Math 4030 – 9bComparing Two Means
1
• Dependent and independent samples
• Comparing two means
Choice of test depend on
2
• Independent samples design or Matched pairs design (Sec. 8.1)
• Size of samples• Equal variances• Normality
3
Independent Samples
Matched Pairs Sample
One sample of differences (Sec.
8.4)
Z test (Sec. 8.2)
Large samples?
(≥ 30)
Y
Normality?N
Y
Equal variance?
Yt test with df = n1 + n2 -2, using pooled estimator for the
common variance (Sec. 8.3)
Y
t test with estimated degree of freedom (Sec. 8.3)
N
Nonparametric Tests (Ch.14)
N
Population 1 (may or may not be normally distributed), with mean 1 (to be estimated and compared) and variance 2
1 (may or may not known).
Sample of size n1:
With sample mean and sample variance:
4
Data format for independent samples:
1,...,, 21 nXXX
2,...,, 21 nYYY
21 and SX
Population 2 (may or may not be normally distributed), with mean 2 (to be estimated and compared) and variance 2
2 (may or may not known).
Sample of size n2:
With sample mean and sample variance:
22 and SY
5
Sampling distribution of : YX
21 YEXEYXE
2
22
1
21
nnYVarXVarYXVar
2
22
1
21
n
S
n
S
Distribution? CLT still apply?
Case 1: both samples are large (n1 ≥ 30, n2 ≥ 30) (Sec. 8.2)
6
,,~2
22
1
21
21
nn
NYX
or 1,0~
2
22
1
21
21 N
nn
YXZ
2
22
1
21
n
S
n
S
Example 1:
7
It is believed that the resistance of certain electric wire can be reduced by 0.05 ohm by alloying. (Assuming standard deviation of resistance of any wire is 0.035 ohm.)A sample of 32 standard wires and 32 alloyed wires are sampled. Question: Find the probability that average resistance of 32 standard wires is at least 0.03 ohm higher than that of 32 alloyed wires.
X
Confidence interval for :
8
21
2
22
1
21
2/ n
s
n
szyx
Test statistic for H0: 021
2
22
1
21
0
nS
nS
YXZ
Example 2:
9
It is believed that the resistance of certain electric wire can be reduced by alloying. To verify this, a sample of 32 standard wires results the sample mean 0.136 ohm and sample sd 0.034 ohm, and a sample of 32 alloyed wires results the sample mean of 0.083 ohm and sample sd 0.036 ohmQuestion: Construct a 95% confidence interval for the mean resistance reduction due to alloying.
X
Example 3:
10
It is claimed that the resistance of certain electric wire can be reduced by more than 0.05 ohm by alloying. To verify this, a sample of 32 standard wires results the sample mean 0.136 ohm and sample sd 0.004 ohm, and a sample of 32 alloyed wires results the sample mean of 0.083 ohm and sample sd 0.005 ohm.Question: Can we support the claim at = 0.05 level?
X
Case 2.1: Small sample(s), normal populations with known equal variance 2 (Sec. 8.3)
11
,11
,~21
221
nn
NYX
or 1,0~
11
21
21 N
nn
YXZ
Example 4:
12
It is claimed that the resistance of certain electric wire can be reduced by more than 0.05 ohm by alloying. To verify this, a sample of 15 standard wires results the sample mean 0.136 ohm, and a sample of 15 alloyed wires results the sample mean of 0.083 ohm. (Assume that the resistance has normal distribution with standard deviation 0.0049 ohm for any types of wire.)Question: Can we support the claim at = 0.05 level?
X
Case 2.2: Small sample(s), normal populations with unknown equal variance (Sec. 8.3)
13
1~11
21
21
21
nnt
nnS
YXt
p
2
11
21
222
2112
nn
SnSnS p
Example 5:
14
It is claimed that the resistance of certain electric wire can be reduced by more than 0.05 ohm by alloying. To verify this, a sample of 15 standard wires results the sample mean 0.136 ohm and sample sd 0.0049, and a sample of 15 alloyed wires results the sample mean of 0.083 ohm and sample sd 0.0052 . (Assume that the resistance has normal distribution with the same variance)Question: Can we support the claim at = 0.05 level?
X
Case 2.3: Small sample(s), normal populations with unequal variance (Sec. 8.3)
15
2
22
1
21
21'
nS
nS
YXt
11 2
2
2
22
1
2
1
21
2
2
22
1
21
n
ns
n
ns
ns
ns
df
has t distribution with estimated degree of freedom
Only one population, and one sample of size n, but two measurements:
16
nXXX ,...,, 21
nYYY ,...,, 21
Matched Pairs Samples (Sec. 8.4)
Since we are interested in the differences, this is really a one sample problem:
where
nDDD ,...,, 21
iii YXD
17
Sampling distribution of : D
DYEXEYXEDE 21
YVarXVarDVar
Test the hypothesis D = 0 vs. Confidence interval containing 0.
.1
1
1
22
n
iiD DD
nS
Example 6:
18
It is claimed that the resistance of certain electric wire can be reduced by more than 0.05 ohm by alloying. To verify this, a sample of 15 wires are tested before the alloying and again after the alloying, we find the mean reduction 0.063 ohm, and the sd of the reductions 0.025. (Assume that the resistance has normal distribution)Question: Can we support the claim at = 0.05 level?
X
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