Download Introduction to t-test

INTRODUCTION TO T-TEST
Chapter 7
Hypothesis testing
• Critical region
• Region of rejection
• Defines “unlikely event” for H0
distribution
• Alpha (
)
• Probability value for critical region
• If set .05 = probability of result
occurred by chance only 5x out of 100
• Critical value (cv)
• Value of the statistic for alpha
• p-value
• Actual probability of result occurring
Review: z-test
• Hypothesis: Special training in reading skills will produce
change in scores.
• Reading skills for population: N(65, 15)
• Treatment group: N = 25; M = 70
• Is there evidence that training has an effect?
z
M 

N
Review: z-test
• State hypotheses
• H0: µ = 65
• H1: µ ≠ 65 (2-tailed)
• α = .05; so critical region = ?
• zcv = 1.96
• Calculate test statistic (z-test = ?)
M 
70  65
z
 1.67

15
N
25
• Make decision and state conclusion
z
• Fail to reject H0
• “No evidence that special training changed scores, z (n = 25) =
1.67, p = ns.”
Review: z-test
• BUT if expect training to improve scores…
• State hypotheses
• H0: µ = 65
• H1: µ > 65 (1-tail)
• α = .05; so critical region zcv = 1.645
• Calculate test statistic
70  65
z
 1.67
15
25
• Make decision and state conclusion
• Reject H0 : “Training significantly improved scores, z (n = 25) =
1.67, p < .05.”
Errors
Conclude there was
an effect when there
actually wasn’t – the
risk of that is
Reject H0
Experimenter’s
Decision
Retain H0
Actual situation
NO Effect
Effect
H0 True
H0 False
Type I
error
Correct
decision
Correct
decision
Type II
error
Conclude there wasn’t
an effect when there
actually was an effect –
also called
Type I and Type II errors
• Type I: Incorrectly conclude significant difference
• Conclude treatment has an effect but really doesn’t
• Type II: Miss a significant result
• Conclude no effect of treatment when it really does
• Which is worse error to make?
• Examples:
• Law:
• Type I: Jury says guilty when innocent
• Type II: Jury says innocent when guilty
• Medicine:
• Type I: Doctor says cancer present when isn’t
• Type II: Doctor says no cancer when it is there
• Answer: it depends!
Setting your alpha level
• Lower alpha to
minimize chance of
Type I error
• But, then increase
chance of Type II
error!
Concerns with Alpha
• All-or-none decision
• Reject or accept null hypothesis
• Alpha (criteria) is set arbitrarily
• Null hypothesis logic is artificial
• No such thing as “no effect”
• Doesn’t give size of effect
• p-value is chance of occurrence
• Cannot say “very significant”!
• Sample size changes p-value (probability of rejecting H0)
Statistical Power
• What is the probability of making the correct decision??
• If treatment effect truly exists either…
• We correctly detect the effect or…
• We fail to detect the effect (Type II error or )
• So, the probability of correctly detecting is 1 • Power: probability that test will correctly reject null hypothesis
(i.e. will detect effect)
• Power depends on:
• Size of effect
• Alpha level
Reject H0
• Sample size
-3
-2
-1
0-3
1-2
2-1
30
1
2
3
Small
effect size
Large
effect size
Small
sample
size ->
large SD
Large
sample
size ->
small SD
Concerns with z-test
• Assumptions
• Must have a normal distribution
• Must know population mean and deviation
• Population mean (µ)
• Population standard deviation (σ)
• If don’t know above info or can’t make
assumptions need to use other statistics!
• Use “statistics” to make inferences
• Inferential statistics test: t-test
T-test: Estimates
• We’ve been using the z-score or z-test
formulas…
Z
• Where
X 
M=

OR
Z
M 
M
Z
M 

N
standard error of the mean
• Instead…
• Use t-test
• Where M is a sample mean
• Where µ can be the population mean or
hypothetical H0 mean (i.e. chance or 0)
• Where sM = estimated standard error
M 
t
sM
OR
s
n
s2
n
From z to t
• We use the “one-sample t-test” when we don’t
know .
z
M 


n
t
M 
s2
n
OR
M 
t
s
N
• Use the sample variance instead of population SD
parameter ( )
• And, use hypothesized µ from the H0
t-distribution
•
“Student’s t-distribution”
similar in shape to N(0,1)
•
•
•
Symmetric around 0
Spread of t is greater
than N(0,1)
As N increases, curve
approaches N(0,1)
•
Different distributions
for “df” or n -1
•
•
“Degrees of freedom”
Table A.3 pp384
t table
• Is the t* (tcv) at
or higher than t
(statistic) given
df and α level
selected?
Finding the Critical t*
• Find the df in the left column
• Go across top to find the selected a level.
• Find the critical value, t*, for the df and a.
• If the absolute value of t is greater than or equal
to t* then the test is significant at the a level.
• *Not all df are provided, so use the smaller df
(larger t*) for a more conservative estimate.
Eating behaviors
• “ATE”: positive attitude toward eating scale
• 3-point scale: -1 neg, 0 neutral, +1 positive
• 5 items: Eating desserts, snacks, etc.
• Minimum: -5; Maximum: +5
• Null hypothesis
• ATE = 0
• Alternative hypothesis
• ATE ≠ 0
• What values are needed to calculate the t-test?
•N
• Sample M and SD
t-test:
M 
t
sM
• N = 40 women
• ATE results: M = 0.525, SD = 2.16
• H0: ATE = 0
• Calculate t-test:
0.525  0 0.525
t

 1.5
2.16
0.35
40
• 1.5?? Look up in t-table tcv
• Not every df so use df = 30:
• tcv = 2.042 = p = .05, 2-tailed
• Does not exceed critical value, so not significant
• Reporting a t-statistic: t(df) = value, p = value
• Women were found to have a neutral feeling toward eating, t(39) = 1.5,
p = ns.
Eating behaviors
• Women rated their feelings toward eating on a 3-point
scale.
• Participants’ average response (M = 0.5, SD = 2.16)
was not found to be significantly different from a neutral
rating of zero, t(39) = 1.5, p = ns.
• The results suggest that women do not have an overly
positive or negative feeling toward eating.
Confidence intervals
• CI: % confident that interval contains population
mean (µ)
• % is determined by researcher (e.g. 85, 90, 95%)
• Formula for z-test and t-test:
• CI = M +/- z*(σM)
• CI = M +/- t*(sM)
• Where z* or t* is the critical value
• Example:
• CI = 86 +/- 1.96(1.7) = 82.67 to 89.33
• 95% confident pop mean in this range
Spatial map study
Spatial Map Study
• % correct: academic, athletic, residence halls, social, administrative,
parking locations on campus
• H0 = 50% correct, H1 > 50% correct (1-tailed)
• Information: N = 11, M = 0.36, SD = 0.15
M 
t
s
N
t
.36  .50  0.14

.15
.045
11
t = -3.11
• df 10; for α=.05 (1 tail): t* or tcv= 1.812
(tcv = 3.169 for p = .0005 (1tail))
Participants remembered significantly less than 50% of the campus
map (M = .36), t(10) = -3.11, p < .01.
CI = M +/- t*(sM) = 0.36 +/- 1.812 (.045) = 0.68 to 0.28
95% confident that true mean falls within that range
Lateralization in Perception of Emotion
• Two chimeric faces – which one is
happier (or younger example)?
• Emotion is processed in the right
hemisphere
• Dependent variable:
• Total score: -36 to +36
• where 0 = no lateralization
• What are the hypotheses?
• Null hypothesis: Total score = 0
• Alternative hypothesis?
Total score ≠ 0
t-test example
• Lateralization study (N = 173)
• H0: totscore = 0
• Totscore M = -11.7, SD = 16.6, S2 = 276
• t-test: -11.7 – 0
=
sqrt(276/173)
-11.7
1.26
=
- 9.28
• -9.28?? Look up tcv in table…
• Use 120: tcv = 2.576 = p = .005
• Participants show a significant lateralization to detect emotion
using the right hemisphere, t(172) = 9.28, p < .005
• CI = M +/- t*(sM) = -11.7 +/- 2.576 (1.26) = -8.45 to -14.95
Lateralization results
• Participants demonstrated a statistically
significant lateralization effect, t(172) = -9.28,
p < .05. Emotion was more influenced by the
right hemisphere (M = -11.7, CI (.95) = -8.45
to -14.95), as opposed to what would be
expected by chance (M = 0).