Post on 29-Dec-2015
Making estimations
Statistics for the Social SciencesPsychology 340
Spring 2010
PSY 340Statistics for the
Social Sciences
Statistical analysis follows design
• Are you looking for a difference between groups?
• Are you estimating the mean (or a mean difference)?
• Are you looking for a relationship between two variables?
PSY 340Statistics for the
Social SciencesEstimation
• So far we’ve been dealing with situations where we know the population mean. However, most of the time we don’t know it.
μ = ?
• Two kinds of estimation– Point estimates
• A single score
– Interval estimates• A range of scores
PSY 340Statistics for the
Social SciencesEstimation
μ = ?
Two kinds of estimation– Point estimates
– Interval estimates
Advantage Disadvantage
A single score
A range of scoresConfidence of the estimate
Little confidence of the estimate “the mean is 85”
“the mean is somewhere between 81 and 89”
PSY 340Statistics for the
Social SciencesEstimation
• Both kinds of estimates use the same basic procedure– The formula is a variation of the test statistic formula (so far we
know the z-score)
zX
X
X
X
zX
(X
) X X
€
X
= X ± zX
(σX
)
PSY 340Statistics for the
Social SciencesEstimation
• Both kinds of estimates use the same basic procedure– The formula is a variation of the test statistic formula (so far we
know the z-score)
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.
Why the sample mean?
€
X
= X ± zX
(σX
)
PSY 340Statistics for the
Social SciencesEstimation
• Both kinds of estimates use the same basic procedure– The formula is a variation of the test statistic formula (so far we
know the z-score)
1) A test statistic value (e.g., a z-score)2) The standard error (the difference that you’d expect by chance)
Margin of error
€
X
= X ± zX
(σX
)
PSY 340Statistics for the
Social SciencesEstimation
– Step 1: 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
– What do I mean by “confident”?
» 90% confidence means that 90% of confidence interval estimates of this sample size will include the actual population mean
€
X
= X ± zX
(σX
)
• Both kinds of estimates use the same basic procedure
PSY 340Statistics for the
Social SciencesEstimation
– Step 1: 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
– Step 2: You take your “reasonable” estimate for your test statistic, and put it into the formula and solve for the unknown population
parameter.
€
X
= X ± zX
(σX
)
• Both kinds of estimates use the same basic procedure
PSY 340Statistics for the
Social Sciences Estimates with z-scores
Make a point estimate of the population mean given a sample with a X = 85, n = 25, and a population σ = 5.
X
X zX
(X
)
85 (0)5
25
85 So the point estimate is the sample mean
PSY 340Statistics for the
Social Sciences Estimates with z-scores
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.
X
X zX
(X
)
-1-2 1 2
95%
What two z-scores do 95% of the data lie between?
PSY 340Statistics for the
Social Sciences Estimates with z-scores
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.
What two z-scores do 95% of the data lie between?
So the confidence interval is: 83.04 to 86.96
X
X zX
(X
)
85 (1.96)5
25
86.96
From the table: z(1.96) =.0250
-1-2 1 2
95%
2.5% 2.5%
83.04
or 85 ± 1.96
PSY 340Statistics for the
Social Sciences Estimates with z-scores
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.
What two z-scores do 90% of the data lie between?
So the confidence interval is: 83.35 to 86.65
X
X zX
(X
)
85 (1.65)5
25
86.65
From the table: z(1.65) =.0500
83.35
or 85 ± 1.65
-1-2 1 2 -1-2 1 2
5% 5%
90%
PSY 340Statistics for the
Social Sciences Estimates with z-scores
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.
What two z-scores do 90% of the data lie between?
So the confidence interval is: 80.88 to 89.13
X
X zX
(X
)
85 (1.65)5
4
89.13
From the table: z(1.65) =.0500
80.88
or 85 ± 4.13
-1-2 1 2 -1-2 1 2
5% 5%
90%
PSY 340Statistics for the
Social Sciences Estimation in other designs
Confidence interval
sX =sn
Diff. Expected by chance
X = X ± (tcrit )(sX )
Estimating the mean of the population from one sample, but we don’t know the σ
How do we find this?
How do we find this?
Use the t-table
PSY 340Statistics for the
Social Sciences Estimates with t-scores
Proportion in one tail0.10 0.05 0.025 0.01 0.005
Proportion in two tailsdf 0.20 0.10 0.05 0.02 0.01: : : : : :5 1,476 2.015 2.571 3.365 4.0326 1.440 1.943 2.447 3.143 3.707: : : : : :
-1-2 1 2
Confidence intervals always involve + a margin of errorThis 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?
95%
95% in middle
2.5% 2.5%
so two tails with 2.5% in each
2.5%+2.5% = 5% or α = 0.05, so look here
PSY 340Statistics for the
Social Sciences
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.
Estimates with t-scores
What two critical t-scores do 95% of the data lie between?
So the confidence interval is: 82.94 to 87.06
X = X ± tcrit (sX )85 ± (2.064)5
25
⎛⎝⎜
⎞⎠⎟
87.06
From the table:
tcrit =+2.064
-1-2 1 2
95%
2.5% 2.5%
82.94
or 85 ± 2.064
df n−125 −124
95% confidence
Proportion in one tail 0.10 0.05 0.025 0.01 0.005
Proportion in two tails df 0.20 0.10 0.05 0.02 0.01 : : : : : :
24 1.318 1.711 2.064 2.492 2.797 25 1.316 1.708 2.060 2.485 2.787 : : : : : :
PSY 340Statistics for the
Social Sciences Estimation in other designs
€
sD
=sD
nD
Confidence interval
Diff. Expected by chance €
D
= D ± (tcrit )(sD
)
Estimating the difference between two population means based on two related samples
PSY 340Statistics for the
Social Sciences Estimation in other designs
Confidence interval
€
A − μB = (X A − X B ) ± (tcrit )(sX A −X B
)
Estimating the difference between two population means based on two independent samples
sXA −XB=
sP2
nA
+sP
2
nB
Diff. Expected by chance
PSY 340Statistics for the
Social Sciences Estimation Summary
Design Estimation (Estimated) Standard error
€
A − μB = (X A − X B ) ± (tcrit )(sX A −X B
) sXA −XB=
sP2
nA
+sP
2
nB€
sD
=sD
nD
€
D
= D ± (tcrit )(sD
)
sX =sn
X = X ± (tcrit )(sX )
€
X
= X ± zX
(σX
) X =σ
nOne sample, σ known
One sample, σ unknown
Two related samples, σ unknown
Two independent samples, σ unknown
PSY 340Statistics for the
Social Sciences
Statistical analysis follows design
• Questions to answer:• Are you looking for a
difference, or estimating a mean?
• Do you know the pop. SD (σ)?
• How many samples of scores?
• How many scores per participant?
• If 2 groups of scores, are the groups independent or related?
• Are the predictions specific enough for a 1-tailed test?
PSY 340Statistics for the
Social Sciences Design Summary
DesignOne sample, σ known, 1 score per sub
One sample, σ unknown, 1 score per
2 related samples, σ unknown, 1 score per - or –1 sample, 2 scores per sub, σ unknown
Two independent samples, σ unknown,1 score per sub
Independent samples-t
€
A − μB = (X A − X B ) ± (tcrit )(sX A −X B)
t =(XA −XB)−(A −B)
sXA −XB
sXA −XB=
sP2
nA
+sP
2
nB
Related samples t
€
D
= D ± (tcrit )(sD
)t =
D−D
sD
€
sD
=sD
nD
One sample t
X = X ± (tcrit )(sX )t =X −X
sX
sX =sn
One sample z
€
X
= X ± zX
(σX
)zX =X −X
X
X =σ
n
• Questions to answer:• Are you looking for a
difference, or estimating a mean?
• Do you know the pop. SD (σ)?
• How many samples of scores?
• How many scores per participant?
• If 2 groups of scores, are the groups independent or related?
• Are the predictions specific enough for a 1-tailed test?
PSY 340Statistics for the
Social Sciences Estimates with z-scores
Researchers used a sample of n = 16 adults. Each person’s mood was rated while smiling and frowning. It was predicted that moods would be rated as more positive if smiling than frowning. Results showed Msmile = 7 and Mfrown = 4.5.
Are the groups different?
• Questions to answer:• Are you looking for a
difference, or estimating a mean?
• Do you know the pop. SD (σ)?
• How many samples of scores?
• How many scores per participant?
• If 2 groups of scores, are the groups independent or related?
• Are the predictions specific enough for a 1-tailed test?
Researcher measures reaction time for n = 36 participants. Each is then given a medicine and reaction time is measured again. For this sample, the average difference was 24 ms, with a SD of 8. With 95% confidence estimate the population mean difference.
A teacher is evaluating the effectiveness of a new way of presenting material to students. A sample of 16 students is presented the material in the new way and are then given a test on that material, they have a mean of 87. How do they compare to past
classes with a mean of 82 and SD = 3?
t =D−D
sD
Related samples t
Related samples CI
€
D
= D ± (tcrit )(sD
)
1 sample z zX =X −X
X