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PSY 307 – Statistics for the
Behavioral Sciences
Chapter 11-12 – Confidence
Intervals, Effect Size, Power
Point Estimates

The best estimate of a population
mean is the sample mean.


When we use a sample to estimate
parameters of the population, it is
called a point estimate.
How accurate is our point estimate?

The sampling distribution of the mean
is used to evaluate this.
Confidence Interval

The range around the sample mean
within which the true population
mean is likely to be found.



It consists of a range of values.
The upper and lower values are the
confidence limits.
The range is determined by how
confident you wish to be that the
true mean falls between the values.
What is a Confidence Interval?

A confidence interval for the mean
is based on three elements:




The value of the statistic (e.g., the
mean, m).
The standard error (SE) of the measure
(sx).
The desired width of the confidence
interval (e.g., 95% or 99%, 1.96 for z).
To calculate for z: m ± (zconf)(sx)
Levels of Confidence


A 95% confidence interval means
that if a series of confidence
intervals were constructed around
different means, about 95% of
them would include the true
population mean.
When you use 99% as your
confidence interval, then 99%
would include the true pop mean.
Demos


http://www.stat.sc.edu/~west/java
html/ConfidenceInterval.html
http://www.ruf.rice.edu/~lane/stat_
sim/conf_interval/
Calculating Different Levels

For 95% use the critical values for z
scores that cutoff 5% in the tails:


533 ± (1.96)(11) = 554.56 & 511.44
where M = 533 and sM = 11
For 99% use the critical values that
cutoff 1% in the tails:

533 ± (2.58)(11) = 561.38 & 504.62
Sample Size

Increasing the sample size
decreases the variability of the
sampling distribution of the mean:
Effect of Sample Size

Because larger sample sizes
produce a smaller standard error of
the mean:


The larger the sample size, the
narrower and more precise the
confidence interval will be.
Sample size for a confidence
interval, unlike a hypothesis test,
can never be too large.
Other Confidence Intervals

Confidence intervals can be
calculated for a variety of statistics,
including r and variance.


Later in the course we will calculate
confidence intervals for t and for
differences between means.
Confidence intervals for percents or
proportions frequently appear as
the margin of error of a poll.
Effect Size

Effect size is a measure of the
difference between two populations.



One population is the null population
assumed by the null hypothesis.
The other population is the population
to which the sample belongs.
For easy comparison, this difference
is converted to a z-score by dividing
it by the pop std deviation, s.
Effect Size
Effect Size
X1
X2
A Significant Effect
Effect Size
X1
Critical Value
X2
Critical Value
Calculating Effect Size


Subtract the means and divide by
the null population std deviation:
Interpreting Cohen’s d:



Small = .20
Medium = .50
Large = .80
Comparisons Across Studies


The main value of calculating an
effect size is when comparing
across studies.
Meta-analysis – a formal method for
combining and analyzing the results
of multiple studies.

Samples sizes vary and affect
significance in hypothesis tests, so test
statistics (z, t, F) cannot be compared.
Probabilities of Error

Probability of a Type I error is a.


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Most of the time a = .05
A correct decision exists .95 of the time
(1 - .05 = .95).
Probability of a Type II error is b.


When there is a large effect, b is very
small.
When there is a small effect, b can be
large, making a Type II error likely.
When there is no effect…
a = .05
Hypothesized
and true
distributions
coincide
Sample means
that produce a
type I error
COMMON
1.65
.05
Effect Size and Distribution
Overlap

Cohen’s d is a measure of effect
size.


The bigger the d, the bigger the
difference in the means.
http://www.bolderstats.com/gallery/normal/cohenD.html
Power

The probability of producing a
statistically significant result if the
alternative hypothesis (H1) is true.


Ability to detect an effect.
1- b (where b is the probability of
making a Type II error)
Small Effects Have Low Power
Effect Size
X1
X2
Power
Critical value
Large Effects Have More Power
Power
Effect Size
X1
Critical Value
X2
Critical Value
Calculating Power


Most researchers use special
purpose software or internet power
calculators to determine power.
This requires input of:

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Population mean, sample mean
Population standard deviation
Sample size
Significance level, 1 or 2-tailed test
http://www.stat.ubc.ca/~rollin/stats/ssize/n2.html
Sample Power Graph 1
Sample Power Graph 2
How Power Changes with N

WISE Demo

http://wise.cgu.edu/powermod/exercise1b.asp
Effect of Larger Sample Size
Larger samples
produce smaller
standard deviations.
Smaller standard
deviations mean less
overlap between two
distributions.
b Decreases with Larger N’s
Note: This is for an effect in the negative direction
(H0 is the red curve on the right).
Increasing Power




Strengthen the effect by changing
your manipulation (how the study is
done).
Decrease the population’s standard
deviation by decreasing noise and
error (do the study well, use a
within subject design).
Increase sample size.
Change the significance level.