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