Comparing Two Populations or Treatments

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Chapter 11

• Comparing Two Populations or Treatments
• Homework for 1st half
• 11. 3, 11.11, 11. 23, 11.19 c

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A sampling method is independent when the
individuals selected for one sample does not
dictate which individuals are to be in a second
sample. A sampling method is dependent when the
individuals selected to be in one sample are used
to determine the individuals to be in the second
sample. Dependent samples are often referred to
as matched pairs samples.
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EXAMPLE Independent versus Dependent Sampling
For each of the following, determine whether the
sampling method is independent or dependent. (a)
A researcher wants to know whether a particular
diet program was effective, so 20 individuals
were randomly chosen and their weights were
recorded before the program and after the
program. (b) A researcher wishes to determine
the effects of alcohol on reaction time. 50
people were randomly put into two groups of 25.
Group 1 is given 2 drinks of alcohol, and Group 2
is given 2 drinks of alcohol-tasting placebo.
Thirty minutes after the drinks, the subjects are
given a test to determine reaction time.
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Hypothesis Tests Comparing Two Means Independent
sampling
Large size sample techniques allow us to test the
null hypothesis H0 m1 - m2 hypothesized
value against one of the usual alternate
hypotheses using the statistic
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Two-Sample t Test for Comparing Two Population
Means Independent Samples
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Two-Sample t Test for Comparing Two Population
Means
If you use the classical method, this formula can
be used to find the df, or you may use the
conservative method of using the smallest n-1.
We will only focus on the p-value method in this
course for 2-sample tests.
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Independent Samples P-Value Method (Minitab
will also give you the same results.)

• Once the assumptions have been confirmed
  (independent random samples, normally
  distributed), use these steps
• Stat, Tests, 2-SampZTest (if s is Known)
• Stat, Tests, 2-SampTTest (if s is unknown)
• Choose Data or Stats
• Freq 1 and Freq 2 are always equal to 1
• Pooled No
• To use pooled we must show that the standard
  deviations are the same using a different test.
  In this course, we will always use the more
  conservative method Pooled No.
• Calculate
• Compare p to a. If p lt a, reject the null.

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Hypothesis Test Brand A B
In an attempt to determine if two competing
brands of cold medicine contain, on the average,
the same amount of acetaminophen, twelve
different tablets from each of the two competing
brands were randomly selected and tested for the
amount of acetaminophen each contains. The
results (in milligrams) follow. Use a
significance level of 0.01.
Brand A Brand B 517,
495, 503, 491 493, 508, 513, 521 503, 493, 505,
495 541, 533, 500, 515 498, 481, 499, 494 536,
498, 515, 515 State and perform an appropriate
hypothesis test.
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Hypothesis Test Brand A B
m1 the mean amount of acetaminophen in cold
tablet brand A m2 the mean amount of
acetaminophen in cold tablet brand B H0 m1 m2
(m1 - m2 0) Ha m1 ? m2 (m1 - m2 ?
0) Significance level a 0.01
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Hypothesis Test Example
AssumptionsThe samples were selected
independently and randomly. Since the samples are
not large, we need to be able to assume that the
populations (of amounts of acetaminophen are both
normally distributed.
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Hypothesis Test Example
Assumptions (continued)
As we can see from the normality plots and the
boxplots, the assumption that the underlying
distributions are normally distributed appears to
be quite reasonable.
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Hypothesis Test Brand A B Solution
Stat-gtTests, 2-SampTTest Data, enter appropriate
lists, Freq 1 Select alternative
hypothesis Pooled No Calculate
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• Conclusion Since P-value 0.002 lt 0.01 a, H0
  is rejected. The data provides strong evidence
  that the mean amount of acetaminophen is not the
  same for both brands. Specifically, there is
  strong evidence that the average amount per
  tablet for brand A is less than that for brand B.

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EXAMPLE Comparing Two Means Independent
Sampling A researcher wanted to know whether
state quarters had a weight that is more than
traditional quarters. He randomly selected 18
state quarters and 16 traditional quarters,
weighed each of them and obtained the following
data.
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STATE
TRAD
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Confidence Intervals Comparing Two Means

• The general formula for a confidence interval for
  m1 m2 when
• The two samples are independently chosen random
  samples, and
• The sample sizes are both large (generally n1 ?
  30 and n2 ? 30) OR the population distributions
  are approximately normal is
• TI-83/84 Stat, Tests, 2-SampTInt

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Confidence Intervals Comparing Two Means
A student recorded the mileage he obtained while
commuting to school in his car. He kept track of
the mileage for twelve different tanks of fuel,
involving gasoline of two different octane
ratings. Compute the 95 confidence interval for
the difference of mean mileages. His data had the
following results
The 95 confidence interval for the true
difference of the mean mileages is (-3.99, -0.57).