Upload
hoangmien
View
220
Download
3
Embed Size (px)
Citation preview
1
Independent Samples T-test
• With previous tests, we were interested incomparing a single sample with a population
• With most research, you do not haveknowledge about the population -- you don’tknow the population mean and standarddeviation
INDEPENDENT SAMPLES T-TEST:• Hypothesis testing procedure that uses
separate samples for each treatment condition(between subjects design)
• Use this test when the population mean andstandard deviation are unknown, and 2separate groups are being compared
Example: Do males and females differ in termsof their exam scores?
• Take a sample of males and a separate sampleof females and apply the hypothesis testingsteps to determine if there is a significantdifference in scores between the groups
2
Formula:
• We are interested in a difference between 2populations (females, µ1, and males, µ2) andwe use 2 samples (females, x1, and males, x2)to estimate this difference
( ) ( )21
2121
xxs
xxt
−
−−−=
µµ
ESTIMATED STANDARD ERROR OF THEDIFFERENCE:
• Gives us the total amount of error involved inusing 2 sample means to estimate 2population means. It tells us the averagedistance between the sample difference (x1-x2)and the population difference (µ1-µ2)
• As we’ve done previously, we have toestimate the standard error using the samplestandard deviation or variance and, sincethere are 2 samples, we must average thetwo sample variances.
3
POOLED VARIANCE: The average of the twosample variances, allowing the larger sampleto weighted more heavily
Formulae:
OR
df1=df for 1st sample; n1-1df2=df for 2nd sample; n2-1
21
22
212
12 )()(
dfdf
sdfsdfs pooled +
+=
21
212
dfdf
SSSSspooled +
+=
Estimated Standard Error of theDifference
Degrees of freedom (df) for the Independentt statistic is n1 + n2 - 2 or df1+df2
2
2
1
2
21 n
s
n
ss pooledpooled
xx +=−
€
sx 1−x 2 =SS1 + SS2n1 + n2 − 2
1n1
+1n2
book formula
4
Hypothesis testing using an IndependentSamples t-Test:
Example: Do males and females differ in their testscores for exam 2? The mean test score forfemales is 27.1 (s=2.57, n=19), and the meantest score for males is 26.7 (s=3.63, n=20)
Step 1: State the hypothesesH0: µ1-µ2=0 (µ1=µ2)H1: µ1-µ2≠0 (µ1 ≠µ2)
• This is a two-tailed test (no direction is predicted)
Step 2: Set the criterion• α=?
• df= n1+n2-2=?• Critical value for the t-test ?
Step 3: Collect sample data, calculate xand s
From the example we know the mean test scorefor females is 27.1 (s=2.57, n=19), and themean test score for males is 26.7 (s=3.63,n=20)
5
Step 4: Compute the t-statistic
where
( ) ( )21
2121
xxs
xxt
−
−−−=
µµ
2
2
1
2
21 n
s
n
ss pooledpooled
xx +=−
• Calculate the estimated standard error of thedifference
21
22
212
12 )()(
dfdf
sdfsdfs pooled +
+=
1918
63.3)19(57.2)18( 222
+
+=pooleds
37
18.13)19(61.6)18( +=
37
36.25098.118 +=
98.937
34.369==
6
• Compute the standard error (continued)
2
2
1
2
21 n
s
n
ss pooledpooled
xx +=−
20
98.9
19
98.921
+=−xxs 499.525. += 01.1=
( ) ( )21
2121
xxs
xxt
−
−−−=
µµ
396.01.1
4.
01.1
)7.261.27(==
−=t
*This alwaysdefaults to 0
•Calculate the t statistic
Step 5: Make a decision about thehypotheses
• The critical value for a two-tailed t-test withdf=37 (approx. 40) and α=.05 is 2.021
• Will we reject or fail to reject the nullhypothesis?
7
Assumptions for the Independent t-Test:
• Independence: Observations within eachsample must be independent (they don’tinfluence each other)
• Normal Distribution: The scores in eachpopulation must be normally distributed
• Homogeneity of Variance: The twopopulations must have equal variances (thedegree to which the distributions are spreadout is approximately equal)
Repeated Measures T-test
• Uses the same sample of subjects measuredon two different occasions (within-subjectsdesign)
• Use this when the population mean andstandard deviation are unknown and you arecomparing the means of a sample of subjectsbefore and after a treatment
• We are interested in finding out how muchdifference exists between subjects’ scoresbefore the treatment and after the treatment
8
DIFFERENCE SCORE (or D)• The difference between subjects’ scores before
the treatment and after the treatment• It is computed as x2-x1, where x2 is the subjects’
score after the treatment and x1 is the subjects’score before the treatment
• We use the sample of difference scores toestimate the population of difference scores (µD)
Example:
Does alcohol affect a person’s ability to drive?
A researcher selects a sample of 5 people and setsup an obstacle course.
– Each subject drives the course and the number ofcones he or she knocks over is counted.
– Next, the researcher has each subject drink a six-packof beer, then drive the course again, counting thenumber of cones each subject knocks over.
NOTE: Theory has shown that alcohol decreasesmotor and cognitive skills
9
Step 1: State the hypothesesH0: µD=0H1: µD ≠0
Step 2: Set the criterion• One-tail test or two-tail test?• α=?
• df=n-1• Critical value for t ?
Step 3: Collect sample data, calculate D• Once the difference scores are obtained, all further
statistics are calculated using these scores instead ofthe pretest / posttest or before / after scores
Before After DSubject (x1) (x2) (x2 - x1) 1 2 8 62 0 4 43 4 11 74 2 5 35 3 8 5
D∑ = 25
10
• Find the mean (average) difference score (D)
• The average difference of the number ofcones knocked down from before drinking toafter drinking is 5 cones. Remember, we arehypothesizing the difference to be zero.
D =D∑
n
55
25
5
53746==
++++=D
Step 4: Calculate the t-statisticFormula:
where andDs
Dt =
n
ss DD= D =
D∑n
estimated std.errorof mean diff. scores
the meandifferencescore
estimated std. deviationof diff. scores
11
• Compute the estimated standard deviation of thedifference scores (sD)
6 1 14 -1 17 2 43 -2 45 0 0
• The average deviation of the difference scores (D) about the meandifference score (D) is 1.58 cones
D D − D (D− D)2
SSD =10 σ = SSn -1 =1.58
€
sD =SSn −1
=104
=1.58
• Compute the estimated standard error of the meandifference scores
– The average deviation of the sample mean differencescores (D) from the population mean difference score(µD) is .707 cones
n
ss DD= 707.
5
58.1==
Ds
• Compute the t-statistic
Ds
Dt = 07.7
707.
5==t
12
Step 5: Make a decision• The critical value for a one-tailed t-test with
df=4 and α=.05 is 2.132
• Will we reject or fail to reject the nullhypothesis?
Advantages and Disadvantages of theRepeated Measures t-Test:
Advantages:• Controls for pre-existing individual
differences between samples (because only 1sample of people are being used
• More economical (fewer subjects are needed)
Disadvantages:• Subject to practice effects - the subjects are
performing the measurement task (i.e.driving the obstacle course, taking an exam)twice - scores may improve due to thepractice
13
Assumptions of the Repeated Measures t-Test:
Independent Observations: The scores frombefore and after the treatment must not berelated (no practice effects)
Normal Distribution: The population ofdifference scores must be normallydistributed
Summary of HypothesisTesting through t-statistic
• We have looked at four inferential statistics:– z-score statistic– single sample t-statistic– independent samples t-statistic– repeated measures t-statistic, or matched subjects
• the generic formula for these statistics is:
z or t = sample statistic - population parameterstandard error
14
Summary of HypothesisTesting through t-statistic
• z-score statistic compares a sample to apopulation when the population s.d. is known
• t-statistic compares a sample to a populationwhen the population s.d. is unknown
• independent samples t-statistic compares 2independent samples
• repeated measures t-statistic compares 1sample measured on 2 occasions