Statistical Hypothesis Testing q A statistical hypothesis is

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Statistical Hypothesis Testing q A statistical hypothesis is an assertion concerning one or more

Statistical Hypothesis Testing q A statistical hypothesis is an assertion concerning one or more populations. q In statistics, a hypothesis test is conducted on a set of two mutually exclusive statements: H 0 : null hypothesis H 1 : alternate hypothesis q Example H 0 : μ = 17 H 1 : μ ≠ 17 q We sometimes refer to the null hypothesis as the “equals” hypothesis. JMB Ch 10 Lecture 1 9 th ed. v Oct 2013 EGR 252 2013 Slide 1

Tests of Hypotheses - Graphics I q We can make a decision about our

Tests of Hypotheses - Graphics I q We can make a decision about our hypotheses based on our understanding of probability. q We can visualize this probability by defining a rejection region on the probability curve. q The general location of the rejection region is determined by the alternate hypothesis. H 0 : μ = _____ H 1 : μ < _____ One-sided JMB Ch 10 Lecture 1 9 th ed. v Oct 2013 H 0 : μ = _____ H 1 : μ ≠ _____ H 0 : p = _____ H 1 : p > _____ One-sided Two-sided EGR 252 2013 Slide 2

Choosing the Hypotheses q Your turn … Suppose a coffee vending machine claims it

Choosing the Hypotheses q 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? H 0 : μ = _____ H 1 : μ ≠ _____ H 0 : μ = _____ H 1 : μ < _____ JMB Ch 10 Lecture 1 9 th ed. v Oct 2013 EGR 252 2013 Slide 3

Potential errors in decision-making q α Ø Probability of committing a Type I error

Potential errors in decision-making q α Ø Probability of committing a Type I error Ø Probability of rejecting the null hypothesis given that the null hypothesis is true Ø P (reject H 0 | H 0 is true) JMB Ch 10 Lecture 1 9 th ed. v Oct 2013 qβ Ø Probability of committing a Type II error Ø Power of the test = 1 - β (probability of rejecting the null hypothesis given that the alternate is true. ) Ø Power = P (reject H 0 | H 1 is true) EGR 252 2013 Slide 4

Hypothesis Testing – Approach 1 q Approach 1 - Fixed probability of Type 1

Hypothesis Testing – Approach 1 q 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. JMB Ch 10 Lecture 1 9 th ed. v Oct 2013 EGR 252 2013 Slide 5

Hypothesis Testing – Approach 2 Significance testing based on the calculated P-value 1. State

Hypothesis Testing – Approach 2 Significance testing based on the calculated P-value 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 H 0, based on the P-value. 5. Make an engineering or scientific conclusion. Three potential test results (actual test results should include only one arrow per plot): p = 0. 02 P-value 0 ↓ p = 0. 45 0. 25 JMB Ch 10 Lecture 1 9 th ed. v Oct 2013 ↓ 0. 50 p = 0. 85 ↓ 0. 75 EGR 252 2013 1. 00 P-value Slide 6

Hypothesis Testing Tells Us … q Strong conclusion: Ø If our calculated t-value is

Hypothesis Testing Tells Us … q Strong conclusion: Ø If our calculated t-value is “outside” tα, 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. q Weak conclusion: Ø If our calculated t-value is “inside” tα, ν (approach 1) or we have a “large” p-value (approach 2), then we cannot reject H 0: μ = μ 0. q 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. JMB Ch 10 Lecture 1 9 th ed. v Oct 2013 EGR 252 2013 Slide 7

Example: Single Sample Test of the Mean P -value Approach A sample of 20

Example: Single Sample Test of the Mean P -value Approach 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 JMB Ch 10 Lecture 1 9 th ed. v Oct 2013 EGR 252 2013 Slide 8

Single Sample Example (cont. ) Approach 2: = -1. 11842 Find probability from chart

Single Sample Example (cont. ) Approach 2: = -1. 11842 Find probability from chart or use Excel’s tdist function. P(x ≤ -1. 118) = TDIST (1. 118, 19, 1) = 0. 139665 p = 0. 14 0_______1 Decision: Fail to reject null hypothesis Conclusion: The mean is not significantly less than 35 mpg. JMB Ch 10 Lecture 1 9 th ed. v Oct 2013 EGR 252 2013 Slide 9

Example (concl. ) Approach 1: Predetermined significance level (alpha) Step 1: Use same hypotheses.

Example (concl. ) Approach 1: Predetermined significance level (alpha) Step 1: Use same 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, ” tcrit = -1. 729 Step 4: tcalc = -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 is 35 mpg. ****** JMB Ch 10 Lecture 1 9 th ed. v Oct 2013 EGR 252 2013 Slide 10