Statistical Inference – Comparing Two Means PDF

Statistical Inference – Comparing Two Means Two Independent Sample t-Test  2 Test of Significance (2 Independent Samples t-test) The General four steps f...

Statistical Inference – Comparing Two Means Two Independent Sample t-Test  2 Test of Significance (2 Independent Samples t-test) The General four steps for a significance test: Step 2a: Verify conditions OK AS AL Situation 1: Population is approximately normal … BOTH are 1. State the null and alternative hypotheses. W 2. Check conditions and then Calculate the test AY random sample AND any size is OK. S! statistic. Situation 2: Population is not approximately normal, and 3. Find the P-value using the appropriate there is skewness or extreme outliers in distribution. 4. State your conclusion in the context of the EITHER sample specific setting of the test. … a large random sample (n1+n2 ≥ 40) is needed. Step 1: Determine null and alternative hypotheses for µ0=0 Step 2b: The Test Statistic … a standardized score for how extreme the difference between the sample means is from H0: µ1 – µ2 = 0 vs Ha: µ1 – µ2 ≠ 0 (two-sided) the null hypothesis value of the population difference. or Ha: µ1 – µ2 < 0 i.e. µ1 < µ2 (one-sided) The test statistic for two independent mean: or Ha: µ1 – µ2 > 0 i.e. µ1 > µ2 (one-sided) The t test statistic follows a t distribution with degrees of freedom of the smaller (n – 1).  3 Test of Significance cont. (2 Independent Samples t-test) Step 3: Find the p-value For example: t-statistic = -2.50 for a 2 independent sample difference n1 =12, n2 = 10 , Degrees of freedom, df = 9 If 1-sided test, P-value is between 0.01 and 0.02. If 2-sided test, P-value is between 0.02 and 0.04. Step 4: Decision and Conclusion Choose a level of significance a, and if the p-value ≤ α 🡪 reject null concluding there is significant evidence for the LWAYS! alternative. A SA Otherwise, if p-value > α 🡪 cannot reject null with conclusion that there is not enough evidence to support the alternative hypothesis.  4 Example - Exercise and Pulse Rates Part 1 (t-test) A study was performed to compare the mean Step 1: Hypotheses: resting pulse rate of adult subjects who regularly NULL: H0: μNon - μEx = 0 exercise to the mean resting pulse rate of those who do not regularly exercise. The mean resting pulse rate of adult subjects who regularly exercise is the same as the mean resting pulse rate of those Summary statistics (sample data) of the who do not regularly exercise. comparative experiment were: n Sample Mean Sample St Dev ALTERNATIVE: Ha: μNon - μEx ≠ 0 Non-exercisers 31 75 9.0 The mean resting pulse rate of adult subjects who regularly Exercisers 29 66 8.6 exercise is different from the mean resting pulse rate of Do these two populations differ in their mean those who do not regularly exercise. resting pulse rates? Step 2a: Verify conditions OK Perform a test of significance at α=0.05. Large random sample (n1+n2 ≥ 40), we can use t-Test even if skewed.  5 Example - Exercise and Pulse Rates Part 2 (t-test) Step 3: Find the p-value The t test statistic follows a t distribution with degrees of freedom of the smaller (n – 1). Df = n2 – 1 = 29 – 1 = 28. The t statistic is larger than the t critical value of 3.674. P-value < 0.001 (2-sided P) Step 4: Decision and Conclusion P-value < α = 0.05, we can reject null hypothesis. We conclude that there is significant evidence at 5% level that the mean resting pulse rate of adult subjects who do not regularly exercise is different to the mean resting pulse rate of those who regularly exercise.