Mathematical statistics

Transcription

1 November 15 th, 2018 Lecture 21: The two-sample t-test

2 Overview Week 1 Week 2 Week 4 Week 7 Week 10 Week 14 Probability reviews Chapter 6: Statistics and Sampling Distributions Chapter 7: Point Estimation Chapter 8: Confidence Intervals Chapter 9, 10: Test of Hypothesis Regression

3 Inferences based on two samples 10.1 Difference between two population means z-test confidence intervals 10.2 The two-sample t test and confidence interval 10.3 Analysis of paired data

4 Two-sample inference

5 Settings Independent samples 1 X 1, X 2,..., X m is a random sample from a population with mean µ 1 and variance σ Y 1, Y 2,..., Y n is a random sample from a population with mean µ 2 and variance σ The X and Y samples are independent of each other. Paired samples 1 There is only one set of n individuals or experimental objects 2 Two observations are made on each individual or object

6 Properties of X Ȳ Proposition

7 Confidence intervals

8 Testing the difference between two population means Setting: independent normal random samples X 1, X 2,..., X m and Y 1, Y 2,..., Y n with known values of σ 1 and σ 2. Constant 0. Null hypothesis: Alternative hypothesis: (a) H a : µ 1 µ 2 > 0 (b) H a : µ 1 µ 2 < 0 (c) H a : µ 1 µ 2 0 H 0 : µ 1 µ 2 = 0 When = 0, the test (c) becomes H 0 : µ 1 = µ 2 H a : µ 1 µ 2

9 Case 1: Normal distributions with known variances

10 Testing the difference between two population means Proposition

11 Sample solution

12 Sample solution

13 Case 2: Large-sample tests/confidence intervals

14 Principles Central Limit Theorem: X and Ȳ are approximately normal when m, n > 30 so is X Ȳ. Thus ( X Ȳ ) (µ 1 µ 2 ) σ1 2 m + σ2 2 n is approximately standard normal When n is sufficiently large S 1 σ 1 and S 2 σ 2 Conclusion: ( X Ȳ ) (µ 1 µ 2 ) S1 2 m + S2 2 n is approximately standard normal when n is sufficiently large If m, n > 40, we can ignore the normal assumption and replace σ by S

15 Large-sample tests Proposition

16 Large-sample CIs Proposition Provided that m and n are both large, a CI for µ 1 µ 2 with a confidence level of approximately 100(1 α)% is x ȳ ± z α/2 s 2 1 m + s2 2 n where gives the lower limit and + the upper limit of the interval. An upper or lower confidence bound can also be calculated by retaining the appropriate sign and replacing z α/2 by z α.

17 The two-sample t test and confidence interval

18 Remember Chapter 8? Section 8.1 Normal distribution σ is known Section 8.2 Normal distribution Using Central Limit Theorem needs n > 30 σ is known needs n > 40 Section 8.3 Normal distribution σ is known n is small Introducing t-distribution

19 Principles For one-sample inferences: For two-sample inferences: X µ S/ n t n 1 ( X Ȳ ) (µ 1 µ 2 ) S1 2 m + S2 2 n t ν where ν is some appropriate degree of freedom (which depends on m and n).

20 Chi-squared distribution Proposition If Z has standard normal distribution Z(0, 1) and X = Z 2, then X has Chi-squared distribution with 1 degree of freedom, i.e. X χ 2 1 distribution. If Z 1, Z 2,..., Z n are independent and each has the standard normal distribution, then Z Z Z 2 n χ 2 n

21 t distributions Definition Let Z be a standard normal rv and let W be a χ 2 ν rv independent of Z. Then the t distribution with degrees of freedom ν is defined to be the distribution of the ratio T = Z W /ν

22 Definition of t distributions: Z W /ν t ν Our statistic: ( X Ȳ ) (µ 1 µ 2 ) S 2 1 m + S2 2 n = [ ( X Ȳ ) (µ 1 µ 2 ) ] σ1 / 2 ( ) ( ) S 2 1 m + S2 2 σ 2 n / 1 m + σ2 2 n m + σ2 2 n What we need: ( S 2 1 m + S 2 2 ) ( ) σ 2 / 1 n m + σ2 2 = W n ν

23 Quick maths What we need: ( S 2 1 m + S 2 2 ) ( ) σ 2 = 1 n m + σ2 2 W n ν What we have E[W ] = ν, Var[W ] = 2ν E[S1 2] = σ2 1, Var[S 1 2] = 2σ4 1 /(m 1) E[S2 2] = σ2 2, Var[S 2 2] = 2σ4 2 /(n 1) Variance of the LHS [ S 2 Var 1 m + S 2 2 ] 2σ1 4 = n (m 1)m 2 + 2σ2 4 (n 1)n 2 Variance of the RHS Var [( σ 2 1 m + σ2 2 n ) W ν ] ( ) σ 2 2 = 1 m + σ2 2 2ν n ν 2

24 2-sample t test: degree of freedom

25 CIs for difference of the two population means

26 2-sample t procedures

27 Example Example A paper reported the following data on tensile strength (psi) of liner specimens both when a certain fusion process was used and when this process was not used: The authors of the article stated that the fusion process increased the average tensile strength. Carry out a test of hypotheses to see whether the data supports this conclusion (and provide the P-value of the test)

28 t-table

29 Solution

30 Solution

31 The paired samples setting 1 There is only one set of n individuals or experimental objects 2 Two observations are made on each individual or object

32 Settings Independent samples 1 X 1, X 2,..., X m is a random sample from a population with mean µ 1 and variance σ Y 1, Y 2,..., Y n is a random sample from a population with mean µ 2 and variance σ The X and Y samples are independent of each other. Paired samples 1 There is only one set of n individuals or experimental objects 2 Two observations are made on each individual or object

33 Example Example Consider two scenarios: A. Insulin rate is measured on 30 patients before and after a medical treatment. B. Insulin rate is measured on 30 patients receiving a placebo and 30 other patients receiving a medical treatment.

34 Notes In the independent case, we construct the statistics by looking at the distribution of X Ȳ which has E[ X Ȳ ] = µ 1 µ 2, Var[ X Ȳ ] = Var( X )+Var(Ȳ ) = σ2 1 m +σ2 2 n With paired data, the X and Y observations within each pair are not independent, so X and Ȳ are not independent of each other the computation of the variance is in valid could not use the old formulas

35 The paired t-test Because different pairs are independent, the D i s are independent of each other We also have E[D] = E[X Y ] = E[X ] E[Y ] = µ 1 µ 2 = µ D Testing about µ 1 µ 2 is just the same as testing about µ D Idea: to test hypotheses about µ 1 µ 2 when data is paired: 1 form the differences D 1, D 2,..., D n 2 carry out a one-sample t-test (based on n 1 df) on the differences.

36 Settings Assumption 1 The data consists of n independently selected pairs of independently normally distributed random variables (X 1, Y 1 ), (X 2, Y 2 ),..., (X n, Y n ) with E(X i ) = µ 1 and E(Y i ) = µ 2. 2 Let D 1 = X 1 Y 1, D 2 = X 2 Y 2,..., D n = X n Y n, so the D i s are the differences within pairs.

37 Confidence intervals A t confidence interval for for µ D = µ 1 µ 2 can be constructed based on the fact that T = D µ D S D / n follows the t distribution with degree of freedom n 1. The CI for µ D is d ± t α/2,n 1 s D n A one-sided confidence bound results from retaining the relevant sign and replacing t α/2,n 1 by t α,n 1.

38 The paired t-test

39 Example Example Consider two scenarios: A. Insulin rate is measured on 30 patients before and after a medical treatment. B. Insulin rate is measured on 30 patients receiving a placebo and 30 other patients receiving a medical treatment. What type of test should be used in each cased: paired or unpaired?

40 Example Example Suppose we have a new synthetic material for making soles for shoes. We d like to compare the new material with leather using some energetic kids who are willing to wear test shoes and return them after a time for our study. Consider two scenarios: A. Giving 50 kids synthetic sole shoes and 50 kids leather shoes and then collect them back, comparing the average wear in each group B. Give each of a random sample of 50 kids one shoe made by the new synthetic materials and one shoe made with leather What type of test should be used in each cased: paired or unpaired?

41 Example Consider an experiment in which each of 13 workers was provided with both a conventional shovel and a shovel whose blade was perforated with small holes. The following data on stable energy expenditure is provided: Calculate a confidence interval at the 95% confidence level for the true average difference between energy expenditure for the conventional shovel and the perforated shovel (assuming that the differences follow normal distribution).

42 Example Consider an experiment in which each of 13 workers was provided with both a conventional shovel and a shovel whose blade was perforated with small holes. The following data on stable energy expenditure is provided: Carry out a test of hypotheses at significance level.05 to see if true average energy expenditure using the conventional shovel exceeds that using the perforated shovel; include a P-value in your analysis.

43 t-table