Tests Involving Paired Differences of Dependent Samples

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Tests Involving Paired Differences of Dependent
Samples

• Section 9.4

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Paired Data Dependent Samples

• Members of each pair have a natural matching of
  characteristics.
• Example weight before and weight after a diet.

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Advantages of Using Paired Data Sets

• May reduce the danger of introducing extraneous
  or uncontrollable factors into sample
  measurements.
• Reduces measurement variability.
• Increases accuracy of statistical conclusions.

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In a random sample of n pairs where the
differences d between the first and second
members of each pair are (approximately) normal
distributed with population mean ?d...
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then the t values shown below follow a Students
t distribution with d.f. n 1.
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When testing the mean of the difference of paired
values

• The null hypothesis is that there is no
  difference among the pairs.
• H0 ?d 0

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Left-Tailed Test

• H0 ?d 0
• H1 ?d lt 0

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Right-Tailed Test
H0 ?d 0 H1 ?d gt 0
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Two-Tailed Test
H0 ?d 0 H1 ?d ? 0
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We wish to determine (using ? 0.05) whether a
particular diet causes a significant weight loss.
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Calculate the differences in weight.
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Do a t-Test on The DifferencesUsing TI-83
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We wish to determine (using ? 0.05) whether a
particular diet causes a significant weight loss.

• H0 ?d 0
• H1 ? d lt 0
• ? 0.05

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Calculate P
P
t 1.17
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Hypothesis Testing of Paired Differences using
the Ti-83

• Press STAT and ENTER
• Enter the before data in L1
• Enter the after data in L2
• Move cursor over L3 Title, input L1-L2 and press
  ENTER
• Press STAT, Move cursor over to TESTS
• Move cursor down to 2 T-Test, Press ENTER
• Highlight Data, Enter µ0 0, the Null hypothesis
• List L3, Freq1
• Highlight the appropriate alternative
• Highlight Calculate and press ENTER
• Read P, Reject H0 if P ? ? ,
• Do Not Reject if P gt ?

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Hypothesis Testing of Paired Differences using
the Ti-83The Weight Loss Example

• Enter Data As Described in Previous Slide
• Read P .8364 gt ? 05
• We conclude that we cannot reject the hypothesis
  that there is no significant difference in the
  weights before and after the diet.
• Weights before and after the diet are essentially
  the same.

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L1 L2 L3 L1-L2 H0
µd 0 H1 gt 0 Use 2T Test with Data P
.0012 lt a .01 - We reject H0 Anxiety level
is lower
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Use a 5 Test Is There a Difference?
L1 L2 L3
L1-L2 H0 µd 0 H1 ? 0 Use 2T Test with
Data P .0456 lt a .05 - We reject
H0 Reading age is different