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# Paired Samples t tests.docx

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Ohio State University

Psychology

PSYCH 2220

dr.weiss

Fall

Description

T tests
Single Sample t-Test - compares sample mean to a specified value (population)
Independent Samples t-test - compares means of two independent samples
Paired samples t-test - comparing means of two dependent samples
Each case consists of a pair of data points - one data point from each of the two samples
Data points can be paired by:
Repeated-measures design - participants provide data at two points in time
o Longitudinal research - follows participants over time
o
Pre-post design - participants measured on dependent variable before and after an
intervention
Within subjects design- same participants measured in two different situations or under two
different conditions
o
Ex: measure study retention in silence and with music with same participants
Matched pairs - participants are grouped by the researcher into sets of two based on their being
similar on potential confounding variables
o Dean comparing GPAs of male and female students may match students based on IQ
into male/female pairs - can't argue that intelligence was a confounding variable if one
sex has higher GPA
Characteristics of Paired Samples Tests
Each condition in a dependent samples study is considered a "sample"
Adv. Over independent samples design - controls for individual differences - attributes that vary
from case to case
Same participants are in both groups, research can be sure two samples are comparable in
terms of background characteristics, more confident observed differences between groups on
dependent variable is due to independent variable and not confounding variable
Multiple names for the same test - paired samples t test, dependent samples t test, correlated
samples t test, related samples t test, matched pairs t test, within subjects t test, repeated
measures t test
Often used
o Pre-post design - where dependent variable is measured before and after the
independent variable is applied
o Controlling individual differences makes studies that use paired samples more powerful
than studies that use independent samples - means probability of being able to reject
null is higher - need a smaller sample size for paired samples than independent samples
Example: compare wound healing time. Matched pairs of women with same socioeconomic status and
age, but one person is a caregiver for someone with alzheimers. B/c they are matched on possible
confounding variables (age, socioeconomic status) can't say that those would be cause of differences.
Looking at stress of caring for another person as independent variable on dependent variable of wound
healing time.
9.2 Calculating the Paired Samples t Test
Study example: Six volunteers tested in temperature and humidity controlled chamber twice over a
period of two days. Chamber set at 76 degrees, once with low humidity, once with high humidity (condition counter-balanced). Participant's perceived temperature is the dependent variable measured.
Difference between perceived low-humidity and perceived high-humidity temperatures calculated,
means, and standard deviations.
Step 1. Pick a Test
Within-subjects design
Sample of people measured in two different conditions
Two-dependent samples
Comparing means of dependent samples - paired samples t-test
Step 2. Check the Assumptions
Random sample: sample is random from population Robust to
violation
Independence of observations: each case within a group or condition Not robust to
is independent of the other cases in that group or condition violation
Normality - population of difference scores is normally distributed Robust to
violation
Random - no, convenience sample
Observations are independent within a sample - refers to independence within a sample, not between
samples - same case in both samples or conditions, two samples are NOT independent. But each person
is tested individually and each person is only tested once in each condition - so Independence of
observations I NOT violated.
Normality - that in the larger population, difference scores are normally distributed - difference scores
are needed to calculate paired-samples t-test. 6 is a small sample size for making decisions about parent
population, based on previous research, she proceeds.
Calculation for Difference Score (D)
D = X1 - X2
D = difference score being calculated
X1 = a case's score in condition 1
X2 = a case's score in condition 2
Does not matter which value is subtracted from the other, as long as same order is followed for all
cases.
Step 3: List the Hypotheses
Statements about populations
Same as they were for independent samples t-test
Null says population means are the same
Alternative says two population means differ - won't indicate whether difference is large/small,
positive/negative
Non-directional, two tailed
Two possibilities:
o
Low-humidity is perceived as hotter than high humidity
o High humidity is perceived as hotter than low humidity o
Ho:ulowhumidityuhighhumidity
o Ha:ulowhumidityuhighhumidity
If a prediction had been made before-hand that high humidity would lead to higher perceived
temperature, then it would be
o Ho:ulowhumidityuhighhumidity
o Ha:u < u
lowhumidity highhumidity
Step 4: Set the Decision Rule
Formulated same as for independent samples t test
Critical value based on:
o Number of tails
o How willing you are to make a Type I error (alpha level)
o How many degrees of freedom there are
Degrees of freedom
o df = N-1
o N = the number of pairs of cases
Critical value marks rare zone - where null hypothesis is rejected and common zone - where it is
not
When drawing on a chart - t=0
Step 5: Calculate the Test Statistic
Same general procedure as was used in independent samples t
Divide the difference between the two sample means by the standard error of the difference
Standard error of the difference is the standard error of the difference for difference scores
abbreviated SMD
Standard error of the difference for difference scores is the standard deviation of the sampling
distribution of difference scores
Calculating the Standard Error of the Difference for Difference Scores (S )
MD
SMD =
SMD = standard error of the difference for difference scores
SD= standard deviation(s) of the difference scores
N = number of pairs of cases
Calculate SDby doing the following equations using the dif

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