Two-Sample Inference Procedures with Means

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Two-Sample Inference Procedures with Means
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Remember
We will be interested in the difference of means,
so we will use this to find standard error.
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Suppose we have a population of adult
men with a mean height of 71 inches
and standard deviation of 2.6
inches. We also have a population of adult women
with a mean height of 65 inches and standard
deviation of 2.3 inches. Assume heights are
normally distributed. Describe the distribution
of the difference in heights between males and
females (male-female).
Normal distribution with mx-y 6 inches sx-y
3.471 inches
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s 3.471
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• What is the probability that the height of a
  randomly selected man is at most 5 inches taller
  than the height of a randomly selected woman?
• b) What is the 70th percentile for the difference
  (male-female) in heights of a randomly selected
  man woman?

P((xM-xF) lt 5) normalcdf(-8,5,6,3.471) .3866
(xM-xF) invNorm(.7,6,3.471) 7.82
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• a) What is the probability that the mean height
  of 30 men is at most 5 inches taller than the
  mean height of 30 women?
• b) What is the 70th percentile for the difference
  (male-female) in mean heights of 30 men and 30
  women?

6.332 inches
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Two-Sample Procedures with means
When we compare, what are we interested in?

• The goal of these inference procedures is to
  compare the responses to two treatments or to
  compare the characteristics of two populations.
• We have INDEPENDENT samples from each treatment
  or population

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Assumptions

• Have two SRSs from the populations or two
  randomly assigned treatment groups
• Samples are independent
• Both distributions are approximately normally
• Have large sample sizes
• Graph BOTH sets of data
• ss unknown

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Hypothesis Statements

• H0 m1 - m2 0
• Ha m1 - m2 lt 0
• Ha m1 - m2 gt 0
• Ha m1 - m2 ? 0

H0 m1 m2
Be sure to define BOTH m1 and m2!
Ha m1lt m2
Ha m1gt m2
Ha m1 ? m2
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Formulas

• Since in real-life, we will NOT know both ss, we
  will do t-procedures.

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Hypothesis Test
Since we usually assume H0 is true, then this
equals 0 so we can usually leave it out
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Degrees of Freedom

• Option 1 use the smaller of the two values n1
  1 and n2 1
• This will produce conservative results higher
  p-values lower confidence.
• Option 2 approximation used by technology

Calculator does this automatically!
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Dr. Phils Survey
If there were such a thing as a personality test,
do you think that the guys personalities would
be different from the girls?
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Two competing headache remedies claim to give
fast-acting relief. An experiment was performed
to compare the mean lengths of time required for
bodily absorption of brand A and brand B. Assume
the absorption time is normally distributed.
Twelve people were randomly selected and given an
oral dosage of brand A. Another 12 were randomly
selected and given an equal dosage of brand B.
The length of time in minutes for the drugs to
reach a specified level in the blood was
recorded. The results follow
mean SD n Brand A 20.1 8.7 12
Brand B 18.9 7.5 12 Is there sufficient
evidence that these drugs differ in the speed at
which they enter the blood stream?
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Have 2 independent randomly assigned treatments
Given the absorption rate is normally
distributed ss unknown
State assumptions!
Hypotheses define variables!
Where mA is the true mean absorption time for
Brand A mB is the true mean absorption time for
Brand B
Formula calculations
Conclusion in context
Since p-value gt a, I fail to reject H0. There is
not sufficient evidence to suggest that these
drugs differ in the speed at which they enter the
blood stream.
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Suppose that the sample mean of Brand B is 16.5,
then is Brand B faster?
No, I would still fail to reject the null
hypothesis.
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A modification has been made to the process for
producing a certain type of time-zero film (film
that begins to develop as soon as the picture is
taken). Because the modification involves extra
cost, it will be incorporated only if sample data
indicate that the modification decreases true
average development time by more than 1 second.
Should the company incorporate the
modification? Original 8.6 5.1 4.5 5.4 6.3 6.6 5.
7 8.5 Modified 5.5 4.0 3.8 6.0 5.8 4.9 7.0 5.7
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Assume we have 2 independent SRS of film
Both distributions are
approximately normal due to approximately
symmetrical boxplots
ss unknown
H0 mO- mM 1 HamO- mM gt 1
Where mO is the true mean developing time for
original film mM is the true mean developing
time for modified film
Since p-value gt a, I fail to reject H0. There is
not sufficient evidence to suggest that the
company incorporate the modification.
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Confidence intervals
Called standard error
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Two competing headache remedies claim to give
fast-acting relief. An experiment was performed
to compare the mean lengths of time required for
bodily absorption of brand A and brand B. Assume
the absorption time is normally distributed.
Twelve people were randomly selected and given an
oral dosage of brand A. Another 12 were randomly
selected and given an equal dosage of brand B.
The length of time in minutes for the drugs to
reach a specified level in the blood was
recorded. The results follow
mean SD n Brand A 20.1 8.7 12
Brand B 18.9 7.5 12 Find a 95
confidence interval difference in mean lengths of
time required for bodily absorption of each brand.
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Assumptions Have 2 independent randomly assigned
treatments Given the absorption rate is
normally distributed ss unknown
State assumptions!
Think Price is Right! Closest without going
over
Formula calculations
Conclusion in context
From calculator df 21.53, use t for df 21
95 confidence level
We are 95 confident that the true difference in
mean lengths of time required for bodily
absorption of each brand is between 5.685
minutes and 8.085 minutes.
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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.
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 Compute a 95 confidence interval
for the mean difference in amount of
acetaminophen in Brand A and Brand B.
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Input Brand A data into L1 and Brand B data into
L2. I am computing a 2-sample T interval for
means at a 95 confidence level. Confidence
level (-28.48, -7.189) with 17 df I am 95
confident that the true difference in the mean
amount of acetaminophen in Brand A is between
28.5 and 7.2 mg lower than in Brand B.
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Note confidence interval statements

• Matched pairs refer to mean difference
• Two-Sample refer to difference of means

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Pooled procedures

• Used for two populations with the same variance
• When you pool, you average the two-sample
  variances to estimate the common population
  variance.
• DO NOT use on AP Exam!!!!!

We do NOT know the variances of the population,
so ALWAYS tell the calculator NO for pooling!
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Robustness

• Two-sample procedures are more robust than
  one-sample procedures
• BEST to have equal sample sizes! (but not
  necessary)