Chapter 11 Inferences on Two Samples

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Chapter 11Inferences on Two Samples

• 11.1
• Inference about Means
• Dependent Sampling

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Objectives

• Distinguish between independent and dependant
  sampling
• Test claims made regarding matched-pairs data
• Construct confidence intervals about the
  population mean difference of matched-pairs data

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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 the price of a
one night stay at a Holiday Inn Express Hotel is
less than the price of a one night stay at a Red
Roof Inn Hotel. She randomly selects 8 towns
where the location of the hotels is close to each
other and determines the price of a one night
stay.
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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 the price of a
one night stay at a Holiday Inn Express Hotel is
less than the price of a one night stay at a Red
Roof Inn Hotel. She randomly selects 8 towns
where the location of the hotels is close to each
other and determines the price of a one night
stay. This is dependant sampling. Since the
hotels are in the same town then the Holiday Inn
chosen affects the Red Rood Inn chosen.
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EXAMPLE Independent versus Dependent Sampling
For each of the following, determine whether the
sampling method is independent or dependent. (b)
A researcher wants to know whether the newly
issued state quarters have a mean weight that
is different from traditional quarters. He
randomly selects 18 state quarters and 16
traditional quarters. Their weights are
compared.
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EXAMPLE Independent versus Dependent Sampling
For each of the following, determine whether the
sampling method is independent or dependent. (b)
A researcher wants to know whether the newly
issued state quarters have a mean weight that
is different from traditional quarters. He
randomly selects 18 state quarters and 16
traditional quarters. Their weights are
compared. This is independent sampling since the
state quarters chosen do not determine which
traditional quarters are chosen.
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The t- distribution
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Hypothesis Testing

• In order to test the hypotheses regarding the
  mean difference, we need certain requirements to
  be satisfied.
• A simple random sample is obtained
• the sample data is matched pairs
• the differences are normally distributed or the
  sample size, n, is large (n gt 30).

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Hypothesis Testing-Classical Method
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Hypothesis Testing-Classical Method
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Hypothesis Testing- Classical Method
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Hypothesis Testing-Classical Method
Step 4 Compare the critical value with the test
statistic
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Hypothesis Testing- Classical Method
Step 5 State the conclusion.
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EXAMPLE Testing a Claim Regarding Matched
Pairs Data- Using the P-value method
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Source Expedia.com
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EXAMPLE Testing a Claim Regarding Matched
Pairs Data- Using the p-value method

• Step 1- Make the Claim
• The null hypothesis is that µ1 µ2 and therefore
  that µ1- µ2 µd 0.
• Since we are testing the claim that the prices
  are different then it is a two-tailed test and
  the alternative hypothesis is µd ? 0

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EXAMPLE Testing a Claim Regarding Matched
Pairs Data

• Step 2- Select the level of significance
• According to the given information a .05
• Step 3- Find the p-value
• See page 448 for step by step directions.
• Using the T-Test we obtain that p 0.31

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EXAMPLE Testing a Claim Regarding Matched
Pairs Data

• Step 4- Determine if the p-value is lower than a
  (for a one tail test) or a/2 (for a two tailed
  test).
• Since p (.31) is not smaller than a/2 (.025) then
  we can not reject the null.
• Step 5- Make your conclusions
• There is not sufficient evidence to support the
  claim that the price of La Quinta Inns is
  different than the price of Hampton Inns.

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EXAMPLE Testing a Claim Regarding Matched
Pairs Data- Using the P-value method

• The following data represents the reaction time
  (in seconds) to press a key.
• Participants must press a key on seeing either a
  blue or red screen.
• Test the claim that the reaction time to the red
  screen is less than the reaction time to the blue
  screen at the a .05 level of significance.
• Note since the sample size is less than 30 then
  we must test for normality.

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EXAMPLE Testing a Claim Regarding Matched
Pairs Data- Using the p-value method
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EXAMPLE Testing a Claim Regarding Matched
Pairs Data- Using the p-value method

• Step 1- Make the Claim
• The null hypothesis is that µb µr and therefore
  that µr- µb µd 0.
• Since we are testing the claim that the reaction
  time of the red screen is less than the blue then
  it is a one-tailed test.
• We say µr lt µb and µd µr µb lt 0 the
  alternative hypothesis is µd lt 0

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EXAMPLE Testing a Claim Regarding Matched
Pairs Data

• Step 2- Select the level of significance
• According to the given information a .05
• Step 3- Find the p-value
• See page 448 for step by step directions.
• Using the T-Test we obtain that p .0294

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EXAMPLE Testing a Claim Regarding Matched
Pairs Data

• Step 4- Determine if the p-value is lower than a
  (for a one tail test) or a/2 (for a two tailed
  test).
• Since p (.0294) is smaller than a (.05) then we
  can reject the null hypothesis.
• Step 5- Make your conclusions
• There is sufficient evidence to support the claim
  that the reaction time to the red screen is less
  than the reaction time to the blue screen.

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Confidence Intervals for Matched Paris Data
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EXAMPLE Constructing a Confidence Interval
for the Mean Difference Construct a 90
confidence interval for the mean difference in
price of Hampton Inn versus La Quinta hotel
rooms. See page 448 for step by step directions.
By selecting the TInterval we obtain (-2.181,
8.381)