So, let’s get
Unless
you real
live in
to the
myworld….
animated
world, Z-Testing
Welcome
to
with population σ
T
Myisn’t
World
Sucka!!
reality…
Confidence Intervals and Significance Testing in the World of T
Z-test for population mean
Don’t
Conditions we need for
my end up
like Caped
Z-procedure:
Data from a SRS and
Boy…
Do
a
TPopulation is approximately
Test Fooh!!
NORMAL (Large n)
Yeah,
Fooh…
But
you boy
have to
Look’s
like
fancy
That’s
assume the POPULATION
needs
a
taste
of
my
world!!
UNREALISTIC!!!!
standard deviation, σ.
Why a T-Test? What’s T-like?
Wow! A tailor made T for each
Check
Me
Out!!
sample!!
me break
ForLet
T- Testing,
we’re
 T is a density curve Let’s See
still
testing
this
downfor
 Symmetric about Zero, single peaked, “bell” shaped
What
Makes
Population
Means,
really
simply
 T’s variation depends
on sample
size but
T,
T…SAMPLE
webecome
only
need
 Remember, samples
less
variable
as they get larger
for
you…
data!!
 Degrees of Freedom
 T makes an adjustment for each sample size with by changing
the degrees of freedom
 Basically gives us a new T to work with for each sample size!!
T’s Statistic
Since you’re not using a Z distribution
anymore, you’ll need a different
statistic!! NO more Z!!
One Sample T Statistic
__
Sample
Standard
Dev.
x
s

n
Standard Error
With n-1 Degrees of Freedom
Reading the T-Table
Let’s Practice Using the
Table…
 Degrees of Freedom (df)
 Left hand column of chart
 Different T-Distribution for each sample size
 Larger the sample, the closer to Normal the T distribution
 T-Statistic
 Leads to the p-value or vice versa
 P-Value
 Area to the right of t
 Area to the left of –t
 2(P) for two-sided
Table Practice
What happens if you get a T
 Find
the t-statistic
for the
following:
that’s
not on your
table?
 1) 5 dof;Then
p = .05What?
(right) t = 2.015
 2) n = 22; p = .99 (left) t = 2.518
t = 1.333
 3) 80% CI; n = 18
 Find the p-value for the following:
 1) 5 dof; t = 3.365
 2) n = 12; t = 1.856
 3) n = 67; t = 2.056
p = .01
.025 < p <.05
.02 < p <.025
the t-statistic
YouNotice
will simply
say you’re is
p-value
limited
to certain2values
on
is BETWEEN
values!!
your table!!!
Confidence Intervals in
the T - Distribution
 s 
x t 

 n
Confidence Interval = __
With n-1
Degrees of
Freedom
*
This
Confidence
Ex:
90% CIInterval
for will
be approximately correct for
nlarge
= 10n.
The t* Pcomes
= .05from p
area on the right half of
= 1.833%...
the T
Confidence
Confidence in T’s Bling?!!
 Mr. T’s looking to get into the Golden Circle bling
business. He’s doing some research to find the
overall average weight of Golden Circle bling so he
can plan his gold needs. He bought a random
sample of 32 Golden Circles from different stores
and found they had an average weight of 4.6 lbs
with a standard deviation of .45 lbs.
IFind
am 95%
confident
the mean
Goldenfor
Circle
a 95%
Confidence
Interval
thebling
weight is
between
4.376
& 4.7625
lbs.
average
Golden
Circle
bling
weight
.
P = .025 for 95% CI
df = 31 (round down to 30 w/ chart)
 .45 
4.6  2.042

 32 
4.376 - 4.7624
Significance Testing for
Population Mean (unknown σ)
 With these tests you are given an alpha level
against which you test your p-value or you use
.05 if nothing’s given:
 p ≤ a – Reject the null; accept the Ha
 p > a – Fail to reject the null
Ha: µ > µ0
Ha: µ < µ0
Ha: µ ≠ µ0
Significance Testing for
Population Mean
 For T – Tests…
 Assumptions
Compare
Find
State
the
the
your
p-value
Hop-value
and
from
Hato
 SRS
the
in
t-statistic
symbols
and
w/size
n-1
the
specified
a,
and
make
Find
the
T-Statistic
 Approx
normal;
large
sample
(*show normality with
graphs)
your
degrees
decision
context
of freedom
in context
 Conditions
 Testing for Population Mean w/ unknown σ
We use the Same Basic
steps as in all Hypothesis
Testing
Investigating T’s Ice Cream
Mr. T has asked his factories to be sure the average
ounceage
of Mr.
icing on .01
a Mr.
Ice Cream cone is 4.5
Since
p isTbetween
andT.02,
ounces. which
Because
Mr. than
T is so
hard,
he has decided to take a
is less
.05,
I have
simple statistically
random sample
of 50 cones
from his Lexington
significant
evidence
factory to see
if they’re
falling
in line. His sample
reject
that the
Lexington
produces
an average
of 4.42
ounces with
Factory
is putting
an average
of a standard
deviation
ounces.
Does
Mr.and
T have enough evidence
4.5of.235
oz of T
on their
cones,
to prove
Lexington’s
been
skimpin’
their T and shut
SQUAK
them for
putting
on on
less.
them down? ( < 4.5oz)
4.42  4.5 (n – 1) df = 49
Ho: µ = 4.5oz
t
Ha: µ < 4.5oz
.235
(round down to
40 for table)
50
t = -2.4072
P is between .01 & .02…
Matched Pairs T-Testing
 Matched Pairs Test
 Match data values of different distributions based
on similar characteristics
 “Difference” between values is THE data
 Ho = µdiff = 0 [µdiff = (µ1 - µ2)]
 Ha = µdiff < or > or ≠
Male Rat Weight (g)
Female Rat Weight (g)
Difference (M – F) g
12
9.5
2.5
11
9
8
5
3
4
When to Use T
(or not to)
 Most important
assumption
that the
If your
HISTOGRAM
This
means
you’ll ishave
to data is from an
doesn’t
lookstep
is skewed,
add
1 more
to your
 Except for small samples sizes (then you need normality)
either
scrap the
t-test
Significance
Tests!!!
 Sample size
or< 15
talk about the
 Data
close toyour
normaldistribution
- USE T
Graph
for
questionability
of
the
 Data has outliers, skewness - DON’T USE T or state
samples results
less than 40 to
questionability of results!!
determine
level
of
 Sample Size ≥ 15
normality!!!
 USE T – except for strong outliers or strong skewness
SRS
 Large Samples (USE T)
 Even if it is skewed, you can use T as long as n ≥ 40
Practice
 Today’s Work
 Height Revisited
 11.1 Worksheet
 Ch 11 #’s 27 – 33
Get
To
Work