109 Example (cont.) Approach 2: = _________________ Using Excel’s tdist function, P(x ≤ ) = _____________ Conclusion: __________________________________

EGR Ch Example (concl.) Approach 1: t 0.05,19 = _____________ Since H 1 specifies “< μ,” t crit = ___________ t calc = _________ Conclusion: _________________________________

EGR Ch Your turn … A sample of 20 cars driven under varying highway conditions achieved fuel efficiencies as follows: Sample mean x = mpg Sample std dev s = mpg Test the hypothesis that the population mean equals 35.0 mpg vs. μ ≠ 35 at an α level of Draw the picture.

EGR Ch Two-Sample Hypothesis Testing Example: A professor has designed an experiment to test the effect of reading the textbook before attempting to complete a homework assignment. Four students who read the textbook before attempting the homework recorded the following times (in hours) to complete the assignment: 3.1, 2.8, 0.5, 1.9 hours Five students who did not read the textbook before attempting the homework recorded the following times to complete the assignment: 0.9, 1.4, 2.1, 5.3, 4.6 hours

EGR Ch Two-Sample Hypothesis Testing Define the difference in the two means as: μ 1 - μ 2 = d 0 What are the Hypotheses? H 0 : _______________ H 1 : _______________ or H 1 : _______________ or H 1 : _______________

EGR Ch Our Example Reading: n 1 = 4x 1 = 2.075s 1 2 = No reading: n 2 = 5x 2 = 2.860s 2 2 = If we assume the population variances are “equal”, we can calculate s p 2 and conduct a __________. = __________________

EGR Ch Your turn … Lower-tail test ((μ 1 - μ 2 < 0) –“Fixed α” approach (“Approach 1”) at α = 0.05 level. – “p-value” approach (“Approach 2”) Upper-tail test (μ 2 – μ 1 > 0) –“Fixed α” approach at α = 0.05 level. – “p-value” approach Two-tailed test (μ 1 - μ 2 ≠ 0) –“Fixed α” approach at α = 0.05 level. – “p-value” approach Recall 

EGR Ch Lower-tail test ((μ 1 - μ 2 < 0) Draw the picture: Solution: Decision: Conclusion:

EGR Ch Upper-tail test (μ 2 – μ 1 > 0) Draw the picture: Solution: Decision: Conclusion:

EGR Ch Two-tailed test (μ 1 - μ 2 ≠ 0) Draw the picture: Solution: Decision: Conclusion:

EGR Ch Another Example Suppose we want to test the difference in carbohydrate content between two “low-carb” meals. Random samples of the two meals are tested in the lab and the carbohydrate content per serving (in grams) is recorded, with the following results: n 1 = 15x 1 = 27.2s 1 2 = 11 n 2 = 10x 2 = 23.9s 2 2 = 23 t calc = ______________________ ν = ________________ (using equation in table 10.2)

EGR Ch Example (cont.) What are our options for hypotheses? At an α level of 0.05, – One-tailed test, t 0.05, 15 = ________ – Two-tailed test, t 0.025, 15 = ________ How are our conclusions affected?

EGR Ch Special Case: Paired Sample T-Test Examples Paired-sample? A.CarRadialBelted 1 ** **Radial, Belted tires 2 ** ** placed on each car. 3 ** ** 4 ** ** B.Person Pre Post 1 ** **Pre- and post-test 2 ** **administered to each 3 ** **person. 4 ** ** C.Student Test1 Test2 1 ** **5 scores from test 1, 2 ** **5 scores from test 2. 3 ** ** 4 ** **

EGR Ch Example* Nine steel plate girders were subjected to two methods for predicting sheer strength. Partial data are as follows: GirderKarlsruheLehighdifference, d Conduct a paired-sample t-test at the 0.05 significance level to determine if there is a difference between the two methods. * adapted from Montgomery & Runger, Applied Statistics and Probability for Engineers.

EGR Ch Example (cont.) Hypotheses: H 0 : μ D = 0 H 1 : μ D ≠ 0 t __________ = ______ Calculate difference scores (d), mean and standard deviation, and t calc … d = s d = t calc = ______________________________

EGR Ch What does this mean? Draw the picture: Decision: Conclusion:

EGR Ch Goodness-of-Fit Tests Procedures for confirming or refuting hypotheses about the distributions of random variables. Hypotheses: H 0 : The population follows a particular distribution. H 1 : The population does not follow the distribution. Examples: H 0 : The data come from a normal distribution. H 1 : The data do not come from a normal distribution.

EGR Ch Goodness of Fit Tests (cont.) Test statistic is χ 2 –Draw the picture –Determine the critical value χ 2 with parameters α, ν = k – 1 Calculate χ 2 from the sample Compare χ 2 calc to χ 2 crit Make a decision about H 0 State your conclusion

EGR Ch Tests of Independence Hypotheses H 0 : independence H 1 : not independent Example Choice of pension plan. 1. Develop a Contingency Table Worker Type Pension Plan Total #1#2#3 Salaried Hourly Total

EGR Ch Example 2. Calculate expected probabilities P(#1 ∩ S) = _______________E(#1 ∩ S) = _____________ P(#1 ∩ H) = _______________E(#1 ∩ H) = _____________ (etc.) Worker Type Pension Plan Total #1#2#3 Salaried Hourly Total #1#2#3 S (exp.) H (exp.)

EGR Ch Hypotheses 3.Define Hypotheses H 0 : the categories (worker & plan) are independent H 1 : the categories are not independent 4. Calculate the sample-based statistic = ________________________________________ = ______

EGR Ch The Test 5. Compare to the critical statistic, χ 2 α, r where r = (a – 1)(b – 1) for our example, say α = 0.01 χ 2 _____ = ___________ Decision: Conclusion:

EGR Ch Homework for Thursday, March 23 3, 6, 7 (pg. 319) (Refer to your updated schedule for future homework assignments.)