Comparison of Two Population Means

Transcription

1 Comparison of Two Population Means Esra Akdeniz March 15, 2015

2 Independent versus Dependent (paired) Samples We have independent samples if we perform an experiment in two unrelated populations. We have dependent or paired samples if we perform two experiments on the same population or if we perform an experiment in two populations that their subjects are paired.

3 Normal Populations Let X 1,..., X m N(µ 1, σ1) 2 Let Y 1,..., Y n N(µ 2, σ2) 2 The two samples are independent. What is an estimator of µ 1 µ 2? What is the sampling distribution of that estimator? How do we construct a (1 α)100% confidence interval for µ 1 µ 2?

4 Case I: Test Procedures for Normal Populations with Known Variances Null hypothesis: H 0 : µ 1 µ 2 = 0 OR H 0 : µ 1 µ 2 0 OR H 0 : µ 1 µ 2 0 Test statistic: z = x ȳ 0 σ 2 1 m + σ2 2 n N(0, 1)

5 Case II: Test Procedures for Normal Population with Unknown Variances and both n > 30, m > 30 Null hypothesis: H 0 : µ 1 µ 2 = 0 OR H 0 : µ 1 µ 2 0 OR H 0 : µ 1 µ 2 0 Test statistic: z = x ȳ 0 S 2 1m + S2 2n N(0, 1)

6 Example Patients in a breast cancer treatment facility has been randomly assigned to one of the two groups: control and experimental. The control group was given placebo. The experimental group was given a treatment drug. We measured the survival time for 79 patients in the control group and 85 patients in the experimental group. The sample mean and sample standard deviation for the control group are and 11.60, respectively. The sample mean and sample standard deviation for the experimental group are and 8.85, respectively. Assuming we have a normal population, does this information suggest that the patients in the experimental group did better than the patients in the control group at the significance level of 0.05? Construct a 95% CI for the difference between two means.

7 Cases III and IV: Test Procedures for Normal Population with Unknown Variances and at least one of sample sizes is less than 30 Unpooled Case. Pooled Case (we assume σ 2 1 = σ 2 2).

8 Case III: Unpooled Since the population variances are unknown, our test statistic will have the following form: T = X Ȳ (µ1 µ2). S1m 2 + S2 2n The degrees of freedom is found by rounding the result of the following formula down to the nearest integer: ( ) s s2 2 m n ν = ( s 2 ) 2 1 m + m 1 ( s 2 ) 2 2 n n 1

9 Confidence Interval and Test Procedure A (1 α)100% CI for µ 1 µ 2 is... Null hypothesis: H 0 : µ 1 µ 2 = 0 OR H 0 : µ 1 µ 2 0 OR H 0 : µ 1 µ 2 0 Test statistic: t = x ȳ 0 S 2 1m + S2 2n t ν

10 Case III: Pooled S 2 p is calculated as follows: Distribution of T: S 2 p = (m 1)S (n 1)S 2 2 n + m 2 T = X Ȳ (µ 1 µ 2) ( 1 + ) t m+n 2 1 m n S 2 p (1 α)100% CI for µ 1 µ 2 is...

11 Test Procedure Null hypothesis: H 0 : µ 1 µ 2 = 0 OR H 0 : µ 1 µ 2 0 OR H 0 : µ 1 µ 2 0 Test statistic: t = x ȳ 0 S 2 p( 1 m + 1 n ) tm+n 2

12 Nonparametric Tests for Independent Samples Until now we assumed X 1,..., X m and Y 1,..., Y n are from a normal distribution. What if we don t have normal distributions?

13 Wilcoxon Rank Sum Test (Mann Whitney Test) Null hypothesis: H 0 : median 1 median 2 = 0 OR H 0 : median 1 median 2 0 OR H 0 : median 1 median 2 0 To find the test statistic one needs to the following: Label the small sample as your Xs and the large sample as your Ys. Subtract 0 from the values in your X sample. Combine the samples X 0 and Y s. Fill the table with the ranks. The test statistic W equals to the sum of the ranks of the X 0 in the combined sample.

14 Paired data Until now we assumed X 1,..., X m and Y 1,..., Y n are independent. Now, what happens if the two samples are dependent? First: The observations are paired, so the samples should have the same number of observations. Second: Since we have the same number of observations and each observation from the X sample can be paired with an observation from the Y sample, that means we can calculate the difference D = X Y in each pair.

15 D is assumed to follow a normal distribution with mean µ D and variance σ 2 D.

16 Test Procedure and Confidence Interval Null hypothesis: H 0 : µ D = 0 OR H 0 : µ D 0 OR H 0 : µ D 0 Test statistic: t = d 0 s D / n tn 1 The CI for µ D is d ± t α/2,n 1.s D / n.

17 Example Consider the data taken from a study in which 8 adult males with coronary heart disease is subject to a series of exercise tests on a number of different occasions. On one day, a patient undergoes an exercise test on a treadmill; the length of time from the start of the test until the patient experiences angina pain or spasms in the chest. He is then exposed to plain room air for approximately one hour. At the end of this time, he performs a second exercise test; time until the onset of angina is again recorded. Assuming this population is normal, test the hypothesis that plain room air improves the results of the exercise test. Test st test nd test

18 Nonparametric Test for Paired Samples: Wilcoxon Signed-Rank test Null hypothesis: H 0 : median 1 median 2 = 0 OR H 0 : median 1 median 2 0 OR H 0 : median 1 median 2 0 To find the test statistic make a table of differences for each pair of observations, ignore the signs of these differences, rank their absolute values from smallest to largest. Tied observations are assigned an average rank. Find the signed rank, i.e., each rank gets the sign of the difference it is corresponding to. Sum of the ranks is the test statistic.