Chapter 8

Interval Estimates For Proportions,

Mean Differences And Proportion Differences

Figure 8.1 The Sampling Distribution of for Our Small-Scale Illustration

p

0.0 1/6 = .167

.5 4/6 = .667

1.0 1/6 = .167

p f( )p

1.00

Figure 8.2 A Bar Chart Showing the

Sampling Distribution of pP( )p

p

.167

.667

Sample Proportion of Favorable Responses

0 .5 1.0

The Sampling Distribution of the Sample Proportion

The sampling distribution of the sample proportion is the probability distribution of all possible values of the sample proportion, , when a sample of size n is taken from a particular population.

p

Standard Deviation of the Sampling (8.1) Distribution of the Sample Proportion

n

)1(p

Figure 8.3 The Sampling Distribution of the Sample Proportion

np

)1(

p

Interval Estimate for a (8.2)

Population Proportion

nzp

)1(

Interval Estimate for a (8.3) Population Proportion using the Estimated Standard Error

p zp p

n

( )1

Determining Sample Size for (8.4) Estimating a Population Proportion

2

)1)((z

E

n =

The sampling distribution of the sample mean difference is the probability distribution of all possible values of the sample mean difference,

- , when a sample of size n1 is taken from one population and a sample of size n2 is taken from another.

The Sampling Distribution of the Sample Mean Difference

1x 2x

Standard Deviation of the Sampling (8.5) Distribution of the Sample Mean Difference

2

22

1

21

nn

2x1x - =

Figure 8.4 The Sampling Distribution of Sample Mean Difference

1 – 2

2

22

1

21

21 nnxx

21 xx

Estimating the Difference Between (8.6) the Means of Two Populations

2

22

1

21

21 )(nn

zxx

Estimating the Difference between the (8.7) Means of Two Populations: Large Sample Sizes, Population Standard Deviations are Unknown

2

22

1

21

21 n

s

n

sz)xx(

Pooled Sample Standard Deviation (8.8)

2

11s

21

222

211

-nn

)s-(n)s-(npooled

Estimating the Difference between (8.9) Means of Two Populations: Small Sample Sizes, Population Standard Deviations are Unknown but equal

( )x x ts

n

s

n1 2

2

1

2

2

pooled pooled

The sampling distribution of the sample proportion difference is the probability distribution of all possible values of the sample proportion difference, - , when a sample of size n1 is taken from one population and a sample of size n2 is taken from another.

The Sampling Distribution of the Sample Proportion Difference

p1 p2

Standard Deviation of the (8.10) Sampling Distribution of the Sample Proportion Difference

p p n n1 2

1 1

1

2 2

2

1 1

Figure 8.5 The Sampling Distribution of Sample Proportion Difference

p p n n1 2

1 1

1

2 2

2

1 1

21 pp 1 – 2

Estimating the Difference between (8.11) Two Population Proportions

( )( ) ( )

p pn n1 2

1 1 2 21 1

z

1 2

Estimating the Difference between (8.12) Two Population Proportions, s replace sp

( )( ) ( )

p p zp p

n

p p

n1 21 1

1

2 2

2

1 1

Estimating the Difference in (8.13) Two Population Means: Matched Samples

n

std d

Standard Deviation of Matched (8.14) Sample Differences

1

)( 2

n

dd isd =