Introduction to Hypothesis Testing

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Chapter 10 Introduction to Hypothesis Testing Pop dist: Normal  known μ? Z-dist n  30 Pop dist: Non-Normal ? n > 30 Z-dist The purpose of hypothesis testing is to determine whether there is enough statistical evidence supporting a certain belief (or claim) about a parameter Examples • Is there statistical evidence that support the hypothesis that more than p% of all potential customers will purchase a new products? • Is the hypothesis that a certain drug is effective? 11.1 Concepts of Hypothesis Testing The Reject-Region method Step 1: Two hypotheses are defined H0: The null hypothesis specifies our current belief about the parameter we test ( = 170, p = 4, etc.) Must be a specific value H1: The alternative hypothesis specifies a range of values for the parameter tested ( > 170; p .4; etc.) effected by the belief A person is assumed to be innocent before court trial H0 Needed in the test Not guilty Do not reject H0 Reject H0 guilty Claim Claim HH000 HHAA =A =A =A =A ≠A ≠A –tail 22-tail -tail or or 2-side 2-sidetest test >A >A =A =A A A ≥A ≥A =A =A A 11 –tail –tail or or 1-side 1-sidetest test –tail or 1-side test A A A =A =A =A A –tail or 1-side test ≤A ≤A ≤A =A =A =A A –tail or 1-side test 10.2 Testing the Population Mean when the Population Standard Deviation is Known Example 1: The manager of a department determines that new billing system will be cost-effective only if the mean monthly account is more than $170 A random sample of 400 monthly accounts is drawn, for which the sample mean is $178 Assume a standard deviation of $65 Can the manager conclude from this that the new system will be cost-effective? Step 1: H0: µ = 170 H1: µ > 170 Step 2: Choose the significance level  reject reject H H000 Verdict: Guilty Do not H reject H00 accept accept Verdict:HNot Guilty H00 is really true true HAA is really false false Really innocent Type Type II error error Probability Probability of of committing committing the the type type II error error Send an innocent person to jail OK OK H00 is really false false HAA isisreally true true true Really guilty OK OK Type II error Type II error of committing the Probability Probability type II error of committing the type errorperson go free Let aIIguilty The more severe the consequence of committing the type-I error, the smaller/higher the value of  Example 1:  = 0.05 Step 3: Determine the sample size n and hence the sampling distribution Example 1:n=400; N(0,1) Step 4: Depending on the sampling distribution, the HA, and the value of  , find the suitable critical value and the reject region HA: μ > 170 (1-sided test) N(0,1)  Reject region: Z > 1.645 1.645 Critical value Right-Tail Testing Left-Tail Testing Two–Tail Testing Step 5: Collect data x 178 Calculate the standard test statistic Example 1: HA: μ =170 x   178  170 z  2.46  / n 65 / 400 Step 6: If the test statistic is in the reject region, then reject H0, otherwise not reject H0 Example 1: Reject H0 at  = 05 Reject H0: There is enough statistical evidence to conclude that the alternative hypothesis is true Do not reject H0: There is not enough statistical evidence to conclude that the alternative hypothesis is true A Left Hand Tail Test H0:   H1:  <  Reject H0 if x falls here Critical value 16 The SSA envelop plan example: The chief financial officer in FedEx believes that including a stamped selfaddressed (SSA) envelop in the monthly invoice sent to customers will decrease the amount of time it take for customers to pay their monthly bills Currently, customers return their payments in 22 days on the average, with a standard deviation of days A random sample of 220 customers was selected and SSA envelops were included with their invoice packs The mean time it took customers to pay their bill was 21.63 Can the CFO conclude that the plan will be successful at 10% significance level? 17 Step 1: H0:  H1:   Step 2:  = 0.10 Step 3: n= 220 = 10 Step 4: Step 5: −1.28 Z x  21.63 22 z   91  / n 6/ 220 Step 6: Do not reject the null hypothesis 18 A Two Tail Test H 0:     H1:     Reject H0 if x falls here Critical value Reject H0 if x falls here Critical value 19 Example 2: A statistician believes the monthly mean of the long-distance bills for all AT&T residential customers is $17.09 A random sample of 100 customers of its leading competitor yields x 17.55 Assuming the standard deviation of the bills of the competitors is 3.87, can we infer that there is a difference between AT&T’s bills and the competitor’s bills (on the average)? 20 Step 1: H0:  H1:   Step 2:  = 0.05 Step 3: n= 100 = 025 Step 4: −1.96 Step 5: Critical value = 025 1.96 Critical value x   17.55 17.09 z  1.19  / n 3.87/ 100 Step 6: Do not reject the null hypothesis 21 ... CFO conclude that the plan will be successful at 10% significance level? 17 Step 1: H0:  H1:   Step 2:  = 0 .10 Step 3: n= 220 = 10 Step 4: Step 5: −1.28 Z x  21.63 22 z  ...  Step 2:  = 0.05 Step 3: n= 100 = 025 Step 4: −1.96 Step 5: Critical value = 025 1.96 Critical value x   17.55 17.09 z  1.19  / n 3.87/ 100 Step 6: Do not reject the null... A A A =A =A =A A –tail or 1-side test ≤A ≤A ≤A =A =A =A A –tail or 1-side test 10. 2 Testing the Population Mean when the Population Standard Deviation is Known Example 1: The
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