statistics 300: elementary statistics section 9-3

Statistics 300:Elementary Statistics

Section 9-3

Section 9-3 concernsConfidence Intervals

andHypothesis tests forthe difference of two

means, (1 – 2)

222yxyx

What is the variance of

? )( 21 xx • Apply the new concept here also:

Application: Use sample values

y

y

x

xyx

yxyx

yxyx

n

s

n

ss

sss22

2

222

222

Alternative approach when two samples come from

populations with equal variances

[ “homogeneous variances” or “homoscedastic” ]

22

21

When samples come from populations with equal variances:

222

221

222

21

of estimatean is

of estimatean is

common

common

common

s

s

and

When samples come from populations with equal variances:

21

222

2112

2

: is

of estimatebest the

dfdf

sdfsdfsp

common

Alternative notation for thepreceding formula:

11

11

21

222

2112

nn

snsnsp

When doing confidence intervalsof hypothesis tests involving

(m1 – m2) one must first decide whether the variances are

different or the same,heterogeneous or homogeneous.

If the variances are different(heterogeneous), then use thefollowing expression as part

of the confidence interval formulaor the test statistic:

2

22

1

21

n

s

n

s

If the variances are the same(homogeneous), then use thisalternative expression as part

of the confidence interval formulaor the test statistic:

pooled"" means

"" where2

2

1

2

p

n

s

n

s pp

When the variances are pooledto estimate the common

(homogeneous) variance, then the degrees of freedom for both

samples are also pooled by adding them together.

Application to CI(1 - 2)

212/

2121 )()(

xxstE

ExxCI

Application to CI(1 - 2)when variances are different

2

22

1

21

2/

2121 )()(

n

s

n

stE

ExxCI

In this case, the degrees of freedom for “t” will bethe smaller of the two sample degrees of freedom.

Application to CI(1 - 2)when variances are the same

2

2

1

2

2/

2121 )()(

n

s

n

stE

ExxCI

pp

In this case, the degrees of freedom for “t” will bethe sum of the two sample degrees of freedom.

Tests concerning (1 - 2)Test Statistic

)(

02121

21

)()(

xxs

xx

Test statistic for (1 - 2)when variances are different

2

22

1

21

02121 )()(

ns

ns

xx

In this case, the degrees of freedom for “t” will bethe smaller of the two sample degrees of freedom.

Test statistic for (1 - 2)when variances are the same

2

2

1

2

02121 )()(

n

s

n

s

xx

pp

In this case, the degrees of freedom for “t” will bethe ssum of the two sample degrees of freedom.

Section 9-3Handling Claims / Hypotheses

• Write the claim in a symbolic expression as naturally as you can

• Then rearrange the expression to have the difference between the two means on one side of the relational operator (< > = …)

Section 9-3Handling Claims / Hypotheses

• Statement: Mean #2 is less than four units more than Mean #1

• So:

• Rearrange:

• H0:

• H1:

412 412

412 412

Section 9-3Handling Claims / Hypotheses

• Statement: On average, treatment A produces 18 more than treatment B

• So:

• Rearrange:

• H0:

• H1:

18 BA 18 BA

18 BA

18 BA