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Reasoning in Psychology
Using Statistics
Psychology 138
2013
• Describe the typical college student
μ= ?
– If we can’t measure the entire population of
students, how do we get the population means
for these variables?
Age
μ= ?
hours of studying
per week
μ= ?
μ=?
hours of sleep
per night
Reasoning in Psychology
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pizza consumption
– If we can’t measure the entire population of
students, how do we get the population means
for these variables?
μ= ?
• Estimate it based on what
we do know
– On information from a sample
• Two kinds of estimation
– Point estimates
• A single score
– Interval estimates
• A range of scores
X
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• Describe the typical college student
– Point estimates
– Interval estimates
“12 hrs”
“2 to 21 hrs”
“19 yrs”
“17 to 21 yrs”
Age
“8 hrs”
“4 to 10 hrs”
hours of sleep
per night
Reasoning in Psychology
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hours of studying
per week
“1 per wk”
“0 to 8 per wk”
pizza consumption
Estimate the number of people attending lecture today
How confident are you that your estimate is correct?
“Not real confident,
maybe 20%”
“50 students”
“Somewhere between
20 and 80 students”
Estimation
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“Fairly confident,
maybe 90%”
Kinds of estimation
Point estimate
“50 students”
Interval estimate
“Somewhere between
20 and 80 students”
Estimation
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Advantage
Disadvantage
A single score
Little confidence in
the estimate
Confidence in
the estimate
A range of scores
• Both kinds of estimates use the same basic procedure
– The formula is a variation of the test statistic formulas (we’ll start
with the z-score)
mX = X ± zX (s X )
Do some adding/subtracting
zX (s X ) = X - mX
Multiply both sides by
sX
zX =
Estimation
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X - mX
sX
This is what we
want to estimate
• Both kinds of estimates use the same basic procedure
– The formula is a variation of the test statistic formulas (we’ll start
with the z-score)
mX = X ± zX (s X )
Why the sample mean?
1)
It is often the only piece of evidence that we have, so it is our best
guess.
2)
Most sample means will be pretty close to the population mean, so we
have a good chance that our sample mean is close.
Estimation
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• Both kinds of estimates use the same basic procedure
– The formula is a variation of the test statistic formulas (we’ll start
with the z-score)
mX = X ± zX (s X )
Margin of error
1)
2)
A test statistic value (e.g., a z-score)
based on design (z or t) and level of confidence
The standard error (the difference that you’d expect by
chance)
Based on sample size (n) and population standard
deviation (σ)
Estimation
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• Finding the right test statistic for your estimate
mX = X ± zX (s X )
– You begin by making a reasonable estimation of what the z (or t)
value should be for your estimate.
• For a point estimation, you want what? z (or t) = 0, right in the
middle
mX
μ= ?
zX =
Raw scores
X - mX
sX
transform
z scores
Z=0
Estimation
Reasoning in Psychology
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• Finding the right test statistic (z or t)
mX = X ± zX (s X )
– You begin by making a reasonable estimation of what the z (or t)
value should be for your estimate.
• For a point estimation, you want what? z (or t) = 0, right in the middle
• For an interval, your values will depend on how confident you want to be in
your estimate
Actual population mean
– What do I mean by “confident”?
μ
» 90% confidence means that 90% of
confidence interval estimates of this
sample size will include the actual
population mean
9 out of 10 intervals contain μ
Estimation
Reasoning in Psychology
Using Statistics
• Finding the right test statistic (z or t)
mX = X ± zX (s X )
– You begin by making a reasonable estimation of what the z (or t)
value should be for your estimate.
• For a point estimation, you want what? z (or t) = 0, right in the middle
• For an interval, your values will depend on how confident you want to be in
your estimate
– Computing the point estimate or the confidence interval:
• Step 1: Take your “reasonable” estimate for your test statistic
• Step 2: Put it into the formula
• Step 3: Solve for the unknown population parameter
Estimation
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Make a point estimate of the population mean given a sample with a X =
85, n = 25, and a population σ = 5.
æ 5 ö
mX = X ± zX (s X ) = 85 ± (0)ç
÷ = 85
è 25 ø
sample mean serves
as the center
So the point estimate
is the sample mean
z (or t) = 0, right in
the middle
Estimates with z-scores
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sX =
s
n
Make an interval estimate with 95% confidence of the population mean
given a sample with a X = 85, n = 25, and a population σ = 5.
mX = X ± zX (s X )
What two z-scores
do 95% of the data
lie between?
95%
Estimates with z-scores
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Make an interval estimate with 95% confidence of the population mean
given a sample with a X = 85, n = 25, and a population σ = 5.
æ 5 ö = 86.96 What two z-scores
do 95% of the data
mX = X ± zX (s X ) = 85 ± (1.96)ç
÷
è 25 ø= 83.04 lie between?
So the 95% confidence interval is: 83.04 to 86.96
or 85 ± 1.96
z
:
1.8
1.9
2.0
:
0.02
:
.0344
.0274
.0217
:
0.03
:
.0336
.0268
.0212
:
0.04
:
.0329
.0262
.0207
:
0.05
:
.0322
.0256
.0202
:
0.06
:
.0314
.0250
.0197
:
From the table:
z(1.96) =.0250
2.5%
2.5%
95%
Estimates with z-scores
Reasoning in Psychology
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Make an interval estimate with 90% confidence of the population mean
given a sample with a X = 85, n = 25, and a population σ = 5.
æ 5 ö = 86.65 What two z-scores
do 90% of the data
mX = X ± zX (s X ) = 85 ± (1.65)ç
÷
è 25 ø= 83.35 lie between?
So the 90% confidence interval is: 83.35 to 86.65
or 85 ± 1.65
From the table:
z(1.65) =.0500
5%
5%
90%
Estimates with z-scores
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Make an interval estimate with 90% confidence of the population mean
given a sample with a X = 85, n = 4, and a population σ = 5.
æ 5 ö = 89.13 What two z-scores
do 90% of the data
mX = X ± zX (s X ) = 85 ± (1.65)ç ÷
è 4 ø = 80.88 lie between?
So the confidence interval is: 80.88 to 89.13
or 85 ± 4.13
From the table:
z(1.65) =.0500
5%
5%
90%
Estimates with z-scores
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• The size of the margin of error related to:
mX = X ± zX (s X )
– Sample size
• As n increases, the margin of error gets narrower (changes the
standard error)
– Level of confidence
• As confidence increases (e.g., 90%-> 95%), the margin of error gets
wider (changes the critical test statistic values)
Estimation
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• Two kinds of estimates that use the same basic procedure
– The formula is a variation of the test statistic formulas
Different Designs: Estimating the mean of the population
from one or two samples, but we don’t know the σ
m X = X ± zX (s X )
mX = [sample mean(s)] ± (tcrit )(estimated standard error )
Center/point estimate?
How do we find
How do we find this?
this?
Depends on the design
Use the t-table & your
Depends on the design
(what is being estimated) confidence level
sX
D
X
XA - XB
sD
Estimation in other designs
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sXA - XB
Confidence intervals always involve + a margin of error
This is similar to a two-tailed test, so in the t-table, always use
the “proportion in two tails” heading, and select the α-level
corresponding to (1 - Confidence level)
What is the tcrit needed for a 95% confidence interval?
so two tails with
2.5%+2.5% = 5% or
95% in middle 2.5% in each
α = 0.05, so look here
0.10
df
:
5
6
:
0.20
:
1,476
1.440
:
Proportion in one tail
0.05
0.025
Proportion in two tails
0.10
0.05
:
:
2.015
2.571
1.943
2.447
:
:
0.01
0.005
0.02
:
3.365
3.143
:
0.01
:
4.032
3.707
:
2.5%
Estimates with t-scores
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2.5%
95%
Estimating the difference between the population mean
and the sample mean based when the population
standard deviation is not known
Confidence
interval
Diff.
Expected by
chance
m X = X ± (tcrit )(sX )
s
sX =
n
Estimation in other designs
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Make an interval estimate with 95% confidence of the population mean
given a sample with a X = 85, n = 25, and a sample s = 5.
two critical tæ 5 ö = 87.06 What
scores do 95% of the
m X = X ± tcrit (sX )= 85 ± (2.064) ç
÷
è 25 ø = 82.94 data lie between?
df = n - 1 = 25 - 1 = 24
From the table:
tcrit =+2.064
0.10
df
:
24
25
:
0.20
:
1.318
1.316
:
Proportion in one tail
0.05
0.025
0.01
Proportion in two tails
0.10
0.05
0.02
:
:
:
1.711
2.064
2.492
1.708
2.060
2.485
:
:
:
So the confidence interval is: 82.94 to 87.06
or 85 ± 2.064
0.005
0.01
:
2.797
2.787
:
2.5%
2.5%
95%
Estimation in one sample t-design
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Estimating the difference between two population
means based on two related samples
Confidence
interval
mD = D ± (t crit )(sD )
Diff.
Expected by
chance
sD =
sD
nD
Estimation in related samples design
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•
Dr. S. Beach reported on the effectiveness of
cognitive-behavioral therapy as a treatment for
anorexia. He examined 12 patients, weighing
each of them before and after the treatment.
Estimate the average population weight gain for
those undergoing the treatment with 90%
confidence.
Differences (post treatment - pre treatment weights):
10, 6, 3, 23, 18, 17, 0, 4, 21, 10, -2, 10
Related samples estimation
mD = D ± (t crit )(sD )
sD =
D = (XA - XB ) Confidence level 90%
dfD = nD - 1 = 11
D
D=å
tcrit = ±1.796
nD
sD =
mD = 10 ± (1.796)(2.38)
CI(90%)= 5.72 to 14.28
SSD
nD - 1
SSD = å ( D - D )
Estimation in related samples design
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sD
nD
2
Estimating the difference between two population means
based on two independent samples
Confidence
interval
mA - mB = (X A - X B ) ± (tcrit )(sX
Diff.
Expected by
chance
sXA - XB
A -X B
)
sP2 sP2
=
+
nA nB
Estimation in independent samples design
Reasoning in Psychology
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•
Dr. Mnemonic develops a new treatment for
patients with a memory disorder. He randomly
assigns 8 patients to one of two samples. He then
gives one sample (A) the new treatment but not
the other (B) and then tests both groups with a
memory test. Estimate the population difference
between the two groups with 95% confidence.
X A = 44.5 X B = 50
sA = 7.19
sB = 9.13
Independent samples t-test situation
mA - mB = (X A - X B ) ± (tcrit )(sX
Confidence level 95%
df = nA + nB - 2 = 6
tcrit = ±2.45
mA - mB = 5.5 ± (2.45)(5.81)
CI(95%)= -8.73to 19.73
s XA - XB
A -X B
)
sP2 sP2
=
+
= 5.81
nA nB
s 2p
s df ) + ( s df )
(
=
2
A
A
2
B
B
dfA + dfB
Estimation in independent samples design
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• Notice that this interval includes zero
-8.73
19.73
0
• If we had instead done a hypothesis test with an
α = 0.05, what would you expect our conclusion
to be?
H0: “there is no difference between the groups”
- Fail to reject the H0
CI(95%)= -8.73to 19.73
Relating estimates to hypothesis tests
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Design
Estimation
One sample, σ
known
mX = X ± zX (s X )
One sample, σ
unknown
m X = X ± (tcrit )(sX )
Two related
samples, σ
unknown
mD = D ± (t crit )(sD )
Two independent
samples, σ
unknown
mA - mB = (X A - X B ) ± (tcrit )(sX
(Estimated) Standard error
sX =
n
s
sX =
n
sD
sD =
nD
)
A -X B
Estimation Summary
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s
sXA - XB
sP2 sP2
=
+
nA nB
• Practice computing and interpreting confidence
intervals
In labs
Reasoning in Psychology
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