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Chapter 24: Comparing Means
Key Vocabulary:


two-sample t-interval
two-sample t-test
Calculator Skills:



pooled t-test
2-SampTInt
2-SampTTest
Comparative studies are more convincing than single single-sample investigations, so
____________________ inference is not as common as comparative ____________________
inference.
In a comparative study, we may want to compare two ____________________ or we may want
to compare two ____________________ . In either case, the samples must be chosen
____________________ and ____________________ in order to perform statistical inference.
Two Samples: Independence
The sample values selected from one population are ___________ __________________
or _____________________ _________________________ with the sample values selected
from the other population.
If the values in one sample are _________________________ to the values in the other
sample, the samples are ________________________. Such samples are often referred to as
________________________________ or ________________________________.
We may use ____________________ inference to compare two ____________________ or two
____________________.
Chapter 24: Comparing Means
Remember that, for independent random quantities, variances _______________________.
So, the standard deviation of the difference between two sample means is:
__________________
We still don’t know the _____________________ _______________________
_______________________ of the two groups, so we need to estimate and use the standard
__________________________:
______________________________
Because we are working with __________________ and estimating the standard error of their
difference using the data, we shouldn’t be surprised that the sampling model is a
____________________________________.
The confidence interval we build is called a __________________________________________
(for the difference in means).
The corresponding hypothesis test is called a _________________________________________.
The null hypothesis is that there is no difference between the two parameters.
____________________
or
____________________
The alternative hypothesis could be that
____________________ (two-sided)
____________________ or ____________________ (one-sided)
____________________ or ____________________ (one-sided)
Before you begin, check your ____________________ ! For comparing two means, both
samples must be an ____________________ and must be chosen ____________________ .
Also, both populations must be ____________________ _______________. (Check the data for
____________________ or ____________________.)
Chapter 24: Comparing Means
Independence Assumption (Each condition needs to be checked for both groups.):
__________________________Condition: Were the data collected with suitable
randomization (representative random samples or a randomized experiment)?
____________ Condition: We don’t usually check this condition for differences of
means. We will check it for means only if we have a very small population or an
extremely large sample.
Normal Population Assumption:
__________________________________ Condition: This must be checked for both
groups. A violation by either one violates the condition. Both samples come from a
normal population, or samples are large (>40), or samples are medium (15-40) and plots
show little skewness and no outliers, or samples are small (<15) and plots show no
skewness and no outliers.
Independent Groups Assumption:
__________________________________ Condition: The two groups we are comparing
must be independent of each other.
If these assumptions hold, then the difference in sample means is an unbiased estimator of the
difference in population means, so ____________________ is equal to
____________________.
Also, the variance of ____________________ is the sum of the variances of _________ and
_________, which is ____________________.
Furthermore, if both populations are normally distributed, then ____________________ is also
normally distributed.
Chapter 24: Comparing Means
In order to standardize ____________________, subtract the __________ and divide by the
_________________________:
z
 x1  x2    1  2 
 12
n1

 22
n2
If we do not know _____ and _____, we will substitute the standard error _____ and _____ for
the standard deviation. This gives the standardized t value:
t
 x1  x2    1  2 
s12 s22

n1 n2
CAUTION! This statistic does _____ have a t distribution.
Therefore, a special formula must be used to calculate the degrees of freedom for the difference
of two means. The special formula for the degrees of freedom for our t critical value is a bear:
2
 s12 s22 
n n 
2 
 1
(you do not need to know)
df 
2
2
2
1  s1 
1  s22 

n1  1  n1  n2  1  n2 
Because of this, we will use this estimate: df = ___________________ of __________________
and ___________________. Because we estimate the df and the TI-84 calculator uses the actual
df, the p-values and intervals we calculate will be slightly larger (more conservative values) then
the p-values and intervals calculated on the TI-84.
Two Sample t-interval
When the conditions are met, we are ready to find the confidence interval for the difference
between means of two independent groups, 1 – 2.
The confidence interval is
_______________________________________
Chapter 24: Comparing Means
where the standard error of the difference of the means is
_______________________________
The critical value t*df depends on the particular confidence level, C, that you specify and on the
number of degrees of freedom, which we get from the sample sizes and use the smaller of
________________ and _________________as the df.
Problem
Manufacturers make claims about their products and usually try to convince you that their
product is better than that of a competitor. Most brands of paper towels claim to pick up more
liquid than any other brand. How much of a difference, on average, can be expected between
Brand A and Brand B? This calls for a confidence interval to find the true difference, μA-μB,
between the mean number of milliliters of water absorbed by each towel. A random sample of 16
of each type of towel was tested for absorbency. The mean number of ml. for Brand A was
15.625 ml. with a standard deviation of 3.12 ml. while for Brand B the mean was 14 ml. with a
standard deviation of 2.53 ml. Find 95% confidence interval for μA-μB. Use Panic!!
Chapter 24: Comparing Means
Two Sample t-Test
The hypothesis test we use is the ________________________________________ for means.
The conditions for the two-sample t-test for the difference between the means of two
independent groups are the ______________ as for the two-sample
___________________________.
We test the hypothesis H0:1 – 2 = 0, where the hypothesized difference, 0, is almost always
0, using the statistic:
____________________________________
The standard error is
____________________________________
When the conditions are met and the null hypothesis is true, this statistic can be closely modeled
by a t-disttribution with a number of degrees of freedom given by the estimate: df =
____________________ of n1 – 1 and n2 – 1. We use that model to obtain a P-value.
Problem
It is a common belief that women tend to live longer than men. Random samples from the death
records for men and women in Montgomery County were taken and age at the time of death
recorded. The average age of the 48 males was 68.33 years with a standard deviation of 12.49
years, while the average age of the 40 females was 78.7 years with a standard deviation of 16.43
years. Do women in this country tend to live longer than men? Use PHANTOMS!!
Chapter 24: Comparing Means
Pooling
When should you use pooled-t methods rather than two-sample t methods? _________________
We will use the TI-84 to perform two-sample T tests.
2-SampTTest for a ___________________________________
2-SampTInt for a ____________________________________
Chapter 24: Comparing Means