6.3 TwoSample Inference for Means

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6.3 Two-Sample Inference for Means

• November 17, 2003

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Paired Differences

• Matched Pairs Design
• experimental plan where the experimental units
  are divided into halves and two treatments are
  randomly assigned to the halves
• Attempting to determine if there is a significant
  difference between the mean responses of the
  treatments

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Procedure

• Obtain the difference between responses for each
  experimental unit
• Analyze the differences using a one-sample
  approach
• If a large sample is obtained, use critical
  values from the Standard Normal distribution (z)
• Otherwise, use critical values from the
  corresponding t distribution

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Example 9 pg 370

• Students worked with a company on the monitoring
  of the operation of an end-cut router in the
  manufacture of a wood product. They measured the
  critical dimensions of a number of pieces of a
  type as they came off the router. Both a
  leading-edge and a trailing-edge measurement were
  made on each piece. Both were to have a target
  value of .172 in.

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Confidence Interval
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Hypothesis Test

• Is there a significant difference between the
  measurements at a .01?
• Ho µ 0
• Ha µ ? 0

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Independent Samples

• The goal of this type of inference is to compare
  the mean response of two variables (or
  treatments) when the data are not paired or
  matched
• A key assumption that will be made is that the
  separate samples used to collect information
  concerning the two variables are independent
• One sample does not influence the other sample in
  any way
• Furthermore, one must assume that both sampled
  populations are normally distributed

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Large Sample Comparison

• The quantity of interest is a linear combination
  of population means, namely µ1 - µ2
• The above quantity will be estimated by
• As a result, various quantities of the sampling
  distribution of the difference, under the
  assumption of equality between the means, need to
  be developed
• Ho µ1 - µ2
• Ha µ1 - µ2 ?

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Test Statistic
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Confidence Interval
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Example

• A company research effort involved finding a
  workable geometry for molded pieces of a solid.
  A comparison was made between the weight of
  molded pieces and the weight of irregularly
  shaped pieces that could be poured into the same
  container. A series of 30 attempts to pack both
  the molded and the irregular pieces of the solid
  were compared. Is there enough evidence to
  suggest that the irregular pieces produced higher
  weights?

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The Data

• 1 molded
• n1 30
• s1 9.31
• 164.65

• 2irregular
• n2 30
• s2 8.51
• 179.65

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Small Samples

• If at least one sample size is small, then use
  critical values from a t distribution for
  constructing confidence intervals and performing
  hypothesis tests with degrees of freedom obtained
  by Satterthwaites Approximation

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Satterthwaites Approximation
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Test Statistic
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Confidence Interval
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Example

• The data shown gives spring lifetimes under two
  different levels of stress (900 and 950 N/mm2).
  Do the data give evidence of a significant
  difference at a .05?

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Minitab Output
Descriptive Statistics 950, 900 Variable
N Mean Median TrMean StDev
SE Mean 950 10 168.3
166.5 167.6 33.1
10.5 900 10 215.1 216.0
211.5 42.9 13.6
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Assignment

• Page 385 3, 4