0' Name populations

1

• 0. Name populations
• H0 m1 m2 0, H1 m1 m2 0
• tobs tdf if H0 is true
  (df rounded down)
• 3. -tdf,a (left tailed), tdf,a (right tailed),
  -tdf,a/2 tdf,a/2 (two tailed)
• 4. Reject H0 if tobs lt -tdf,a (left tailed)
• Reject H0 if tobs gt tdf,a (right tailed)
• Reject H0 if tobs lt -tdf,a/2 or tobs gt tdf,a/2
• 5.
• 6. Reject H0 or Fail to reject H0
• 7. Conclusion

2

• 0. Name populations
• H0 m1 m2 0, H1 m1 m2 0
• tobs tdf if H0 is true
  (df rounded down)
• 3. Reject H0 if p-value lt a
• 4.
• 5. p-value obtained by computer
• 6. Reject H0 or Fail to reject H0
• 7. Conclusion

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4
Several neurosurgeons wanted to determine whether
a dynamic system (Z-plate) reduced the post
operative recovery time relative to a static
system (ALPS plate). Fourteen patients that
received the dynamic system had an average post
operative recovery time of 7.36 days with
standard deviation 1.22 days. The 6 patients that
used the static system had an average post
operative recovery time of 12.17 days with
standard deviation 4.58 days. Test the conjecture
with 95 confidence.
Pop1 Z-plate
Pop2 ALPS plate
Test if m1 lt m2 with 95 confidence
5

• 0. Pop1 Z-plate, Pop2 ALPS plate
• H0 m1 m2 0, H1 m1 m2 lt 0
• tobs t5 if H0 is true
• 3. t5,0.05 -2.015
• Reject H0 if tobs lt -2.015
• 5.
• 6. Reject H0
• 7. With 95 confidence there is sufficient
  evidence to conclude that the mean recovery time
  for the Z-plate is faster than the mean recovery
  time with the ALPS plate.

6
0. Pop1 Z-plate, Pop2 ALPS plate 1. H0 m1 m2
0, H1 m1 m2 lt 0 2. tobs t5 if H0 is
true 3. Reject H0 if p-value lt 0.05 4. 5.
p-value 0.0261 (obtained from Excel) 6. Reject
H0 7. With 95 confidence there is sufficient
evidence to conclude that the mean recovery time
for the Z-plate is faster than the mean recovery
time with the ALPS plate.
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Concentrations of rare earth elements on airborne
particles often indicate the source of
atmospheric materials, such as pollutants. One
such indicator is the ratio of lanthanum to
samarium. Analyses of 14 specimens of Eastern
coal produced an average lanthanum to samarium
ratio of 6.1 with a standard error of 0.3.
Similar statistics on 22 specimens of Western
coal were 8.1 and 0.5, respectively. Test for a
difference in the two coal sources using a 0.10
type I error rate.
Pop1 Eastern
Pop2 Western
Test is m1 and m2 are different with 90
confidence
9

• 0. Pop1 Eastern, Pop2 Western
• H0 m1 m2 0, H1 m1 m2 ? 0
• tobs t32 if H0 is true
• 3. -t32,0.05 -1.697, t32,0.05 1.697 (table
  doesnt have df32 so use df30)
• Reject H0 if tobs lt -1.697 or tobs gt 1.697
• 5. tobs -3.430
• 6. Reject H0
• 7. With 90 confidence there is sufficient
  evidence to conclude that the mean lanthanum to
  samarium ratio is lower in the Eastern source.

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• 0. Pop1 Eastern, Pop2 Western
• H0 m1 m2 0, H1 m1 m2 ? 0
• tobs t32 if H0 is true
• 3. Reject H0 if p-value lt 0.10
• 4. tobs -3.430
• 5. p-value 0.0017 (From Excel using correct
  df)
• 6. Reject H0
• 7. With 90 confidence there is sufficient
  evidence to conclude that the mean lanthanum to
  samarium ratio is lower in the Eastern source.

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CI for mean eastern mean western Since it is
fully negative then we can conclude with 90
confidence that mean eastern is less than mean
western
13
SAS (and most other statistical packages) display
p-values according to a two-tailed alternative so
if the alternative hypothesis tested is not
two-tailed then an adjustment will be necessary
to get the correct p-value.
Let p p-value that SAS supplies

• If H1 is left tailed and tobs is negative then
  p-value p/2
• If H1 is left tailed and tobs is positive then
  p-value 1 p/2
• If H1 is right tailed and tobs is negative then
  p-value 1 p/2
• If H1 is right tailed and tobs is positive then
  p-value p/2

14
Sometimes the samples are not independent. This
is evident because individual observations from
sample 1 will be linked to individual
observations from sample 2. The proper
comparison is called a paired t-test Example
Variable weight loss Subject Before After D
(pop1 pop2) Jim x11 x21 d1 x11
x21 Jasper x12 x22 d2 x12
x22 Sally x13 x23 d3 x13
x23 Roy x14 x24 d4 x14 x24 Jessica x15 x2
5 d5 x15 x25 Rose x16 x26 d6 x16 x26
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• 0. Name populations
• H0 m1 m2 0, H1 m1 m2 0
• tobs tdf if H0 is true df nd - 1
• 3. -tdf,a (left tailed), tdf,a (right tailed),
  -tdf,a/2 tdf,a/2 (two tailed)
• 4. Reject H0 if tobs lt -tdf,a (left tailed)
• Reject H0 if tobs gt tdf,a (right tailed)
• Reject H0 if tobs lt -tdf,a/2 or tobs gt tdf,a/2
• 5.
• 6. Reject H0 or Fail to reject H0
• 7. Conclusion

16

• 0. Name populations
• H0 m1 m2 0, H1 m1 m2 0
• tobs tdf if H0 is true df nd - 1
• 3. Reject H0 if p-value lt a
• 4.
• 5. p-value from computer
• 6. Reject H0 or Fail to reject H0
• 7. Conclusion

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A company was wondering which style of pepperoni
pizza was most popular. It set up an experiment
where ten people were each given two types of
pizza to eat, Type A and Type B. Each pizza was
carefully weighed at exactly 16 oz. After fifteen
minutes, the remainders of the pizza were
weighed, and the amount of each type pizza
remaining per person was calculated. It is
assumed that the subject would eat more of the
type of pizza he or she preferred. Test if either
pizza type is preferred using a 0.1 type I error
rate
Subject Pizza A Pizza B Diff
1 12.9 16.0 -3.1 2 5.7 7.5 -1.8
3 16.0 16.0 0 4 14.3 15.7 -1.4
5 2.4 13.2 -10.8 6 1.6 5.4 -3.8
7 14.6 15.5 -0.9 8 10.2 11.3 -1.1
9 4.3 15.4 -11.1 10 6.6 10.6 -4.0
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• 0. Pop1 Pizza A, Pop2 Pizza B
• H0 m1 m2 0, H1 m1 m2 ? 0
• tobs t9 if H0 is true
• 3. t9,0.05 -1.833 t9,0.05 1.833
• 4. Reject H0 if tobs lt -1.833 or tobs gt 1.833
• 5.
• 6. Reject H0
• 7. With 90 confidence there is sufficient
  evidence to conclude that people eat more pizza
  type A than pizza type B

20

• 0. Pop1 Pizza A, Pop2 Pizza B
• H0 m1 m2 0, H1 m1 m2 ? 0
• tobs t9 if H0 is true
• 3. Reject H0 if p-value lt 0.10
• 4.
• 5. p-value 0.0145
• 6. Reject H0
• 7. With 90 confidence there is sufficient
  evidence to conclude that people eat more pizza
  type A than pizza type B

21
CI for mean pizza type A mean pizza type
B Since it is fully negative then we can
conclude with 90 confidence that people ate more
of A than type B