# Pay Check Samples by nxj76964

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```									 Chapter 24:

Comparing Means
Comparing Two Means

 Population model           For independent random
parameter of interest is    variables, variances add.
the difference between  If we know the population
the means, 1   2 .       means
 12  2 2
 The statistic of interest     SD  y1  y 2       
n1    n2
is the difference in the
 If we estimate, using the sample
two observed means,         means
y1  y 2 .                         SE  y  y  
1   2
s12 s2 2

n1   n2
Comparing Two Means

 Confidence interval is call a two-sample
t-interval.
 The hypothesis test is called a two-
sample t-test.
 y  y   ME
1    2

ME  t  SE  y  y 
*
1   2
A Sampling Distribution for the
Difference Between Two Means
 When the conditions are    Modeled by a Student’s t-model
met, the standardized       with a number of degrees of
sample difference           freedom found with a special
between the means of two    formula.
independent groups,
 We estimate the standard error

t
          
y1  y 2   1  2    with

       
SE y1  y 2                       
SE y1  y 2 
s12 s2 2

n1 n2
Assumptions and Conditions

 Independence Assumption
   Randomization
 Surveys: representative random samples

 Experiments: randomized

   10% Condition
 Normal Population Assumption
   Nearly Normal Condition
 Check both samples.

 Draw pictures!

 Independent Groups Assumption
   Think about how the data were collected.
Two-sample t-interval

 When the conditions are met, find the confidence interval for
the difference between means of two independent groups.
Since the standard error of the difference is


SE y1 - y 2      
s12 s2 2

n1 n2
,

      
*

the interval is y1 - y2  tdf  SE y1 - y 2   
 The critical value depends on the particular confidence level C
that you specify and on the number of degrees of freedom,
which we get from sample size and a special formula.
Comparing Brand Name &
Generic Batteries
 L1: Brand Name    Find the interval that is likely
 L2: Generic        with 95% confidence to
 Plot               contain the true difference
G   B between the mean
AA batteries and the mean
batteries
Comparing Brand Name &
Generic Batteries
 Check the conditions:
   Independent groups assumption: batteries
manufactured by two different companies from
separate packages should be independent.
   Randomization: the batteries were selected at
random from those available for sale. This is not
exactly an SRS, but a reasonably representative
random sample. Since the batteries come in packs,
they may not be independent. Repeat the experiment
for several packages of batteries.
Comparing Brand Name &
Generic Batteries
 Check the conditions:              Histograms
   10%: the number of               Brand Name (L1)
sampled batteries are
certainly less than 10% of
all AA batteries
manufactured by the
companies.                         Generic (L2)
   Nearly Normal condition:
the samples are small, but
the histograms look
unimodal and symmetric.
Comparing Brand Name &
Generic Batteries
 State the sampling                     STAT TESTS 2-SampTInt
distribution model for the
statistic:
   Under these conditions, the
sampling model of the
difference in the sample means
can be modeled by a Student’s
of freedom.
   We will use a two-sample
t-interval.
Comparing Brand Name &
Generic Batteries
 Interpretation: tell what the confidence interval
means
   We are 95% confident that the mean useful life of the
generic batteries is between 2.1 minutes and 35.1 minutes
longer than the mean useful life of the brand-name
   If generic batteries are cheaper, there seems little reason
not to use them. If it is more trouble or costs more to buy
them, then you should consider whether the additional
performance is worth it.
Testing the Difference Between
Two Means
 Two-sample t-test for the difference between the means of
two independent groups:
   The conditions for the two-sample t-test for the difference between
the means of two independent groups are the same as for the two-
sample t-interval.
We test the hypothesis H O : 1   2   O , where
the hypothesized difference is almost always 0, using the

statistic t 
y   1       
 y 2  O
. The standard error is

SE y1  y 2     
        
SE y1  y 2 
s12 s2 2

n1 n2
.
Camera Price Offers

 State the null hypothesis:                  Check to plots:
   We want to know if people are               L1: Friend
more likely to offer a different
amount for a used camera when               L2: Stranger
buying from a friend or a
stranger.
   HO: The difference in mean price
offered to friends and the mean
price offered to strangers is zero:
F  S  0
   HA:The difference in mean price
is not zero:  F   S  0
Camera Price Offers

 Check the conditions:                  Check the conditions:
   Independent groups                     Nearly Normal condition:
assumption: randomizing the             Histograms of the two sets
experiment gives us                     of prices are unimodal and
independent groups.                     symmetric.
   Randomization condition: the           L1                L2
experiment was randomized.
Subjects were assigned to
treatment groups at random.
   10% condition: this is a
randomized experiment, so
this condition does not apply.
Camera Price Offers

 State the sampling distribution model of the
statistic:
   Because the conditions are satisfied, it is appropriate to
model the sampling distribution of the difference in the
means with a Student’s t-model.
   We will perform a two-sample t-test.
Camera Price Offers

 Calculate:               Draw:
   STAT TESTS
   2-SampTTests
Camera Price Offers

 Conclusion:
   The P-value tells us that if there were no difference in the
mean prices, the difference we have observed would
occur only 0.6% of the time. That’s too rare for most
people to believe, so we reject the null hypothesis and
conclude that people are likely to pay a friend for a used
camera a different amount than they would pay a
stranger.
   We may want to take special care not to pay too much
when buying an item such as this from a friend.
Pooled t-test

 If we are willing to assume that means’
variances are equal, we can pool the data
from the two groups to estimate the common
variance and make the degrees of freedom
formula much simpler.
 We are still estimating the pooled standard
deviation from the data, so we use Student’s
t-model, and the test is called a pooled t-test.
Pooled Variance t-test for the Difference
Between Two Independent Means
 The conditions for the pooled t-test for the difference between
two independent means are the same as for the two-sample t-
test with the additional assumption that the variances of the
two groups are the same.
We test the hypothesis H O : 1   2   O , where
the hypothesized difference is almost always 0, using the

statistic t 
 y  y    . The standard error is
1       2            O

SE
pooled y  y 
1         2

s2                s2
        
SE pooled y1  y 2 
pooled

n1

pooled

n2
.
Pooled Variance t-test for the Difference
Between Two Independent Means
 The pooled variance is:                  n1  1 s12   n2  1 s22 .
pooled 
s2
 n1  1   n2  1
 When the conditions are met and the null hypothesis is true,
this statistic follows a Student’s t-model with
 n1  1   n2  1 degrees of freedom.
         *
                      
The corresponding interval is y1 - y 2  tdf  SE pooled y1 - y 2 ,       
where the critical value t * depends on the confidence level
and is found with  n1  1   n2  1 degrees of freedom.
When to Pool?
 The advantage of the pooled method is greatest when the
samples are small.
   But this is when it’s hardest to check conditions.
   When the choice between two-sample t and pooled-t methods make a
difference (sample size is small), the test for whether the variances are
equal hardly works at all.
 In a randomized comparative experiment, we know that each
treatment group is a random sample from the same population.
   So each treatment group begins with the same population variance.
   In this case, assuming equal variances is the same as assuming that the
treatment doesn’t change the variance.
   Check the conditions: Boxplots, Boxplots, Boxplots!!!
When to Pool?

 Because the advantages of pooling are
small, and you are allowed to pool only
rarely – when the equal variances
assumption is met:
DON’T!
 It is never wrong NOT to pool!!
CAUTION!!!

 Watch out for paired data.
   If the samples are not independent, you cannot
use the two-sample methods.
   Two-sample methods can only be used if the
observations in the two groups are independent.
 Look at the plots!
   Check for outliers and non-normal distributions.
   Make and examine boxplots.

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