Chapters 22, 24, 25
Notes Chapter 22
Comparing Two Proportions
• Use when we want to compare the
proportions from two groups of categorical
data:
• Conditions:
*Both are SRS’s from populations of interest
*n1  10% of population 1
n2  10% of population 2
*n1p1, n1(1-p1), n2p2, n2(1-p2) are all  10
For Confidence Intervals:
(we do not assume that p1 = p2 therefore we
do not pool for CI's)
( pˆ1  pˆ 2 )  z *
pˆ1 (1  pˆ1 ) pˆ 2 (1  pˆ 2 )

n1
n2
*choose the non-pooled formula from chart,
calculator will make correct choice automatically
For Confidence Intervals:
• Our confidence interval statements reflect
the true difference in the two proportions
(in context).
For Significance/Hypothesis
Testing
Ho: p1 = p2
Ha: p1  p2 or
p1 < p2
or
p1 > p2
(Since our null states that p1 = p2, we
assume this to be true until proven
otherwise, therefore we automatically pool
here)
z
pˆ1  pˆ 2
1 1
pˆ (1  pˆ )(  )
n1 n2
Where
x1 x2
pˆ 
n1 n2
*choose the pooled formula from the
chart, the calculator will
automatically pool
We rely on the same p-values and alphas to make
our conclusions.
Notes Chapter 24
Comparing Two Means
• When our data is quantitative, then we are
either looking for an interval that contains
their differences or we are comparing
them to one another.
Notes Chapter 24
Comparing Two Means
• Two-Sample t Interval:(Quantitative Data)
s s
x  x   t  
n n
2
1
1
2
2
2
1
2
*choose the non-pooled formula from the formula sheet
*df = there is a nasty formula – ick!
But …calculator gives you this 
Notes Chapter 24
Comparing Two Means
Two-Sample t-test
t
x1  x 2
s12 s 2 2

n1 n 2
*choose the non-pooled formula from the formula
sheet
*df = there is a nasty formula – ick!
But …calculator gives you this 
Conditions:
*The two groups we are comparing must be independent
of each other.
*Both n1 & n2 must be SRS from the populations of interest
*Both n1 & n2 must be < 10% of their respective populations
of interest
*Sample size restriction must be met:
If n1 + n2 < 15, do not use if outliers or severe skewness
are present
If 15 ≤ n1 + n2 < 30, use except in presence of outliers
If n1 + n2  30, sample is large enough to use regardless of
outliers or skewness by CLT
Two-Sample t Procedures:
• We do not pool t-tests.
• Pooling assuming equal σ values. Since
σ1 and σ2 are both unknown, why would
we assume they are equal?
• You must tell the calculator you do not
want to pool. “Just say No”.
• When we interpret confidence intervals we
say “the true difference between two
means” (not the mean difference).
Notes Chapter 25
Paired Samples
• Often we have samples of data that are
drawn from populations that are not
independent. We have to watch carefully
for those!! We can not treat these
samples as two independent samples, we
must consider the fact that they are
related.
• So…what do we do?
Notes Chapter 25
Paired Samples
• If we have two samples of data that are
drawn from populations that are not
independent, we use that data to create a
list of differences. That list of differences
then becomes our data and we will not
use the two individual lists of data
again.
Notes Chapter 25
Paired Samples
• Once we have the list of differences,
everything else is like 1 sample
procedures from the last unit.
• We will treat that list of differences as
our data.
• That list must meet the conditions for a
single sample t-distribution. We will use
the t-interval or the t-test on this list of
differences depending on the question.
• When we interpret our confidence interval
or make our conclusion we will be talking
about the “mean of the differences.”
Reminders for Single Sample t
Conditions:
*Data is from an SRS of size n from the population of
interest
* Sample size < 10% of Population size
*Sample is approx. normal/no outliers or large n
(see guidelines below)
• n < 15: Use t-procedures if the data is close to normal.
If severe skewness or outliers are present, do not use t.
• 15 ≤ n < 30 : The t-procedures can be used except in the
presence of outliers
• large n: The t-procedures can be used even for clearly
skewed distributions when the sample is large (n  30)
by CLT.
The One-Sample t-Procedure:
• A level C confidence interval for  is:
 s 
X t

 n
where t* is the critical value from the t
distribution based on degrees of freedom.
• To test the hypothesis: Ho:  = o and
Ha:  > o
Ha:  < o
Ha:   o
• Calculate the test statistics t and the p
value.
• We make the same conclusions based on
p-values and alpha.
• For a one-sample t statistic:
X 
t
s/ n
has the t distribution with n – 1 degrees of
freedom.
• Just remember that all the variables
represent the “mean difference” for your
populations.