CHAPTER 21 : Comparing Two - PowerPoint Presentation

Slide1

CHAPTER 21:Comparing TwoMeans

Lecture PowerPoint Slides

Basic Practice of Statistics

7

th

EditionSlide2

In Chapter 21, We Cover …Two-sample problems

Comparing two population meansTwo-sample

t

procedures

Using technology

Robustness again

Details of the

t

approximation*

Avoid

the pooled two-sample

t

procedures*

Avoid

inference about standard deviations

*

Permutation tests*Slide3

Two-Sample ProblemsA two-sample problem can arise from a randomized comparative experiment that randomly

divides subjects into two groups and exposes each group to a different treatment.

Comparing

random samples separately selected from two populations

is also

a two-sample problem. Unlike the matched pairs designs studied earlier,

there is

no matching of the individuals in the two samples.

The

two samples are

assumed to

be independent and can be of

different

sizes

.

The most

common goal

of inference is to compare the average or typical responses in the two populations

.Slide4

Comparing Two Population Means

conditions for inference

comparing two means

We have

two SRSs

from two distinct populations. The samples are

independent.

That is, one sample has no influence on the other.

We measure the same response variable for both samples. Both populations are Normally distributed. The means and standard deviations of the populations are unknown. In practice, it is enough that the distributions have similar shapes and that the data have no strong outliers.Call the variable in the first population and in the second because the variable may have different distributions in the two populations.Here is how we describe the two populations:

Population

VariableMeanStandard deviation12

Population

Variable

Mean

Standard deviation

1

2Slide5

Comparing Two Population Means

Here is how we describe the two samples:

To do inference about the difference

between the means of two populations, we start from the difference

between the means of the two samples.

Population

Sample

size

Sample mean

Sample standard deviation

12

Population

Sample

size

Sample mean

Sample standard deviation

1

2Slide6

Two-Sample t ProceduresTo take variation into account, we would like to standardize the observed difference

by subtracting its mean,

, and dividing the result by

its standard

deviation. This standard deviation of the

difference

in sample means is

Because we don't know the population standard deviations, we estimate them

by the

sample standard deviations from our two samples. The result is the

standard error, or estimated standard deviation, of the difference in sample means:

 Slide7

Two-Sample t ProceduresWhen we standardize the estimate by subtracting its mean,

, and dividing

the result

by its standard error, the result is the

two-sample

t

statistic:

The two-sample

t

statistic has approximately a

t

distribution. It does not have exactly a t distribution, even if the populations are both exactly Normal. In practice, however, the approximation is very accurate. There are two practical options for using the two-sample t procedures:

Option 1.

With software, use the statistic

t

with accurate critical values from

the approximating

t

distribution

.Option 2.

Without software, use the statistic

t

with critical values from the t distribution with degrees of freedom equal to the smaller of and . The significance test gives a P-value equal to or greater than the true P-value.

 Slide8

Two-Sample t ProceduresTHE TWO-SAMPLE

t PROCEDURESDraw an SRS of size

from a large Normal population with unknown mean

, and

draw an independent SRS of size

from another large Normal population

with unknown mean . A level C

confidence interval for

is given

by

Here,

is

the critical value for

confidence

level

C

for the

t

distribution with

degrees of

freedom from either Option 1 (software) or Option 2 (the smaller of and ). Slide9

Two-Sample t ProceduresTHE TWO-SAMPLE

t PROCEDURESTo

test the hypothesis

,

calculate the

two-sample

t

statistic:

Find

P

-values from the t distribution with degrees of freedom from either Option 1 (software) or Option 2 (the smaller of and ).

 Slide10

ExampleSTATE: People gain weight when they take in more energy from food than

they expend. James Levine and his collaborators at the Mayo Clinic investigated the link between obesity and energy spent on daily

activity with data from a study with

health volunteers; 10 who were lean, 10 who were mildly obese but still healthy. They wanted to address

the question: D

o

lean and obese people

differ

in the average time they spend standing and walking?PLAN: Give a 90% confidence interval for

, the

difference in average

daily minutes spent standing and walking between lean and mildly obese adults.SOLVE: Examination of the data reveals all conditions for inference can be (at least reasonably) assumed; the distributions are a bit irregular, but with only 10 observations this is to be expected. Slide11

ExampleSOLVE: (cont’d) The descriptive statistics:

For using Option 2 (conservative degrees of freedom in absence of technology),

, and

t

* = 1.833, giving:

= 79.09 to 225.87 minutes

Software using Option 1 gives

df

= 15.174 and

t

* = 1.752, for

a

confidence interval of 82.35 to 222.62 minutes—narrower because Option 2 is

conservative

.

CONCLUDE

: Whichever interval

we report

, we are (at least) 90%

confident

that the mean

difference

in average daily minutes spent standing and walking between lean and mildly obese adults lies in this interval. GroupMean, Std. Dev., s1 (lean)10525.751107.121

2 (obese)

10

373.269

67.498

Group

Std. Dev.,

s

1 (lean)

10

525.751

107.121

2 (obese)

10

373.269

67.498Slide12

ExampleCommunity service and attachment to friends

STATE: Do

college students who have volunteered for community

service

work

differ

from those who have not? A study obtained data

from 57

students who had done service work and 17 who had not. One of the response variables was a measure of attachment to friends. Here are the results:PLAN: The investigator had no specific direction for the difference in mind before looking at the data, so the alternative is two-sided. We will test the following hypotheses:

Group

Conditions1Service57105.3214.682

No service

17

96.82

14.26

Group

Condition

s

1

Service

57

105.3214.682No service1796.8214.26Slide13

ExampleSOLVE: The two-sample t

statistic:

Software

(

Option 1) says that the two-sided

P

-value is 0.0414.

For

using Option

2,

,

and

therefore comparing our test statistic of 2.142 to two-sided critical values of a

t

(16) distribution, Table C shows the

P

-value is between 0.05 and 0.04.

CONCLUDE

: The data give moderately strong evidence (P

<

0.05) that

students who have engaged in community service are, on the average, more attached to their friends. Slide14

Using TechnologyCrunchIt

and the calculator get Option 1 completely right. The accurate approximation uses the

t

distribution with approximately 15.174 (

CrunchIt

rounds this

to 15.17) degrees of freedom. The

P-value is P = 0:0008.Slide15

Using TechnologyMinitab uses Option 1, but it truncates the exact degrees of freedom to the

next smaller whole number to get critical values and P

-values. In this example,

the exact

df

= 15.174 is truncated to

df

=

15 so that Minitab's results are slightly conservative. That is, Minitab's P-value (rounded to P = 0:001 in the output) is slightly larger than the full Option 1 P-value.Excel rounds the exact degrees of freedom to the nearest whole number so that df = 15.174 becomes

df = 15. Excel’s

method agrees with Minitab’s in this example. But when rounding moves the degrees of freedom up to

the next higher whole number, Excel’s P-values are slightly smaller than is correct. This is misleading, another illustration of the fact that Excel is substandard as statistical software.Slide16

Robustness AgainThe two-sample t procedures are more robust than the one-sample

t methods, particularly when the distributions are not symmetric.When the sizes of the two

samples are

equal and the two populations being compared have distributions with

similar shapes

, probability values from the

t

table are quite accurate for a broad range

of distributions when the sample sizes are as small as . When the two population distributions have different shapes, larger samples are needed. As a guide to practice, adapt the

guidelines for one-sample t

procedures to two-sample procedures by replacing “sample size” with

the “sum of the sample sizes,” .Caution: In planning a two-sample study, choose equal sample sizes whenever possible. The two-sample t procedures are most robust against non-Normality in this case, and the conservative Option 2 probability values are most accurate. Slide17

Details of the t Approximation*The exact distribution of the two-sample

t statistic is not a t

distribution. The distribution changes as the unknown population standard deviations change. However, an excellent approximation is available.

Approximate Distribution of the Two-Sample

t

Statistic

The distribution of the two-sample

t statistic is very close to the t distribution with degrees of freedom given by

This approximation is accurate when both sample

sizes

and

are 5 or larger.

 Slide18

Avoid the Pooled Two-Sample t Procedures*Many calculators and software packages offer a choice of two-sample t

statistics. One is often labeled for “unequal” variances; the other for “equal” variances.

The “unequal” variance procedure is our two-sample

t

.

Never use the pooled

t

procedures if you have software or technology that will implement the “unequal” variance procedure.Slide19

Avoid Inference About Standard Deviations*

There are methods for inference about the standard deviations of Normal populations. The most common such method is the “F

test” for comparing the standard deviations of two Normal populations.

Unlike the

t

procedures for means, the

F

test for standard deviations is extremely sensitive to non-Normal distributions.

We do not recommend trying to do inference about population standard deviations in basic statistical practice.Slide20

As an alternative to two-sample t tests, in some instances it may be to our advantage to perform a

permutation test—in general, if experimental units are assigned to two treatment groups

completely at

random, and our null hypothesis is

“no

treatment

effect,”

we can test

hypotheses using sample means as follows:First, list all possible ways units can be assigned to treatment groups.Second, based on the data obtained, for each possible assignment determine what the difference in means (mean for treatment 1 minus mean for treatment 2) would be. Under the null hypothesis, each of these is equally likely.

Permutation Tests*Slide21

Permutation Tests*Permutation test method (cont’d):Third, determine the distribution of these possible outcomes by listing all the

different possible mean differences

and their corresponding probabilities (this can also

be represented

by a histogram). Using this sampling distribution, determine the

P

-value of

the actual mean

difference you obtained.The resulting sampling distribution of outcomes is called the permutation distribution.Practical issues:First: Unless the number of ways units can be assigned is sufficiently large, the smallest possible P-value may not be very small.Second: Listing all possible outcomes can be tediousThird: Permutation method is most likely to be useful with small- to moderate-size experiments because of the robustness of t procedures.