Overview

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Overview

• 10.1 Inference for Mean DifferenceDependent
  Samples
• 10.2 Inference for Two Independent Means
• 10.3 Inference for Two Independent
• Proportions

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p-Value Method

• Method for carrying out hypothesis test
• p-Value measures how well data fits null
  hypothesis.

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p-Value

• Probability of observing a sample statistic (such
  as x or Zdata) at least as extreme as observed
  statistic assuming null hypothesis is true.
• Roughly speaking, represents probability of
  observing sample statistic if the null hypothesis
  is true.
• Since term p-value means probability value,
  always lies between 0 and 1.

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Rejection Rule

• When performing a hypothesis test
• using the p-value method
• Reject H0 when the p-value is less
• than a.

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Assessment of the Strength of Evidence
Table 9.5 Strength of evidence against the null
hypothesis for various levels of p-value
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10.1 Inference for Mean DifferenceDependent
Samples

• Objectives
• By the end of this section, I will be
• able to
• Distinguish between independent samples and
  dependent samples.
• Perform hypothesis tests for the population mean
  difference for dependent samples using the
  p-value method and the critical value method.

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Independent Samples and Dependent Samples

• Two samples are independent when the subjects
  selected for the first sample do not determine
  the subjects in the second sample.
• Two samples are dependent when the subjects in
  the first sample determine the subjects in the
  second sample.
• The data from dependent samples are called
  matched-pair or paired samples.

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Example 10.1 - Dependent or independent sampling?

• Indicate whether each of the following
  experiments uses
• an independent or dependent sampling method.
• a. A study wished to compare the differences in
  price
• between name-brand merchandise and store-brand
• merchandise. Name-brand and store-brand items of
  the
• same size were purchased from each of the
  following six
• categories paper towels, shampoo, cereal, ice
  cream,
• peanut butter, and milk.
• b. A study wished to compare traditional
  acupuncture
• with usual clinical care for a certain type of
  lower-back
• pain. The 241 subjects suffering from persistent
• non-specific lower-back pain were randomly
  assigned to
• receive either traditional acupuncture or the
  usual
• clinical care. The results were measured at 12
  and 24
• months.

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Example 10.1 continued

• Solution
• a. For a given store, each name-brand item in the
  first sample is associated with exactly one
  store-brand item of that size in the second
  sample.
• Items in the first sample determine the items in
  the second sample
• Example of dependent sampling
• b. Randomly assigned to receive either of the two
  treatments
• Thus, the subjects that received acupuncture did
  not determine those who received clinical care,
  and vice versa.
• Example of independent sampling.

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Paired Sample t Test for the Population Mean The
Critical Value Method

• Step 1
• State the hypotheses and the rejection rule.
• Use one of the hypothesis test forms from Table
  10.2 below.
• State clearly the meaning of µd.

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Paired Sample t Test for the Population Mean The
p-value Method

• Step 2
• Enter the columns as lists in your calculator and
  find the difference between the two columns.
• Use the option STATSgtTESTSgt2T-TEST..
• Inpt DATA
• Choose the list with the differences.
• Obtain the p-value.

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Paired Sample t Test for the Population Mean The
Critical Value Method

• Step 3
• If p lt a, reject the null hypothesis.
• Step 4
• State the conclusion and the interpretation.

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10.2 Inference for Two Independent Means
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Sampling Distribution of x1-x2

• Random samples drawn independently from
  populations with population means µ1 and µ2 and
  either
• Case 1
• The two populations are normally distributed,
  or
• Case 2
• The two sample sizes are large (at least 30),
  then the quantity

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Sampling Distribution of x1-x2 continued

• Approximately a t distribution
• Degrees of freedom equal to the smaller of n1 - 1
  and n2 1
• x1 and s1 represent the mean and standard
  deviation of the sample taken from population 1,
• x2 and s2 represent the mean and standard
  deviation of the sample taken from population 2.

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Standard Error of x1-x2

• Standard error of the statistic
  is
• It measures the size of the typical error in
  using to measure µ1- µ2.

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Hypothesis Test for the Difference in Two
Population Means

• p-Value Method
• Step 1
• Select the option STATSgtTESTSgt4 2-SampTTEST.
• Enter the given information.
• Step 2
• Find tdata.

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Hypothesis Test for the Difference in Two
Population Means continued

• Step 3
• Find the p-value using calculator.
• Step 4
• State the conclusion and interpretation.
• Compare the p-value with a.

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10.3 Inference for Two Independent Proportions
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Sampling Distribution of p1 - p2

• Independent random samples from two populations
• The quantity

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Sampling Distribution of p1 - p2 continued

• Has an approximately standard normal distribution
  when the following conditions are satisfied
• x1 5, (n1 - x1) 5, x2 5, (n2 - x2) 5
• p1 and n1 represent the sample proportion and
  sample size of the sample taken from population 1
  with population proportion p1

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Sampling Distribution of p1 - p2 continued

• p2 and n2 represent the sample proportion and
  sample size of the sample taken from population 2
  with population proportion p2
• q1 1 - p1 and q2 1 - p2.

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Standard Error of p1 - p2

• Standard error of the statistic p1 - p2
• Where q1 1 - p1 and q2 1 - p2.
• The standard error measures the size of
  the typical error in using p1 - p2 to estimate p1
  - p2.

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Hypothesis Test for the Difference in Two
Population Proportions p-Value Method

• Two independent random samples
• Taken from two populations
• Population proportions p1 and p2
• Required conditions
• x1 5, (n1 - x1) 5, x2 5,
• and (n2 - x2) 5.

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Hypothesis Test for the Difference in Two
Population Proportions p-Value Method

• Step 1
• State the hypotheses and the rejection rule
• Use one of the forms from Table 10.19 page 576
• State the meaning of p1 and p2
• Reject H0 if the p-value is less than a.

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Hypothesis Test for the Difference in Two
Population Proportions p-Value Method

• Step 2
• Find Zdata.
• Zdata follows an approximately standard normal
• distribution if the required conditions are
  satisfied.

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Hypothesis Test for the Difference in Two
Population Proportions p-Value Method

• Step 3
• Find the p-value using calculator.
• Step 4
• State the conclusion and interpretation.