making estimations statistics for the social sciences psychology 340 spring 2010
TRANSCRIPT
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