Data Decision Making Course: Testing Difference Between Two Population Means - Prof. Torre, Study notes of Mathematics

Download Data Decision Making Course: Testing Difference Between Two Population Means - Prof. Torre and more Study notes Mathematics in PDF only on Docsity!

Training Course on Decision Making with Data

April 26-30, 2010

STATISTICAL RESEARCH AND TRAINING CENTER J and S Building, 104 Kalayaan Avenue, Diliman, Quezon City

Two Population Case

Testing the Difference Between Two Population Means

Sample of Size n _ Sample mean = x Sample s.d.=sx Sample of Size m _ Sample mean = y Sample s.d.=sy

Pop’n mean=

x Pop’n s.d .= X Pop’n mean=Y Pop’n s.d .=  Y Population 1 Population 2 Set-up when testing the difference between two population means.

Types of Sampling Two Independent Samples Two Related Samples or Paired Observations  the selection of the sample in Population 1 is independent of the selection of the sample in Population 2  achieved by either using the same subject in the two samples or pairing of subjects with respect to some extraneous variable that may affect or influence the outcome  used to overcome the difficulty imposed by extraneous differences between the two populations

Testing the Difference Between Two Population Means

Based On Two Independent Samples

Ho: X - Y = 0 Test statistic Ha Region of Rejection Case 1:  X &  Y known zc =  X

•  Y < 0  X
•  Y

0  X

•  Y  0 zc < - z  z c

z  zc<-z/2 & zc >z x y / n m X Y     (^2 )

Case 3:  X &  Y unknown & unequal t c =  X

•  Y < 0  X
•  Y

0  X

•  Y  0 t c < - t (,v) t c

t (,v) t c <- t (/2,v) & t c t (/2,v) where v = x y s n s m X Y   (^2 ) s n s m s n n s m m X Y X Y 2 2 2 2 2 2 2 1 1

                     

Statistical Research and Training Center Training Course on Decision Making with Data April 26-30, 2010

Testing the Difference Between Two Population

Means Based on 2 Independent Samples

 These tests are exact -level tests for independent samples selected from normal populations. However, they provide good approximate -level tests when the distributions are not Normal provided both samples are greater than 30.  Case 4 : If  X and  Y are unknown and both sample sizes are greater than 30 , use the z-test statistic but replacing X 2 and Y 2 by sX 2 and sY 2 , respectively.  If there is no information at all about the population variances, the test based on the t-test statistic will still provide a good approximate -level test so long as the sample sizes are the same and both populations are Normal. This is the reason why the researcher should plan the experiment so that n=m.

Statistical Research and Training Center Training Course on Decision Making with Data April 26-30, 2010 Solution:

1. Here we wish to test Ho:  1 =  2 or equivalently,  1

•  2 = 0 Ha:  1   2 or  1
•  2  0  1 is the mean length of time it took male students to finish a 50 item IQ test  2 is the mean length of time it took female students to finish a 50 item IQ test

Statistical Research and Training Center Training Course on Decision Making with Data April 26-30, 2010

Con’t of Solution:

2) The appropriate test statistic is

since both sample sizes are large,

the normal distribution applies.

m

s

n

s

x y

Z

x y c 2 2

Statistical Research and Training Center Training Course on Decision Making with Data April 26-30, 2010 Con’t of Solution:

5. Since 5.0 > 1.96, we thus reject the null hypothesis at the 5% level of significance. It appears that the difference between completion times of male and female students is significantly different from zero.

Statistical Research and Training Center Training Course on Decision Making with Data April 26-30, 2010

Example of Testing the Difference Between Two

Population Means (Independent Samples)

During 1985 , approximately 31 % of Asian women in their late teens and early 20 s reported that they had used marijuana within the previous year. These are the prime reproductive years for women and researchers were concerned about possible effects of marijuana use during pregnancy on fetal growth and development. A large-scale study of mothers recruited over a three-year period from a general prenatal clinic was done. Among other findings, the researchers reported that the mean birth weight of infants born to 895 mothers who did not use marijuana during pregnancy was 3260 grams, with a standard deviation of 616 grams.

t Test for Differences in Two Means Data Hypothesized Difference 0 Level of Significance 0. Population 1 Sample Sample Size 895 Sample Mean 3260 Sample Standard Deviation 616 Population 2 Sample Sample Size 202 Sample Mean 2980 Sample Standard Deviation 662 Intermediate Calculations Population 1 Sample Degrees of Freedom 894 Population 2 Sample Degrees of Freedom 201 Total Degrees of Freedom 1095 Pooled Variance 390247. Difference in Sample Means 280 t - Test Statistic 5. Lower-Tail Test Lower Critical Value - 2. p - Value 1 Do not reject the null hypothesis

PhStat Output

Statistical Research and Training Center Training Course on Decision Making with Data April 26-30, 2010 Testing the Difference Between two Population Means Based on two Related Samples

Preliminary Steps

1. For each of the n pairs of observations, compute for the

difference in scores.

2. Get the mean of the differences between scores, d:

3. Compute for the standard deviation of the differences between

scores, s

d

d sum of all the differences n  sum of the squared differences nd n

2 1

Example: Testing the Difference Between Two Population Means (Paired Samples) Subjects are tested for reactions times with their left and right hands. (Only right-handed subjects were used.) The results (in thousandths of a second) are given in the accompanying table. Use a 0.05 significance level to test the claim that there is a difference between the mean of the right- and left- hand reaction times. If an engineer is designing a fighter-jet cockpit and must locate the ejection-seat activator to be accessible to either the right or the left hand, does it make a difference which hand she chooses? Subject A B C D E F G H I J K L M N Right 191 97 116 165 116 129 171 155 112 102 188 158 121 133 Left 224 171 191 207 196 165 177 165 140 188 155 219 177 174

Statistical Research and Training Center Training Course on Decision Making with Data April 26-30, 2010 Solution:

1. Ho: vs. Ha: (original claim) where is the mean reaction time of right-hand persons is the mean reaction time of left-hand persons
2. Because we are testing a claim about the means of paired dependent data, we use the Student t distribution. t =  0 d  (^)  0 d  s n d d d /   1  2 