Fabric

1
Fabric DataTried to light 4 samples of 4
different (unoccupied!) pajama fabrics on fire.
18
Higher meanslessflamable
Mean16.85std dev0.94
17
16
15
e
m
i
14
T

n
13
r
u
B
12
Mean10.95std dev1.237
Mean11.00std dev1.299
11
Mean10.50std dev1.137
10
9
4
3
2
1
Fabric
2
Back to burn time example
x s t0.025,3 95 CI Fabric 1 16.85 0.940 3.182 (1
5.35,18.35) Fabric 2 10.95 1.237 3.182 (8.98,
12.91) Fabric 3 10.50 1.137 3.182 (8.69,
12.31) Fabric 4 11.00 1.299 3.182 (8.93, 13.07)
3
Comparison of 2 means

• Example
• Is mean burn time of fabric 2 different from mean
  burn time of fabric 3?
• Why cant we answer this w/ the hypothesis test
• H0 mean of fabric 2 10.5HA mean of fabric
  2 doesnt 10.5
• Whats the appropriate hypothesis test?

x for fabric 3
4

• H0 mean fab 2 mean fab 3 0
• HA mean fab 2 mean fab 3 not 0
• Lets do this w/ a confidence interval.
• Large sample CI
• (x2 x3) /- za/2sqrts22/n2 s23/n3

5

• CI is based on small sample distribution of
  difference between means.
• That distribution is different depending on
  whether the variances of the two means are
  approximately equal equal or not
• Small sample CI
• If var(fabric 2) is approximately var(fabric
  3), then just replace za/2 with ta/2,n2n3-2This
  is called pooling the variances.
• If not, then use software. (Software adjusts the
  degrees of freedom for an appoximate confidence
  interval.)

Rule of thumb OK if 1/3lt(S23/S22)lt3
Read section 10.4
More conservative
6
Minitab example Stat Basic statistics 2 sample
t
Two-sample T for f2 vs f3 N Mean
StDev SE Mean f2 4 10.95 1.24
0.62 f3 4 10.50 1.14
0.57 Difference mu f2 - mu f3 Estimate for
difference 0.450 95 CI for difference
(-1.606, 2.506) T-Test of difference 0 (vs not
) T-Value 0.54 P-Value 0.611 DF 6 Both
use Pooled StDev 1.19
7
Hypothesis test comparison of 2 means

• As in the 1 mean case, replace za/2 with the
  appropriate t based cutoff value.
• When s21 approximately s22 then test statistic
  is
• t(x1x2)/-sqrt(s21/n1s22/n2)
• Reject if t gt ta/2,n1n2-2
• Pvalue 2Pr(T gt t) where Ttn1n2-2
• For unequal variances, software adjusts df on
  cutoff.

8
Paired T-test

• In previous comparison of two means, the data
  from sample 1 and sample 2 were unrelated.
  (Fabric 2 and Fabric 3 observations are
  independent.)
• Consider following
• Investigator wants to see if exercise immediately
  affects the level of a specific chemical in the
  blood. Shell do this by measuring and comparing
  the chemical in two groups a control group that
  has not exercised and a treatment group that
  has.
• Design 1 Two separate groups of 15 people
  each.
• Design 2 People are their own controls.
  Measure before and after exercise.
• Design 1 is like what weve just done. Consider
  design 2 next.

9
Data from design two One Way of Looking At it
noex exercise 1, 117
118 2, 153 156 3, 73
71 4, 64 65 5, 95 109
6, 120 123 7, 94 88 8,
106 121 9, 90 95 10,
96 110 11, 67 66 12, 102
112 13, 111 110 14, 127
133 15, 180 180
exercise
noexercise
180
160
exercise mean 110.47
140
Measure
120
100
No exercise mean 106.33
80
60
2
4
6
8
10
12
14
person
10
noex - ex
noex exercise diff 1, 117
118 -1 2, 153 156 -3 3,
73 71 2 4, 64 65 -1
5, 95 109 -14 6, 120 123
-3 7, 94 88 6 8, 106
121 -15 9, 90 95 -5 10,
96 110 -14 11, 67 66
1 12, 102 112 -10 13, 111
110 1 14, 127 133 -6 15, 180
180 0
5
0
Measure
-5
Mean difference -4.14(noex exercise)
-10
-15
2
4
6
8
10
12
14
Index
Of course, Mean difference mean( noexercise ) -
mean( exercise ) If we want to test difference
0, we need variance of differences too. Note how
difference takes person to person variability
out of the graph. This is a good thing less
variability more power
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Paired t-test

• One persons first observation is dependent on
  the persons second observation, but the
  differences are independent across twins.
• As a result, we can do an ordinary one sample
  t-test on the differences. This is called a
  paired t-test.
• When data naturally come in pairs and the pairs
  are related, a paired t-test is appropriate.

12
Paired T-test

• Minitab basic statistics paired t-test
• Paired T for Noex - Exercise
• N Mean StDev SE Mean
• Noex 15 106.33 31.03 8.01
• Exercise 15 110.47 31.73 8.19
• Difference 15 -4.13 6.46 1.67
• 95 CI for mean difference (-7.71, -0.56)
• T-Test of mean difference 0 (vs not 0)
  T-Value -2.48 P-Value 0.027
• Compare this to a 2-sample t-test

13

• Two-sample T for Noex vs Exercise
• N Mean StDev SE Mean
• Noex 15 106.3 31.0 8.0
• Exercise 15 110.5 31.7 8.2
• Difference mu Noex - mu Exercise
• Estimate for difference -4.1
• 95 CI for difference (-27.6, 19.3)
• T-Test of difference 0 (vs not ) T-Value
  -0.36 P-Value 0.721 DF 28
• Both use Pooled StDev 31.4
• Estimate of difference is the same, but the
  variance estimate is very different
• Paired std dev(difference) 1.67
• 2 sample sqrt (31.02 /15) (31.72/15)
  11.46
• Cutoff is different too
• t0.025,13 for paired
• t0.025,28 for 2 sample

Paired is often better because it often has more
power.