Hypothesis Tests

One-Sample Hypothesis Tests Ho: Null Hypothesis – Benchmark H1: Alternative Hypothesis – The hypothesis being tested Type I Error α P(Reject H0 | H0 is True) False Positive Type II Error β P(Fail to Reject H0 | H0 is False) False Negative Power 1-β P(Reject H0 | H0 is False) Sensitivity For a given type of test and sample size, there is a tradeoff between α and β. The only way to reduce them both simultaneously is to increase the sample size. Left-Tailed Test Two-Tailed Test Right-Tailed Test 𝐻0: 𝜇 ≥ 𝜇0 𝐻0: 𝜇 = 𝜇0 𝐻0: 𝜇 ≤ 𝜇0 𝐻1: 𝜇 < 𝜇0 𝐻1: 𝜇 ≠ 𝜇0 𝐻1: 𝜇 > 𝜇0 The Critical Value is the boundary between the Reject and Do Not Reject regions. It is the standardized score associated with α. The Test Statistic is the standardized score of the sample statistic being tested (in our case, either �̅� 𝑜𝑟 𝑝). If the Test Statistic is more extreme than the Critical Value, there is sufficient evidence to reject the H0. The p-value is the probability associated with the Test Statistic. When the p-value is < than α, the test is significant and you may reject the H0. Case Critical Value Test Statistic P-Value Testing a Mean: Known 𝜎2 𝑍𝛼 𝑂𝑛𝑒 − 𝑇𝑎𝑖𝑙𝑒𝑑 𝑇𝑒𝑠𝑡 𝑍𝛼 2 𝑇𝑤𝑜 − 𝑇𝑎𝑖𝑙𝑒𝑑 𝑇𝑒𝑠𝑡 𝑍𝑐𝑎𝑙𝑐 = �̅� − 𝜇0 𝜎 √𝑛 Probability associated with Z- score. Look it up on statistical table. Testing a Mean: Unknown 𝜎2 𝑡𝛼 𝑂𝑛𝑒 − 𝑇𝑎𝑖𝑙𝑒𝑑 𝑇𝑒𝑠𝑡 𝑡𝛼 2 𝑇𝑤𝑜 − 𝑇𝑎𝑖𝑙𝑒𝑑 𝑇𝑒𝑠𝑡 d.f. = n-1 𝑡𝑐𝑎𝑙𝑐 = �̅� − 𝜇0 𝑠 √𝑛 Probability associated with t- score. Cannot look it up on statistical table. Testing a Proportion 𝑍𝛼 𝑂𝑛𝑒 − 𝑇𝑎𝑖𝑙𝑒𝑑 𝑇𝑒𝑠𝑡 𝑍𝛼 2 𝑇𝑤𝑜 − 𝑇𝑎𝑖𝑙𝑒𝑑 𝑇𝑒𝑠𝑡 𝑍𝑐𝑎𝑙𝑐 = 𝑝 − 𝜋0 √𝜋0(1 − 𝜋0) 𝑛 Probability associated with Z- score. Look it up on statistical table. When testing a proportion, you must first check to determine if you may assume normality of the distribution. You may assume normality when 𝑛𝑝 ≥ 10 AND 𝑛(1 − 𝑝) ≥ 10. Two-Sample Hypothesis Tests Comparing Two Means - Independent Samples Left-Tailed Test Two-Tailed Test Right-Tailed Test 𝐻0: 𝜇1 − 𝜇2 ≥ 𝐷0 𝐻0: 𝜇1 − 𝜇2 = 𝐷0 𝐻0: 𝜇1 − 𝜇2 ≤ 𝐷0 𝐻1: 𝜇1 − 𝜇2 < 𝐷0 𝐻1: 𝜇1 − 𝜇2 ≠ 𝐷0 𝐻1: 𝜇1 − 𝜇2 > 𝐷0 Case – Two Means Critical Value Test Statistic P-Value 1. Known 𝜎2 𝑍𝛼 𝑂𝑛𝑒 − 𝑇𝑎𝑖𝑙𝑒𝑑 𝑇𝑒𝑠𝑡 𝑍𝛼 2 𝑇𝑤𝑜 − 𝑇𝑎𝑖𝑙𝑒𝑑 𝑇𝑒𝑠𝑡 𝑍𝑐𝑎𝑙𝑐 = (𝑥1̅̅̅ − 𝑥2̅̅ ̅) − (𝜇1 − 𝜇2) √ 𝜎1 2 𝑛1 + 𝜎2 2 𝑛2 Probability associated with Z- score. Look it up on statistical table. 2. Unknown 𝜎2- Assumed Equal 𝑡𝛼 𝑂𝑛𝑒 − 𝑇𝑎𝑖𝑙𝑒𝑑 𝑇𝑒𝑠𝑡 𝑡𝛼 2 𝑇𝑤𝑜 − 𝑇𝑎𝑖𝑙𝑒𝑑 𝑇𝑒𝑠𝑡 𝑑. 𝑓. = 𝑛1 + 𝑛2 − 2 𝑡𝑐𝑎𝑙𝑐 = (�̅�1 − �̅�2) − (𝜇1 − 𝜇2) √ 𝑆𝑝2 𝑛1 + 𝑆𝑝2 𝑛2 𝑆𝑝 2 = (𝑛1 − 1)𝑆1 2 + (𝑛2 − 1)𝑆2 2 𝑛1 + 𝑛2 − 2 Probability associated with t- score. Cannot look it up on statistical table. 3. Unknown 𝜎2Assumed Unequal 𝑡𝛼 𝑂𝑛𝑒 − 𝑇𝑎𝑖𝑙𝑒𝑑 𝑇𝑒𝑠𝑡 𝑡𝛼 2 𝑇𝑤𝑜 − 𝑇𝑎𝑖𝑙𝑒𝑑 𝑇𝑒𝑠𝑡 𝑑. 𝑓. = ( 𝑆1 2 𝑛1 + 𝑆2 2 𝑛2 ) 2 ( 𝑆1 2 𝑛1 ) 2 𝑛1 − 1 + ( 𝑆2 2 𝑛2 ) 2 𝑛2−1 𝑡𝑐𝑎𝑙𝑐 = (�̅�1 − �̅�2) − (𝜇1 − 𝜇2) √ 𝑆1 2 𝑛1 + 𝑆2 2 𝑛2 Probability associated with t- score. Cannot look it up on statistical table. In the common situation in which you are testing for a zero difference (D0 = 0), the term (𝜇1 − 𝜇2) is equal to zero and, thus, drops out of the equations above. Hang Text Box weight variance by sample size Confidence Interval for the Difference of Two Means µ1 - µ2 The point estimate for 𝜇1 − 𝜇2 is �̅�1 − �̅�2, where the sample means are calculated from independent random samples. Remember that the confidence interval is the range within which the true difference might fall. If the confidence interval includes zero, you may conclude there is no difference between the population means. Case Confidence Interval 1. Known 𝜎2 (�̅�1 − �̅�2) ± 𝑍𝛼 2 √ 𝜎1 2 𝑛1 + 𝜎2 2 𝑛2 2. Unknown 𝜎2 Assumed Equal (�̅�1 − �̅�2) ± 𝑡𝛼 2 √ (𝑛1 − 1)𝑆1 2 + (𝑛2 − 1)𝑆2 2 𝑛1 + 𝑛2 − 2 √ 1 𝑛1 + 1 𝑛2 With 𝑑. 𝑓. = (𝑛1 − 1) + (𝑛2 − 1) 3. Unknown 𝜎2 Assumed Unequal (�̅�1 − �̅�2) ± 𝑡𝛼 2 √ 𝑆1 2 𝑛1 + 𝑆2 2 𝑛2 With 𝑑. 𝑓. = ( 𝑆1 2 𝑛1 + 𝑆2 2 𝑛2 ) 2 ( 𝑆1 2 𝑛1 ) 2 𝑛1−1 + ( 𝑆2 2 𝑛2 ) 2 𝑛2−1 Comparing Two Means – Paired Samples In the paired sample t test, you calculate a new variable 𝑑 = �̅�1 − �̅�2. The two samples are thus reduced to one sample of n differences. Calculate this new variable d using a table, as you will need to calculate both the mean �̅� and the standard deviation 𝑠𝑑of this frequency distribution. Observations Sample 1 Sample 2 Differences 1 𝑥11 𝑥12 𝑑1 = 𝑥11 − 𝑥12 2 𝑥21 𝑥22 𝑑2 = 𝑥21 − 𝑥22 3 𝑥31 𝑥32 𝑑3 = 𝑥31 − 𝑥32 . . . . . . n 𝑥𝑛1 𝑥𝑛2 𝑑𝑛 = 𝑥𝑛1 − 𝑥𝑛2 You then calculate the mean �̅� and the standard deviation 𝑠𝑑of n differences as follows. �̅� = ∑ 𝑑𝑖 𝑛 𝑖=1 𝑛 𝑠𝑑 = √∑ (𝑑1 − �̅�)2 𝑛 − 1 𝑛 𝑖=1 Then, calculate your test statistic as follows. 𝑡𝑐𝑎𝑙𝑐 = �̅� − 𝜇𝑑 𝑠𝑑 √𝑛 Compare your test statistic to the critical value (the t score for either a one- or two-tailed test at the selected level of significance with n-1 degrees of freedom). Hang Text Box same group being tested 2 different times Hang Text Box test the strength of the difference Comparing Two Proportions Left-Tailed Test Two-Tailed Test Right-Tailed Test 𝐻0: 𝜋1 − 𝜋2 ≥ 𝐷0 𝐻0: 𝜋1 − 𝜋2 = 𝐷0 𝐻0: 𝜋1 − 𝜋2 ≤ 𝐷0 𝐻1: 𝜋1 − 𝜋2 < 𝐷0 𝐻1: 𝜋1 − 𝜋2 ≠ 𝐷0 𝐻1: 𝜋1 − 𝜋2 > 𝐷0 Remember to check for normality for each sample … 𝑛1𝑝1 ≥ 10 AND 𝑛1(1 − 𝑝1) ≥ 10 𝑛2𝑝2 ≥ 10 AND 𝑛2(1 − 𝑝2) ≥ 10 Calculate your test statistic. 𝑍𝑐𝑎𝑙𝑐 = (𝑝1 − 𝑝2) − (𝜋1 − 𝜋2) √ 𝑝1(1 − 𝑝1) 𝑛1 + 𝑝2(1 − 𝑝2) 𝑛2 Compare your test statistic to the critical value, which is the z-score associated with 𝛼 𝑜𝑟 𝛼 2⁄ as appropriate for a one- or two-tailed test. Or, use the p-value method, which involves finding the p-value (the probability associated with the test statistic) and comparing it to 𝛼. Confidence Interval for the Difference of Two Proportions 𝝅𝟏 − 𝝅𝟐 (𝑝1 − 𝑝2) ± 𝑍𝛼 2⁄ √ 𝑝1(1 − 𝑝1) 𝑛1 + 𝑝2(1 − 𝑝2) 𝑛2 Hang Rectangle Hang Text Box combined variance Hang Text Box The difference is not due to sampling error nor randomness.