Download 07 - Inferential Statistics

Inferential Statistics
Normall Curve
• Specific
p f bell-shaped
p curve that is
unimodal, symmetric, and defined
mathematically.
– Ubi
Ubiquitous
i
– Helps clarify the probability of particular
events
– Let’s see an example with a coin toss
Basis off Inferential
f
l Statistics
1. The approximate shape of the normal
curve is everywhere.
2 The bell shape of the normal curve may
2.
be translated into percentages
(
(standardization).
)
3. A distribution of means produces a bellshaped curve even if the original
distribution of individual scores is not
bell-shaped, as long as the means are
from sufficiently large samples (central
li i theorem)
limit
h
)
Sample
l Size and
d Distributions
b
Standardization
d d
• Comparing z scores:
– Statistics Exam
• Mean = 78, SD = 6, Your Score = 88
• z = 1.67
– Cognition Exam
• Mean = 76, SD = 5, Your Score = 85
• z = 1.8
Transforming z scores into
Percentiles
Guinness and
d Normall Curves
• 1900s: W.S. Gosset hired for
qualityy control
q
– Brewing and bottling both
require
q
ap
precise amount of
yeast
– Can’t test every bottle and every
barrel
• Need a sample!
Centrall Limit Theorem
h
• A distribution of sample means
approaches
pp
a normal curve as the
sample size increases.
– Even when the original distribution of
scores is not normally distributed!
Sampling Distribution of Means
• A distribution composed of many
means that are calculated ffrom all
possible samples of a given size, all
taken ffrom the same p
population.
p
– Less variability than the actual scores.
– Why does this distribution have less
variability?
y
Sampling Distribution of Means
• We cannot use the standard deviation
for this distribution.
distribution
Standard
d d Error
• Standard deviation of a distribution of
sample
p means.
• New Symbols:
σM
σM =
μM
σ
N
Quick
k Review
1. As sample size increases, the mean of
the sampling
p g distribution of means
approaches the mean of the
population
p
p
of individual scores
Quick
k Review
2. The standard error is smaller than the
standard deviation and as sample
p size
increases, standard error decreases.
Quick
k Review
3. The shape of the distribution of
means will approximate
pp
normal if the
distribution of the population of
individual scores is normal or if the
size of each sample that comprises it
is sufficientlyy large,
g , at least 30.
–
Central Limit Theorem
Back
k to z Scores
• Remember that we are often working
with samples,
p , not entire p
populations.
p
– We need a new way to create z scores
(
M − μM )
z=
σM
– z statistic: How many standard errors a
sample mean is from the population mean
A Practicall Example
l
• We conduct an IQ test in a class of 40
and find that the class average
g is 106.
– Population: Mean = 100, SD = 15
– How does our class average measure up
when compared
p
with this p
population?
p
A Practicall Example
l
(
M − μM )
z=
σM
μ M = μ = 100
σM =
σ
15
15
=
=
= 2.372
N
40 6.325
A Practicall Example
l
(
M − μ M ) (106 − 100 )
z=
=
= 2.53
σM
2.372
To convert this z statistic to a percentage,
percentage
consult a z table (at the back of the book)