Confidence Intervals for the Difference between Two Population Means µ1 - µ2:

Independent Samples

Confidence Intervals for the Difference between Two Population Means µ1 - µ2:

Independent Samples

Section 6.2

11

6.2 Confidence Intervals for the Difference between Two Population Means µ1 - µ2: Independent Samples

• Two random samples are drawn from the two populations of interest.

• Because we compare two population means, we use the statistic .

22

21 xx

33

Population 1 Population 2

Parameters: µ1 and 12 Parameters: µ2 and 2

2 (values are unknown) (values are unknown)

Sample size: n1 Sample size: n2

Statistics: x1 and s12 Statistics: x2 and s2

2

Estimate µ1 µ2 with x1 x2

2 21 2

1 21 2

( )s s

SE x xn n

s12

n1

s2

2

n2

dfdf

11--22

x 1 x 2

Sampling distribution model for ? 1 2x x

22 21 2

1 22 22 2

1 2

1 1 2 2

1 11 1

s sn n

dfs s

n n n n

Sometimes used (not always Sometimes used (not always very good) estimate of the very good) estimate of the

degrees of freedom isdegrees of freedom is

min(min(nn1 1 − 1, − 1, nn2 2 − 1).− 1).

1 2 1 2

2 21 2

1 21 2

( )

( )

E x x

SD x xn n

Shape?Shape?

Estimate usingEstimate using

Confidence Interval for –

55

*

*

2 21 2( )

1 21 2

where is the value from the t-table

that corresponds to the confidence level

df

df

Confidence interval

s sx x t

n n

t

22 21 2

1 22 22 2

1 2

1 1 2 2

1 11 1

s sn n

dfs s

n n n n

Confidence Interval for –

66

*

*

2 21 2( )

1 21 2

where is the value from the t-table

that corresponds to the confidence level

df

df

Confidence interval

s sx x t

n n

t

22 21 2

1 22 22 2

1 2

1 1 2 2

1 11 1

s sn n

dfs s

n n n n

Example: “Cameron Crazies”. Confidence interval for –

Do the “Cameron Crazies” at Duke home games help the Blue Devils play better defense?

Below are the points allowed by Duke (men) at home and on the road for the conference games from a recent season.

77

Pts allowed at home

44 56 44 54 75 101 91 81

Pts allowed on road

58 56 70 74 80 67 65 79

1 1 1

2 2 2

home: 68.25 21.8 8

road: 68.63 8.9 8

x s n

x s n

Example: “Cameron Crazies”. Confidence interval for –

88

Calculate a 95% CI for Calculate a 95% CI for 11 - - 22 where where 1 1 = mean points per game allowed by Duke at home.= mean points per game allowed by Duke at home.22 = mean points per game allowed by Duke on road = mean points per game allowed by Duke on road

• nn11 = 8, n = 8, n22 = 8; s = 8; s1122= = (21.8)(21.8)22 = = 475.36475.36; ; ss22

2 2 = (8.9)= (8.9)22 = 79.41 = 79.41

2 22 21 2

1 22 2 2 22 2

1 2

1 1 2 2

475.36 79.418 8

9.271 475.36 1 79.411 17 8 7 81 1

s sn n

dfs s

n n n n

• To use the t-table let’s use df = 9; t9* = 2.2622• The confidence interval estimator for the difference between two means is …

99

*9

2 21 2( )

1 21 2

475.36 79.41(68.25 68.63) 2.2622

8 8

.38 18.84 19.22,18.46

s sx x t

n n

Example: “Cameron Crazies”. Confidence interval for –

Interpretation• The 95% CI for 1 - 2 is (-19.22, 18.46).

• Since the interval contains 0, there appears to be no significant difference between1 = mean points per game allowed by Duke at home.

2 = mean points per game allowed by Duke on road

• The Cameron Crazies appear to have no affect on the ABILITY of the Duke men to play defense.

1010

How can How can this be?this be?

Example: confidence interval for –

• Example (p. 6)– Do people who eat high-fiber cereal for

breakfast consume, on average, fewer calories for lunch than people who do not eat high-fiber cereal for breakfast?

– A sample of 150 people was randomly drawn. Each person was identified as a consumer or a non-consumer of high-fiber cereal.

– For each person the number of calories consumed at lunch was recorded.

1111

Example: confidence interval for –

1212

Consmers Non-cmrs568 705498 819589 706681 509540 613646 582636 601739 608539 787596 573607 428529 754637 741617 628633 537555 748

. .

. .

. .

. .

Consmers Non-cmrs568 705498 819589 706681 509540 613646 582636 601739 608539 787596 573607 428529 754637 741617 628633 537555 748

. .

. .

. .

. .

Solution: (all data on p. 6)• The parameter to be tested is the difference between two means. • The claim to be tested is: The mean caloric intake of consumers (1) is less than that of non-consumers (2).• n1 = 43, n2 = 107; s1

2=4,103; s22=10,670

22 21 2

1 22 22 2

1 2

1 1 2 2

122.61 1

1 1

s sn n

dfs s

n n n n

Example: confidence interval for –

• Let’s use df = 120; t120* = 1.9799• The confidence interval estimator for the difference

between two means using the formula on p. 4 is

1313

*120

2 21 2( )

1 21 2

4103 10670(604.02 633.239) 1.9799

43 107

29.21 27.66 56.87, 1.55

s sx x t

n n

Interpretation

• The 95% CI is (-56.87, -1.55).• Since the interval is entirely negative (that is,

does not contain 0), there is evidence from the data that µ1 is less than µ2. We estimate that non-consumers of high-fiber breakfast consume on average between 1.55 and 56.87 more calories for lunch.

1414

• Let’s use df = min(43-1, 107-1) = min(42, 106) = 42;• t42* = 2.0181• The confidence interval estimator for the difference

between two means using the formula on p. 4 is

1515

*42

2 21 2( )

1 21 2

4103 10670(604.02 633.239) 2.0181

43 107

29.21 28.19 57.40, 1.02

s sx x t

n n

Example: (cont.) confidence interval for 1 – 2 using min(n1 –1, n2 -1) to approximate the df

Beware!! Common Mistake !!!

A common mistake is to calculate a one-sample

confidence interval for a one-sample confidence interval for

and to then conclude that and are equal if the

confidence intervals overlap.

This is WRONG because the variability in the sampling

distribution for from two independent samples is more

complex and must take into account variability coming from both

samples. Hence the more complex formula for the standard

error. 2

22

1

21

n

s

n

sSE

21 xx

INCORRECT Two single-sample 95% confidence intervals: The confidence interval for the male mean and the confidence interval for the female mean overlap,

suggesting no significant difference between the true mean for males and the true mean for females.

Male interval: (18.68, 20.12)Male Female

mean 19.4 17.9

st. dev. s 2.52 3.39

n 50 50

Female interval: (16.94, 18.86)

2 2* 1 2

1 2 .025,1 2

The 2-sample 95% confidence interval of the form

( ) for the difference between the means

is . Interval is entirely positive,

dfs sy y t n n

CORRECT

(.313, 2.69) suggesting signi

male female

between the true mean for males and the true mean for females

(evidence that true male mean is larger than true female mean).

ficant difference

0 1.5.313 2.69

Reason for Contradictory Result

1818

2 21 2 1 2

1 2 1 2

1 2 1 2

It's always true that

. Specifically,

( ) ( ) ( )

a b a b

s s s s

n n n n

SE x x SE x SE x

Does smoking damage the lungs of children exposed

to parental smoking?

Forced vital capacity (FVC) is the volume (in milliliters) of

air that an individual can exhale in 6 seconds.

FVC was obtained for a sample of children not exposed to

parental smoking and a group of children exposed to

parental smoking.

We want to know whether parental smoking decreases

children’s lung capacity as measured by the FVC test.

Is the mean FVC lower in the population of children

exposed to parental smoking?

Parental smoking FVC s n

Yes 75.5 9.3 30

No 88.2 15.1 30

x

Parental smoking FVC s n

Yes 75.5 9.3 30

No 88.2 15.1 30

We are 95% confident that lung capacity is between We are 95% confident that lung capacity is between 19.33 and 6.07 milliliters 19.33 and 6.07 milliliters LESSLESS in children of smoking in children of smoking parents. parents.

x

95% confidence interval for (µ1 − µ2), with

df = min(30-1, 30-1) = 29 t* = 2.0452::2 21 2

1 21 2

2 2

( ) *

9.3 15.1(75.5 88.2) 2.0452

30 3012.7 2.0452*3.24

12.7 6.63 ( 19.33, 6.07)

s sx x t

n n

11 = mean FVC of children = mean FVC of children

with a smoking parent;with a smoking parent;

22 = mean FVC of children = mean FVC of children

without a smoking parentwithout a smoking parent

Do left-handed people have a shorter life-expectancy than

right-handed people? Some psychologists believe that the stress of being left-

handed in a right-handed world leads to earlier deaths among

left-handers. Several studies have compared the life expectancies of left-

handers and right-handers. One such study resulted in the data shown in the table.

We will use the data to construct a confidence interval

for the difference in mean life expectancies for left-

handers and right-handers.

Is the mean life expectancy of left-handers less

than the mean life expectancy of right-handers?

Handedness Mean age at death s n

Left 66.8 25.3 99

Right 75.2 15.1 888

x

left-handed presidents

star left-handed quarterback Steve Young

We are 95% confident that the mean life expectancy for left-We are 95% confident that the mean life expectancy for left-handers is between 3.26 and 13.54 years handers is between 3.26 and 13.54 years LESSLESS than the mean than the mean life expectancy for right-handers.life expectancy for right-handers.

95% confidence interval for (µ1 − µ2), with

df = min(99-1, 888-1) = 98 t* = 1.9845::2 21 2

1 21 2

2 2

( ) *

(25.3) (15.1)(66.8 75.2) 1.9845

99 8888.4 1.9845*2.59

8.4 5.14 ( 13.54, 3.26)

s sx x t

n n

11 = mean life expectancy of = mean life expectancy of

left-handers;left-handers;

22 = mean life expectancy of = mean life expectancy of

right-handersright-handers

Handedness Mean age at death s n

Left 66.8 25.3 99

Right 75.2 15.1 888

The “Bambino”,left-handed Babe Ruth, baseball’s all-time best

player.