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Hypothesis Testing Chapter 10 MDH Ch10 Lecture 1 9th ed. v Spring 2015 EGR 252 2015 Slide 1 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 H0 to represent the null hypothesis Alternative hypothesis: A hypothesis to be considered as an alternative to the null hypothesis. We use the symbol H1 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 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 Statistical Hypothesis Testing In statistics, a hypothesis test is conducted on a set of two mutually exclusive statements: H0 : null hypothesis H1 : alternate hypothesis Example H0 : μ = 17 H1 : μ ≠ 17 We sometimes refer to the null hypothesis as the “equals” hypothesis. MDH Ch10 Lecture 1 9th ed. v Spring 2015 EGR 252 2015 Slide 4 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 Hypothesis Testing Basics Test statistic; n is large, standard deviation is known Z-statistic Test statistic: n is small, standard deviation is unknown T-statistic MDH Ch10 Lecture 1 9th ed. v Spring 2015 EGR 252 2015 Slide 6 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 < _____ One-sided MDH Ch10 Lecture 1 9th ed. v Spring 2015 H0 : μ = _____ H1 : μ ≠ _____ H0 : p = _____ H1 : p > _____ One-sided Two-sided EGR 252 2015 Slide 7 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 < _____ MDH Ch10 Lecture 1 9th ed. v Spring 2015 EGR 252 2015 Slide 8 Potential errors in decision-making H0 True H0 False Do not reject H0 Correct Decision Type II error Reject H0 Type I error Correct Decision 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 H0 | H0 is true) MDH Ch10 Lecture 1 9th ed. v Spring 2015 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 H0 | H1 is true) EGR 252 2015 Slide 9 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 H0, based on the location of the test statistic. 6. Make an engineering or scientific conclusion. MDH Ch10 Lecture 1 9th ed. v Spring 2015 EGR 252 2015 Slide 10 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 pvalue. Draw a graphic representation. 4. Make a decision to reject or fail to reject H0, based on the p-value by comparing it to a, the level of significance . ***If not given assume a = 0.05 5. Make an engineering or scientific conclusion. p-value < a Reject Null “p-value is low, null must go” p-value > a Fail to Reject Null (Accept) “p-value is high, null must fly” MDH Ch10 Lecture 1 9th ed. v Spring 2015 EGR 252 2015 Slide 11 Hypothesis Testing Tells Us … Strong conclusion: If our calculated t-value is “outside” tα,ν (approach 1) or we have a small p-value (approach 2), then we reject H0: μ = μ0 in favor of the alternate hypothesis. Weak conclusion: If our calculated t-value is “inside” tα,ν (approach 1) or we have a “large” p-value (approach 2), then we cannot reject H0: μ = μ0. Failure to reject H0 does not imply that μ is equal to the stated value (μ0), only that we do not have sufficient evidence to support H1. MDH Ch10 Lecture 1 9th ed. v Spring 2015 EGR 252 2015 Slide 12 Types of Tests • Non-directional, two-tail test: H0: parameter = value H1: parameter ≠ value • Directional, right-tail test: H0: parameter value H1: parameter > value • Directional, left-tail test: H0: parameter value H1: parameter < value MDH Ch10 Lecture 1 9th ed. v Spring 2015 EGR 252 2015 Slide 13 Non-directional - Two-tail Test Reject Region Reject Region Do Not Reject H Reject H a/2 0 0 Reject H 1-a –z MDH Ch10 Lecture 1 9th ed. v Spring 2015 a/2 0 +z EGR 252 2015 Slide 14 Directional-Right-tail Test Reject Region Do Not Reject H 0 Reject H 1-a 0 a +z MDH Ch10 Lecture 1 9th ed. v Spring 2015 EGR 252 2015 Slide 15 Directional-Left-tail Test Reject Region Reject H Do Not Reject H 0 1-a a 0 –z MDH Ch10 Lecture 1 9th ed. v Spring 2015 EGR 252 2015 Slide 16 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. H0: μ = 35 H1: μ < 35 Step 2: Determine the appropriate test statistic. σ unknown, n = 20 Therefore, use t distribution MDH Ch10 Lecture 1 9th ed. v Spring 2015 EGR 252 2015 Slide 17 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 H0 region from the do not reject H0 region. ta, n-1 = t0.05,19 = 1.729 T X - S/ n Since H1 format is “μ< μ0,” tcrit = -1.729 Step 4: tcalc = -1.11842 Step 5: Decision Fail to reject H0 Step 6: Conclusion: The population mean is not significantly less than 35 mpg. ****Do not conclude that the population mean equals 35 mpg.**** MDH Ch10 Lecture 1 9th ed. v Spring 2015 EGR 252 2015 Slide 18 Single Sample Example (cont.) Approach 2 p-value approach: X - T S/ n = -1.11842 Find probability from chart or use Excel’s tdist function. P(x ≤ -1.118) = TDIST (1.118, 19, 1) = 0.139665 a = 0.05 a < p-value P-value = 0.14 0______________________________1 Decision: Fail to reject null hypothesis (Accept) Conclusion: The mean is not significantly less than 35 mpg. MDH Ch10 Lecture 1 9th ed. v Spring 2015 EGR 252 2015 Slide 19