# EGR252 F11 Ch 10 9th edition rev2 Slide 1 Statistical Hypothesis Testing Review ï± A...

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EGR252 F11 Ch 10 9th edition rev2 Slide 1 Statistical Hypothesis Testing Review A statistical hypothesis is an assertion concerning one or more populations. 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 : = 17 H 1 : 17 We sometimes refer to the null hypothesis as the equals hypothesis. EGR252 F11 Ch 10 9th edition rev2 Slide 2 Potential errors in decision-making Probability of committing a Type I error Probability of rejecting the null hypothesis given that the null hypothesis is true P (reject H 0 | H 0 is true) Probability of committing a Type II error Power of the test = 1 - (probability of rejecting the null hypothesis given that the alternate is true.) Power = P (reject H 0 | H 1 is true) EGR252 F11 Ch 10 9th edition rev2 Slide 3 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 . 3.Specify the appropriate test statistic and establish the critical region based on . Draw a graphic representation. 4.Calculate the value of the test statistic based on the sample data. 5.Make a decision to reject H 0 or fail to reject H 0, based on the location of the test statistic. 6.Make an engineering or scientific conclusion. EGR252 F11 Ch 10 9th edition rev2 Slide 4 Hypothesis Testing Approach 2 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 P-value. Draw a graphic representation. 4.Make a decision to reject H 0 or fail to reject H 0, based on the P-value. 5.Make an engineering or scientific conclusion. P-value p = 0.05 P-value EGR252 F11 Ch 10 9th edition rev2 Slide 5 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 = mpg Sample std dev s = mpg Test the hypothesis that the population mean equals 35.0 mpg vs. < 35. Step 1: State the hypotheses. H 0 : = 35 H 1 : < 35 Step 2: Determine the appropriate test statistic. unknown, n = 20 Therefore, use t distribution EGR252 F11 Ch 10 9th edition rev2 Slide 6 Single Sample Example (cont.) Approach 2: = Find probability from chart or use Excels tdist function. P(x ) = TDIST (1.118, 19, 1) = p = ______________1 Decision: Fail to reject null hypothesis Conclusion: The mean is not significantly less than 35 mpg. EGR252 F11 Ch 10 9th edition rev2 Slide 7 Example (concl.) Approach 1: Predetermined significance level (alpha) Step 1: Use same hypotheses. Step 2: Lets set alpha at 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 = Since H 1 specifies < we declare t crit = Step 4: Using the equation, we calculate t calc = Step 5: Decision Fail to reject H 0 Step 6: Conclusion: The mean is not significantly less than 35 mpg. EGR252 F11 Ch 10 9th edition rev2 Slide 8 Your turn same data, different hypotheses 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. 35 at an level of Be sure to draw the picture. Step 1 Step 2 Step 3 Step 4 Step 5 Step 6 (Conclusion will be different.) EGR252 F11 Ch 10 9th edition rev2 Slide 9 Two-Sample Hypothesis Testing 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 EGR252 F11 Ch 10 9th edition rev2 Slide 10 Two-Sample Hypothesis Testing Define the difference in the two means as: 1 - 2 = d 0 where d 0 is the actual value of the hypothesized difference What are the Hypotheses? H 0 : _______________ H 1 : _______________ or H 1 : _______________ or H 1 : _______________ EGR252 F11 Ch 10 9th edition rev2 Slide 11 Our Example Using Excel Reading:n 1 = 4mean x 1 = 2.075s 1 2 = No reading:n 2 = 5mean x 2 = 2.860s 2 2 = If we have reason to believe the population variances are equal, we can conduct a t- test assuming equal variances in Minitab or Excel. t-Test: Two-Sample Assuming Equal Variances ReadDoNotRead Mean Variance Observations45 Pooled Variance Hypothesized Mean Difference0 df7 t Stat P(T 0) Conclusions The data do not support the hypothesis that the mean time to complete homework is less for students who read the textbook. or There is no statistically significant difference in the time required to complete the homework for the people who read the text ahead of time vs those who did not. or The data do not support the hypothesis that the mean completion time is less for readers than for non-readers. EGR252 F11 Ch 10 9th edition rev2 Slide 16 Our Example Using Excel Reading:n 1 = 4mean x 1 = 2.075s 1 2 = No reading:n 2 = 5mean x 2 = 2.860s 2 2 = What if we do not have reason to believe the population variances are equal? We can conduct a t- test assuming unequal variances in Minitab or Excel. t-Test: Two-Sample Assuming Equal Variances ReadDoNotRead Mean Variance Observations45 Pooled Variance Hypothesized Mean Difference0 df7 t Stat P(T

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