AP STATISTICS LESSON 11

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AP STATISTICSLESSON 11 2(DAY 1)

• Comparing Two Means

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ESSENTIAL QUESTION When can procedures for
comparing two means be used and what are those
procedures?

• Objectives
• To determine if procedures for comparing two
  means should be used.
• To construct two means significance tests
• To construct confidence intervals to make
  inferences when comparing two samples.

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Comparing Two Means

• Comparing two populations or two treatments is
  one of the most common situations encountered in
  statistical practice. We call such situations
  two-sample problems.
• A two sample problem can arise from a
  randomized comparative experiment that randomly
  divides subjects into two groups and exposes each
  group to a different treatment.
• Comparing random samples separately selected
  from two populations is also a two sample
  problem. Unlike the matched pairs designs
  studied earlier, there is no matching of the
  units in the two samples and the two samples can
  be of different sizes.

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Two Sample Problems

• The goal of inference is to compare the responses
  to two treatments or to compare the
  characteristics of two populations.
• We have a separate sample from each treatment or
  each population.

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Example 11.9 Page 648Two-Sample Problems

1. A medical researcher is interested in the effect
   on blood pressure of added calcium in our diet.
   She conducts a randomized comparative experiment
   in which on group of subjects receives a calcium
   supplement and a control group receives a
   placebo.
2. A psychologist develops a test that measures
   social insight. He compares the social insight
   of male college students with that of female
   college students by giving the test to a sample
   of students of each gender.
3. A bank wants to know which of two incentive plans
   will most increase the use of its credit cards.
   It offers each incentive to a random sample of
   credit card customers and compares the amount
   charged during the following six months.

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Conditions for Comparing Two Means

• We have two SRSs, from two distinct populations.
  The samples are independent (That is, one sample
  has no influence on the other.) Matching
  violates independence, for example. We measure
  the same variable for both samples.
• Both populations are normally distributed. The
  means and standard deviations of the populations
  are unknown.

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Organizing the Data

• Call the variable we measure x1 in the first
  population and x2 in the second . We know
  parameters in this situation.

Population Variable Mean Standard deviation
1 x1 µ1 s1
2 x2 µ2 s2
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Organizing Data (part 2)

• There are four unknown parameters, the two
  means and the two standard deviations.
• Population Sample Mean Sample
• size
  Standard deviation
• 1 n1 x1 s1
• 2 n2 x2
  s2

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Example 11.10 Page 650Calcium and Blood
Pressure

• Does increasing the amount of calcium in our
  diet reduce blood pressure? A randomized
  comparative experiment was designed.
• Subjects 21 Healthy Black Men
• A randomly chosen group of 10 of the men received
  a calcium supplement for 12 weeks.
• The control group of 11 men received a placebo.
• The experiment was double-blind.

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The Sampling Distribution of x1 x2

• The mean of x1 x2 is µ1 µ2. That is, the
  difference of sample means is an unbiased
  estimator of the difference of population means.
• The variance of the difference is the sum of the
  variance of x1 x2 which is
• s1 s2
• Note that the variance add. The standard
  deviations do not.
• If the two populations are both normal

n1
n2
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The Sampling Distribution of x1 x2 (continued)

• Then the distribution of x x is also normal.
• The two-sample z statistic is standardized by
• z (x1 x2 ) ( µ1 µ2 )
• v s12/n1 s22/n2

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Standard Deviation of Two-Sample Means

• Whether an observed difference between two
  samples is surprising depends on the spread of
  the observations as well as on the two means.
  This standard deviation is
• v s12/n1 s22/n2

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Standard Error

• Because we dont know the population standard
  deviations, we estimate them by the sample
  standard deviations from our two samples.
• SE v s12/n1 s22/n2
• The two-sample t statistic
• t (x1 x2 ) ( µ1 µ2 )
• v s12/n1 s22/n2

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Two-Sample t Distributions

• The statistic t has the same interpretation as
  any z or t statistic it says how far x1 x2 is
  from its mean in standard deviation units.
• When we replace just one standard deviation in a
  z statistic by a standard error we must replace
  the z distribution with the t distribution.

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Degrees of Freedom for Two-Sample Problems

• Two methods for calculating degrees of
  freedom
• Option 1 Use procedures based on the statistic t
  with critical values from a t distribution (used
  by calculator).
• Option 2 Use procedures based on the based on
  the statistic t with critical from the smaller n
  1.

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Confidence Interval for a Two-Sample t

• ( µ1 µ2 ) tv s12/n1 s22/n2
• Compute the two-sample t statistic
• t (x1 x2 )
• v s12/n1 s22/n2

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Example 11.11 Page 655Calcium and Blood
Pressure, continued
The P-value. This example uses the conservative
method which leads to the t distribution with 9
degrees of freedom.
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Example 11.12 Page 656 Two-Sample t
Confidence Interval

• Sample size strongly influences the P-value of a
  test.
• An effect that fails to be significant at a s
  specified level a in a small sample will be
  significant in a larger sample.

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Robustness Again

• The two-sample t procedures are more robust than
  the one-sample t methods, particularly when the
  distributions are not symmetric.
• When the sizes of the two samples are equal and
  the two populations being compared have
  distributions with similar shapes, probability
  values from the t table are quite accurate.
• When the two populations distributions have
  different shapes, larger samples are needed.