Statistics for business economics 7th by paul newbold chapter 08

Chapter GoalsAfter completing this chapter, you should be able to:  Form confidence intervals for the difference between two means from dependent samples  Form confidence intervals fo

Statistics for Business and Economics

7th Edition

Chapter 8

Estimation: Additional Topics

Chapter Goals

After completing this chapter, you should be able to:

 Form confidence intervals for the difference between

two means from dependent samples

 Form confidence intervals for the difference between

two independent population means (standard deviations known or unknown)

 Compute confidence interval limits for the difference

between two independent population proportions

or proportion within a specified margin of error

Estimation: Additional Topics

Chapter Topics

Population Means, Independent Samples

Group 1 vs

independent Group 2

Same group

before vs after

treatment

Finite Populations

Examples:

Population Proportions

Proportion 1 vs

Proportion 2

Large Populations

Confidence Intervals

Dependent Samples

Tests Means of 2 Related Populations

 Paired or matched samples

 Repeated measures (before/after)

 Use difference between paired values:

1 i

)d

(dS

n

1 i

2 i

Dependent

samples

Confidence Interval for

d

μ n

S t

α/2 1, n d

d α/2

Confidence Interval for

Mean Difference

 The margin of error is

 tn-1,/2 is the value from the Student’s t distribution with (n – 1) degrees of freedom for which

(continued)

2

α )

t P(tn1  n1,α/2 

n

s t

α/2 1, n



Dependent

samples

 Six people sign up for a weight loss program You collect the following data:

Paired Samples Example

Weight :

Person Before (x) After (y) Difference, di

1 136 125 11

2 205 195 10

3 157 150 7

4 138 140 - 2

5 175 165 10

6 166 160 6

42

d =  di

n

4.82

1 n

) d

(d S

2 i

d







 

= 7.0

Dependent

samples

 For a 95% confidence level, the appropriate t value is

tn-1,/2 = t5,.025 = 2.571

 The 95% confidence interval for the difference between means, μd , is

12.06 μ

1.94

6

4.82 (2.571)

7

μ 6

4.82 (2.571)

7

n

S t

d

μ n

S t

d

d d

d α/2 1, n d

d α/2 1, n

Difference Between Two Means:

Population means,

independent

samples

Confidence interval uses z /2

Confidence interval uses a value from the Student’s t distribution

σx2 and σy2assumed equal

σx2 and σy2 known

σx2 and σy2 unknown

σx2 and σy2assumed unequal

(continued)

Difference Between Two Means:

Independent Samples

σx2 and σy2 Known

Population means,

independent

samples

…and the random variable

has a standard normal distribution

When σx and σy are known and both populations are normal, the

variance of X – Y is

y

2 y x

2 x 2

Y

σ n

2 x

Y X

n

σ n

σ

) μ (μ

) y x ( Z

2 X α/2

Y X

y

2 Y x

2 X α/2

n

σn

σz

)yx

(μ

μn

σn

σz

)yx

σx2 and σy2 known

σx2 and σy2 unknown

σx2 and σy2 known

σx2 and σy2 unknown

σx2 and σy2assumed unequal

 The population variances are assumed equal, so use the two sample standard deviations and pool them to estimate σ

 use a t value with (nx + ny – 2) degrees of freedom

*

σx2 and σy2assumed equal

σx2 and σy2 known

σx2 and σy2 unknown

σx2 and σy2assumed unequal

(n s

y x

2 y y

2 x x

2 p

σx2 and σy2 known

σx2 and σy2 unknown

σx2 and σy2assumed unequal

Confidence Interval,

The confidence interval for

μ1 – μ2 is:

Where

*

σx2 and σy2assumed equal

σx2 and σy2 unknown

σx2 and σy2assumed unequal

y

2 p x

2 p α/2

2, n n Y

X y

2 p x

2 p α/2

2, n n

n

s n

s t

) y x ( μ

μ n

s n

s t



2 n

n

1)s (n

1)s

(n s

y x

2 y y

2 x x

2 p

Pooled Variance Example

You are testing two computer processors for speed

Form a confidence interval for the difference in CPU

speed You collect the following speed data (in Mhz):

Calculating the Pooled Variance

4427.03 1)

14 1)

(17

-56 1 14 74

1

17 1)

n (n

S 1 n

S 1

n S

2 2

y

2 y y

2 x x

) 1

x

The pooled variance is:

The t value for a 95% confidence interval is:

2.045 t

tn n 2 ,α/2 29 ,0.025

y

Calculating the Confidence Limits

 The 95% confidence interval is

y

2 p x

2 p α/2

2, n n Y

X y

2 p x

2 p α/2

2, n n

n

s n

s t

) y x ( μ

μ n

s n

s t

) y x

(

y x y

4427.03 (2.054)

2538) (3004

μ μ 14

4427.03 17

4427.03 (2.054)

2538)

515.31μ

μ

We are 95% confident that the mean difference in

CPU speed is between 416.69 and 515.31 Mhz

σx2 and σy2 Unknown, Assumed Unequal

*

σx2 and σy2assumed equal

σx2 and σy2 known

σx2 and σy2 unknown

σx2 and σy2assumed unequal

σx2 and σy2 Unknown, Assumed Unequal

Population means,

independent

samples

(continued)

Forming interval estimates:

 The population variances are assumed unequal, so a pooled variance is not appropriate

 use a t value with  degrees

σx2 and σy2assumed unequal

1)

/(n n

s 1)

/(n n

s

) n

s (

) n

s (

y 2

y

2 y x

2

x

2 x

2

y

2 y x

2 x

Confidence Interval,

The confidence interval for

μ1 – μ2 is:

*

σx2 and σy2assumed equal

σx2 and σy2 unknown

σx2 and σy2assumed unequal

y

2 y x

2 x α/2

, Y

X y

2 y x

2 x α/2

,

n

s n

s t

) y x ( μ

μ n

s n

s t

) y x

1) /(n n

s 1) /(n n s

) n

s ( ) n

s (

y 2

y

2 y x

2

x

2 x

2

y

2 y x

2 x

Two Population Proportions

Goal: Form a confidence interval for the difference between two

Two Population Proportions

 The random variable

is approximately normally distributed

x

x x

y x

y x

n

) p (1

p n

) p (1

p

) p (p

) p p

( Z

ˆ ˆ

ˆ ˆ

ˆ ˆ

Confidence Interval for Two Population Proportions

x

x

x y

x

n

) p (1

p n

) p (1

p Z

) p p

( ˆ  ˆ   /2 ˆ  ˆ  ˆ  ˆ

Example:

Two Population Proportions

Form a 90% confidence interval for the

difference between the proportion of

men and the proportion of women who

have college degrees.

 In a random sample, 26 of 50 men and

28 of 40 women had an earned college

degree

0.70(0.30) 50

0.52(0.48) n

) p (1

p n

) p (1 p

y

y y

26

p ˆ x  

0.70 40

28

p ˆ y  

(continued)

For 90% confidence, Z/2 = 1.645

Example:

Two Population Proportions

The confidence limits are:

so the confidence interval is

.70)(.52

n

)p(1

pn

)p(1

pZ

)pp

(

y

y y

x

x

x α/2

y x

Sample Size Determination

For the

Mean

Determining Sample Size

For the Proportion

Large Populations

Finite Populations

For the Mean

For the Proportion

8.4

Margin of Error

confidence (1 - )

population parameter

form the confidence interval

Sample Size Determination

n

σ z

n

σ z

Margin of Error (sampling error)

For the Mean

Large Populations

Sample Size Determination

n

σ z

For the Mean

Large Populations

Sample Size Determination

 To determine the required sample size for the

mean, you must know:

 The desired level of confidence (1 - ), which determines the z/2 value

(continued)

Required Sample Size Example

If  = 45, what sample size is needed to estimate the mean within ± 5 with 90%

confidence?

(Always round up)

219.19 5

(45)

(1.645) ME

σ

z

2 2

2

2

2 α/2

Sample Size Determination:

Population Proportion

n

) p (1

p z

n

) p (1

p z



Margin of Error (sampling error)

For the Proportion Large

Populations

2 α/2

n to get

(continued)

n

) p (1

p z

p ˆ  ˆ

For the Proportion

Large Populations

Sample Size Determination:

Population Proportion

 The sample and population proportions, and P, are

generally not known (since no sample has been taken yet)

 P(1 – P) = 0.25 generates the largest possible margin

of error (so guarantees that the resulting sample size will meet the desired level of confidence)

proportion, you must know:

 The desired level of confidence (1 - ), which determines the critical z/2 value

 The acceptable sampling error (margin of error), ME

Required Sample Size Example:

Population Proportion

How large a sample would be necessary

to estimate the true proportion defective in

a large population within ±3%, with 95%

confidence?

Required Sample Size Example

6)

(0.25)(1.9 ME

z

0.25

2 2

2 α/2







Sample Size Determination:

Finite Populations

Finite Populations

n

N n

σ )

X Var(

2

A finite population

correction factor is added:

1 Calculate the required

sample size n0 using the prior formula:

2 Then adjust for the finite

population:

2

2

2 α/2 0

N

n n

Sample Size Determination:

Finite Populations

Finite Populations

For the Proportion

n

N n

P)

)

P(1-p Var( ˆ

A finite population

correction factor is added:

1 Solve for n:

2 The largest possible value

for this expression (if P = 0.25) is:

3 A 95% confidence interval

will extend ±1.96 from the sample proportion

P) P(1

1)σ (N

Example: Sample Size to Estimate Population Proportion

Required Sample Size Example

0.05σ

264.8 0.25

551) (849)(0.02

)

(0.25)(850 0.25

1)σ (N

Chapter Summary

 Formed confidence intervals for the paired difference

 Formed confidence intervals for the difference between two means, population variance known, using z

 Formed confidence intervals for the differences between two means, population variance unknown, using t

 Formed confidence intervals for the differences between two population proportions

 Formed confidence intervals for the population variance using the chi-square distribution

and margin of error requirements