Transcript Standard Error of the Mean N 0.4 0.3 0.2 0.1 0.0 SEM • This equation implies that sampling error decreases as sample size increases. • This is important because it suggests that.

Standard Error of the Mean
N
0.4
0.3
0.2
0.1
0.0
SEM
• This equation implies that
sampling error decreases as
sample size increases.
• This is important because it
suggests that if we want to
make sampling error as
small as possible, we need
to use as large of a sample
size as we can manage.
0.5
0.6
SEM x 

0
20
40
60
N
80
100
Standard Error of the Mean
• The SE provides a useful way to quantify sampling error.
• It is useful because it allows us to represent the amount of
sampling error associated with our sampling process—how
much error we can expect on average.
• Example: If we are sampling (N = 7) from a population in
which mu is 3.6 and sigma is 1.36, then the SE will be
approximately .50. We expect, on average, observed
sample means of 3.6, but, when we’re wrong, we expect to
be off by about .5 points, on average.
Confidence Intervals
• Technically, the SE is also called a 68% confidence
interval.
• Recall that the distribution of sampling errors resembles a
normal curve. We learned earlier, in the days of z-scores,
that 68% of scores fall between –1 and + 1 standard
deviations from the mean.
• By extension, then, we can say that 68% of sample means
will fall between – 1 SE and + 1 SE.
0.4
The area under a normal curve
0.2
0.3
Mu = 3.6
SE = .5
0.1
34%
34%
14%
14%
0.0
2%
1.6
2.1
2%
2.6
3.1
3.6
Sample Mean
4.1
4.6
5.1
5.6
68% Confidence Intervals
• Thus, in forward inference, a 68% confidence interval
marks the bounds within which 68% of the sample means
will fall, given certain facts about the population and the
sampling process.
– lower bound of the CI:  – SE
– upper bound of the CI:  + SE
95% Confidence Intervals
• Sometimes people use 95% confidence intervals.
– lower bound:  – 1.96  SE
– upper bound:  + 1.96  SE
0 .6
0 .4
0 .2
3.6 – 1.96  .5 = 2.62
upper bound:
3.6 + 1.96  .5 = 4.58
0 .0
De n sity
lower bound:
2
3
4
5
Sample Means
n=7
mean of sample means = 3.6
SD of sample means = .5
6
Thus, the 95% confidence interval
ranges from 2.62 to 4.58. Under the
sampling conditions described, 95%
of the time we will observe sample
means between 2.62 and 4.58.
Backward Inference
• What we’ve been describing is a variety of inferential
statistics that I call “forward inference.”
• That is, given certain facts of the population and sampling
conditions, we can make inferences about the statistical
properties of samples that might be observed.
Backward Inference
• In many applied research situations, we don’t know the
facts of the population.
• We know the sampling conditions (e.g., random sample of
size 100) and the facts of the sample (i.e., the sample mean
and standard deviation).
• How do we make inference about the properties of the
population from this information?
Chart contrasting forward and backward inference
Kind of inference Population
values
Sample values
Sampling
conditions
Forward inference known (e.g.,
population mean;
population
variance)
inferred (e.g.,
sample mean;
sample variance)
known (e.g.,
sample size;
random sampling)
Backward
inference
known (e.g.,
sample mean;
sample variance)
known (e.g.,
sample size;
random sampling)
inferred (e.g.,
population mean;
population
variance)
Backwards Inference
• Backwards inference is based on a critical assumption:
That the a priori likelihood of all possible population
means are equivalent.
• This assumption tends to be ignored in undergraduate
statistics textbooks (including yours) and in practice.
We’ll discuss this problem in more detail in a few weeks
(Bayesian statistics), but, for now, let’s explore backwards
inference as it typically applied.
Example
• Let’s begin with a situation in which we take a random
sample of 5 people from a population (1, 2, 3, 4, 4, 5, 4, 6,
3, 4, mu = 3.6, sigma = 1.36), compute a sample mean,
and, based on this, try to infer the mean for the population
from which our sample was drawn.
• We get the following sample scores: 4, 4, 6, 4, 3 (M = 4.2,
SD = .98)
• Important assumption: Prior to observing our sample
mean, all possible population means are equally likely. In
other words, a mu of 3.6 is just as likely as a mu of 4.2.
Means: Backward Inference
• Under this assumption, we can simply flip or reverse the
forward inferential logic used earlier.
• The “best guess” for the population mean is the sample
mean (4.2) because we know that the sample mean is
unbiased. If the population mean is the expected value of a
sample mean in forward inference (i.e., the best guess as to
what sample mean we might observe), then the sample
mean is the best guess about the population mean in
backward inference.
ˆ x  M x
Variances: Backward Inference
• For variances, we can again flip the forward inferential logic we used
earlier.
• Recall that the population variance is typically a bit larger than the
expected sample variance. Hence, the variance we observe in our
sample is a little too small to be a good estimate of the population
variance.
• We can correct it by making it a little bit larger by dividing by N – 1
instead of N. (This is exactly how much we need to correct the sample
variance to make it an unbiased estimate of the population variance.)
X  M 


2
ˆ
2
x
N 1
.98 (unadjusted sample mean)
vs. (4.8/4) = 1.2 (adjusted
sample mean)
Confidence Intervals: Backward Inference
• Similarly, we can reconceptualize confidence intervals
within the framework of backwards inference.
• In our example with forward inference, we said that, under
certain sampling conditions, samples drawn from our
population will have means that fall between 3.1 and 4.1
95% of the time.
Confidence Intervals: Backwards Inference
• In backwards inference, we say that this method of
constructing intervals will capture the population mean
95% of the time.
– lower bound: Mx – 1.96  SE
– upper bound: Mx + 1.96  SE
– (We have to use an estimate the population standard
deviation because we do not know it. Recall that the
sample SD is biased, so we have to used the corrected,
unbiased formula to estimate it correctly.)
SE 
ˆ
N
ˆ 
2


X

M

x
N 1
1.2
 .53
5
– lower bound: 4.2 – 1.96  .53 = 3.16
– upper bound: 4.2 + 1.96  .53 = 5.23
– Thus, we say that there is a 95% chance that the
population mean falls between these two values: 3.16
and 5.23.


In forward inference, we can say that
this method of constructing intervals
will capture the observed sample
mean 95% of the time.
samples
In backwards inference, we can say
that this method of constructing
intervals will capture the population
mean 95% of the time.
• Sometimes in backwards inference the CI is interpreted as
implying that there is a 95% chance that the population
mean falls within this interval.
• This is logic is problematic for reasons we’ll discuss later
in the course. But, for now, if we assume that there is no
particular value of the population mean that is more likely
prior to our study, then this logic is sensible.