Inferencing Between Two Samples: Z Tests & Confidence Intervals for Difference of Means - , Study notes of Data Analysis & Statistical Methods

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Lecture 18: Inferences Based on Two Samples

Devore: Section 9.1-9.

Aug, 2006

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z

Tests and Confidence Intervals for a Difference Between Two

Population Means

An example of such hypothesis would be

μ

1

μ

2

or

σ

1

σ

2

. It may also be appropriate to estimate

μ

1

μ

2

and

compute its

α

confidence interval

1. Assumptions

X

1 ,... , X

m

is a random sample from a population with

mean

μ

1

and variance

σ

(^12)

Y

1 ,... , Y

n

is a random sample from a population with mean

μ

2

and variance

σ

(^22)

3. The

X

and

Y

samples are independent of one another

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The standard deviation of

X

Y

is

σ

¯

X

−

Y¯

σ 12

m

σ (^22)

n

The proof is also elementary:

V

X

Y

V

X

V

Y

σ

(^12)

m

σ

(^22)

n

The standard deviation is the root of the above expression

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The Case of Normal Populations with Known Variances

As before, this assumption is a simplification.

Under this assumption,

Z

X

Y¯

− ( μ 1 − μ 2 )

σ (^12)

m

σ (^22)

n

(1)

has a standard normal distribution

The null hypothesis

μ

1

μ

2

is a special case of the more

general

μ

1

μ

2

0

. Replacing

μ

1

μ

2

in (1) with

0

gives us a test statistic.

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Example

Consider Ex. 9.1 in Devore. Sample sizes are

m

and

n

. Note that

m

n

...it is not important now but will be

later...

exploratory data analysisNote that the normality suggestion is based on some

The hypotheses are

H 0 : μ 1 − μ 2

and

H a : μ 1 − μ 2 6

The test statistic is

z

¯x

¯y

σ (^12)

m

σ (^22)

n

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For a level of significance

α

,

z

α/

2

z

. 005

and

the rejection regions is

z

or

z

.

The computed value of

z

-statistic is

which is well within

the rejection region. The

P

(^) -value for this rejection region is

which mean rejection at

any reasonable

level.

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alternatives. In particular, ifSimilar results can be easily obtained for the other two possible

H a : μ 1 − μ 2 < ∆ 0

, we have

β

′ ) = 1

− Φ ( − z α − ∆ ′ − ∆ 0

σ

If

μ 1 − μ 2 6

0 , the probability of Type II Error is

z

α/

σ ) − Φ ( − z

α/

σ

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Example

probability of detecting a differenceConsider Example 9.3 from Devore. Suppose that the

between the two means

should be

. Can the

level test with

m

and

n

support this?

β For a two-sample test we have

β Because the rejection region is symmetric, we have

β

, and, therefore, the probability of detecting a

difference of

is

β

.

We can conclude that slightly larger sample sizes are needed.

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Large-Sample Tests

unnecessary and variancesIn this case, the assumption of normality for the data is

σ

1 2 (^) ,

σ

(^22)

need not be known

This is because for large

n

the variable

Z

X

Y¯

− ( μ 1 − μ 2 )

S

(^12)

m

S

(^22)

n

is approximately standard normal

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Then, if the null hypothesis is

μ

1

μ

2

0 , the test statistic

Z

X

Y¯

0

S

(^12)

m

S

(^22)

n

is approximately standard normal under the null hypothesis

This test is usually appropriate if both

m >

and

n >

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• H 0 : μ 1 − μ 2

and

H a : μ 1 − μ 2 > 0

Reject

H

0

if

Z >

Calculations:

z

27

2

40

31

2

40

Decision:

H

0

cannot be rejected at

α

; the

p

-value is

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Confidence intervals for

μ

1

μ

2

Since the test statistic

Z

that we just described is exactly

normal when

σ

1 2

and

σ

(^22)

are known,

P

z

α/

2

< Z

X

Y − ( μ 1 − μ 2 )

σ 12

m

σ 22

n

< z

α/

2  

(^) α

The

α

CI is easy to derive from this probability

statement; it is

¯x

¯y

z

α/

2 σ

¯

X

−

Y¯

where

σ

¯

X

−

Y¯

is a square root expression.

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Example

An experiment was conducted in which two types of engines,

A

and

B

, were compared. Gas mileage, in miles per gallon, was

measured.

experiments were conducted using engine type

A

and

were done for engine type

B

. The gasoline used and

engineother conditions were held constant. The average mileage for

A

was

mpg and the average for machine

B

was

mpg. Find an approximate

CI on

μ

B

μ

A

, where

μ

A

and

μ

B

are population mean gas mileage for machines

A

and

B

,

respectively. Sample standard deviation are

and

for

machines

A

and

B

, respectively.

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The point estimate of

μ

B

μ

A

is

¯x

B

¯x

A

.

For

α

, we find the critical value

z

. 02

.

Thus, the confidence interval is

Aug, 2006