9.1 Hypothesis Testing


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1 9.1 Hypothesis Testing Define: 1. Null Hypothesis 2. Alternative Hypothesis Null Hypothesis: H 0, statement that the population proportion, or population mean is EQUAL TO a number population proportion population mean Jan 28 9:59 AM Jan 28 10:36 AM Examples of Claims 1 A TV network claims that the average length of time devoted to commercials in a 60 min program is 12 minutes. State the null hypothesis: Alternative hypothesis H 1 or H a a statement to show that the population proportion or population mean is DIFFERENT than the null hypothesis State the claim: Examples 2 Medical researchers claim that the mean body temperature of healthy adults is not equal to 98.6 State the null hypothesis: Proportions Mean State the claim: Jan 28 10:00 AM Jan 28 10:43 AM Example: A study of average shopping times in a large national house ware store gave the following information Women with a female companion: 8.3 min Women with male companion: 4.5 min Suppose you want to set up a statistical test to challenge the claim that a woman with a female friend spends an average of 8.3 min shopping in such a store. a) What is the null and alternative hypotheses if you believe the average time is less than 8.3 minutes? What is type of test is used? b) What is the null and alternative hypotheses if you believe the average shopping time is different from 8.3 min? What type of test is used? Suppose stores installed entertainment centers for the male companions who are shopping with women. c)what would you use for the null and alternative hypotheses if you believe that the average shopping time with a male companion is more than 4.5 min? What type of test is used? Feb 2 2:07 PM Feb 2 2:11 PM 1
2 d)what would you use for the null hypotheses if you believe that the average shopping time with a male companion is the store is different from 4.5 min? What type of test is used? Types of Tests: Determined by the Alternative Hypothesis. Left tailed test: If the alternative hypothesis, H 1, is stating that the parameter is less than the value claimed in the null hypothesis,h 0. Right tailed test:if the alternative hypothesis, H 1, is stating that the parameter is greater than the value claimed in the null hypothesis,h 0. Two tailed test:if the alternative hypothesis, H 1, is stating that the parameter is different than the value claimed in the null hypothesis,h 0. Feb 2 2:17 PM Feb 2 1:58 PM homework BRASE pg 411 #2,5,7,8 Day 2 Hypothesis testing Review: State the null and alternative hypotheses and identify the claim. a) A school publicizes that the proportion of its students who are involved in at least one extracurricular activity is 61% b) A car dealership announces that the mean time for an oil change is less than 15 minutes c) A company advertises that the mean life of its furnances is more than 18 years. Feb 2 2:18 PM Feb 9 8:16 AM Types of Errors and Significance Level When you perform a hypothesis test, your ultimate goal is to make one of two decisions: 1) reject the null hypothesis 2) fail to reject the null hypothesis Type I and Type II Errors Truth of H 0 If H 0 is true we do not reject H0 we do reject H0 Correct decision Type I error If H 0 is false Type II error Correct decision Since your decision to reject H 0 or fail to reject H 0 is based on a sample, you must accept the fact that your decision might be incorrect. You might reject a null hypothesis when it is actually true, or you might fail to reject a null hypothesis when it is actually false. Feb 9 8:18 AM Feb 2 3:37 PM 2
3 Significance Level is the probability of rejecting the null hypothesis when it is actually TRUE (Type I error) β The probability of failing to reject the null hypothesis when it is indeed FALSE (Type II error = β) Identifying Type I and Type II Errors: The USDA limit for salmonella contamination for chicken is 20%. A meat inspector reports that the chicken produced by a company exceeds the USDA limit. You perform a hypothesis test to determine whether the mean inspector's claim is true. When will a Type I or Type II error occur? Which is more serious? A company specialized in parachute assembly states that it's main parachute failure rate is not more than 1%. You perform a hypothesis test to determine whether the company's claim is false. When will a type I or type II error occur? Which is more serious? Jan 28 2:35 PM Feb 9 8:52 AM FAIL TO REJECT VS ACCEPTING THE NULL HYPOTHESIS The phrase fail to reject says that the evidence in the sample isn't strong enough to warrant a rejection of the null hypothesis, it does not say that we 100% without a doubt accept the statement. That cannot be said unless data from an ENTIRE population is used. Example 1: Find the critical z values, and draw a curve with a critical region. Assume a normal distribution a) Two tailed test, α= 0.5 b) Right tailed test, α= 0.1 c) Left tailed test, α = 0.01 d) α=0.10, H 1 is p>0.18 e) α=0.005, H 1 is p 0.18 Feb 9 8:30 AM Feb 9 9:01 AM Day 2 Homework 9.1 Worksheet Page 1 #5 19 odd, Page 2 #31 39 odd 9.1 Day 3 Using P Values to Make Decisions Test Statistic and P Values Test Statistic: Value computed from the sample data. This value is used to make a decision to reject the null hypothesis or fail to reject the null hypothesis. We use: Test statistic for proportion: Test statistic for mean or known standard deviation unknown standard deviation Feb 9 12:31 PM Feb 2 2:24 PM 3
4 TRADITIONAL METHOD Critical Region: If the test statistics falls into the critical region of a normal curve, we reject the null hypothesis. Find the critical z value, and graph the critical region: P Value: The probability of having a sample statistic with a more extreme value than the one determined from the sample data(test statistic) right tailed test: α=0.005 α=0.04, H 1 is p 0.25 Feb 3 2:35 PM Feb 9 2:22 PM Example 2: Find the P Value a) right tailed test, z=0.70 b) left tailed test z= 1.34 c) two tailed test z=1.4 d) With H a >0.3, z=1.54 P VALUE METHOD Compare P Value with α  If P α, then reject H 0  If P> α, then fail to reject H 0 e) With H a 0.25, z=.75 Feb 3 2:47 PM Feb 9 12:44 PM Find the P Value for a Hypothesis Test a) Find the P value for a left tailed hypothesis test with a test statistics of z= Decide whether to reject H 0 if the level of significance is α=0.01 b) Find the P Value for a right tailed hypothesis test with a test statistic of z=1.71. Decide whether to reject H 0 if the level of significance is α=0.05 Solving a Hypothesis Test Question Step 1: Identify the null and alternative hypotheses Example: A pit crew claims that it's mean pit stop time is less than 13 seconds. A random selection of 32 pit stop times has a sample mean of 12.9 and a standard deviation of 0.19 second. Is there enough evidence to support the claim at claim Feb 9 12:47 PM Jan 28 10:52 AM 4
5 Step 2: Specify the level of significance Step 3: Determine the test statistic Used to accept or reject the null hypothesis Step 4: Draw a sketch showing the P Value area on the standard normal distribution. Find the P Value Jan 28 11:09 AM Feb 3 2:31 PM Step 5: Compare the level of significance with the P Value. What is your conclusion? Step 6: Interpret your results in the context of the problem Feb 9 1:33 PM Feb 3 2:34 PM homework 9.1 worksheet page 1 #25 31 odd, pg 413 #9 13 odd Jan 28 1:30 PM Feb 14 11:14 AM 5
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