The Practice of StatisticsThird Edition

Chapter 10:Estimating with Confidence

Copyright © 2008 by W. H. Freeman & Company

Paired t Procedures

• Comparative studies are more convincing that

single-sample investigations

– One sample inference is less common.

• Subjects are matched in pairs and each treatment

is given to one subject in each pair.

• Paired t procedures is before and after

observations on the same subjects.

The parameter µ in a paired t procedure is:

• the mean difference in the responses to the two treatments within matched

pairs of subjects in the entire population (when subjects are matched in pairs),

or

• the mean difference in response to the two treatments for individuals in the

population (when the same subject receives both treatments),

or

• the mean difference between before and after measurements for all

individuals in the population (for before and after observations on the same

individual).

Is Caffeine Dependence Real(page 651)

• Subjects: 11 people diagnosed as dependent

on caffeine

• Each subject is barred for substances

containing caffeine.

• Took a pill instead.

• During different time period took a placebo.

• Order was randomized

Is Caffeine Dependence Real?Subject Depression

(Caffeine)

Depression

(placebo)

# of beats

(caffeine)

# of beats

(placebo)

1 5 16 281 201

2 5 23 284 262

3 4 5 300 283

4 3 7 421 290

5 8 14 240 259

6 5 24 294 291

7 0 6 377 354

8 0 3 345 346

9 2 15 303 283

10 11 12 340 391

11 1 0 408 411

High scores

mean more

symptoms of

depression.

Beats is number

of beats per

minute when

asked to press a

button.

Does being

deprived of

caffeine affect

these

outcomes?

Remember the 4 Steps

or PANIC

• Parameter of Interest

• Conditions

– SRS

– Normality

– Independence

• Calculations

• Interpretation

Parameter

• Population of interest

– People who are dependent on caffeine

• Want to estimate µDIFF = µPLACEBO – CAFFEINE

in depression score if all individuals in the

population took both the caffeine capsule and

placebo.

Conditions - SRS

• If conditions are met, use one-sample t procedures to

construct a confidence interval for µDIFF since σ is

unknown.

• Probably not a SRS.

– Subjects are usually volunteers.

• Not truly representative of population of interest.

• May have trouble generalizing results of study to a larger

population.

• Since researchers randomly assign order in which subject

took tablet, consistent differences in responses should be

due to the treatments.

Conditions - Normality

• Is the population distribution of differences Normal?No obvious Outliers Normality Plot looks pretty

Linear

Conditions - Independence

• Is the difference (placebo – caffeine) values for

the 11 subjects independent?

• Seems reasonable given the design of the study.

• Note that the two depression scores for each

subject are not independent.

– We would expect the same individual to show

similar tendencies under both treatments.

Calculations

• x-barPLACEBO – CAFFEINE = x-barDIFF

• x-barDIFF = 7.364 and sDIFF = 6.918

• Critical t value for 90% CI and 11 – 1 = 10 df

• t* = 1.812 (look in table C), so the CI is:

• 7.364 ± 1.812(6.918/√11) = 7.364 ± 3.780

• The confidence interval is (3.584, 11.144)

Interpretation

• We are 90% confident that the actual mean difference

in depression score for the population is between

3.584 and 11.144 points.

– People would score between 3.584 and 11.144 points higher on the

Depression Inventory when given a placebo instead of caffeine

• This provides evidence that withholding caffeine from

caffeine dependent individual may lead to depression.

• Warning: Subjects were NOT from a SRS of the

population of interest.

– We CANNOT generalize further.

CAUTION

• Random Selection of INDIVIDUALS for a

statistical study allows us to generalize the

results of the study to the larger population.

• Random Assignment of TREATMENTS to

subjects in an experiment lets us investigate

whether there is evidence of a treatment

effect, which may suggest the treatment

caused the observed difference

The t confidence interval is exactly correct when the distribution of the

population is exactly Normal.

No real data are exactly Normal.

The usefulness of the t procedures in practice therefore depends on how

strongly they are affected by lack of Normality.

Procedures that are not strongly affected are called robust.

If outliers are present in the sample data, then the population may not be Normal.

The t procedures are NOT robust against outliers because x-bar and s are NOT

resistant to outliers.

Back to Light-Duty Engine

Emissions (p. 654)

• There was one outlier (2.94 grams/mile).

• If we remove that data point:

• x-bar = 1.293 and s = .424

• This makes the 95% CI (1.165, 1.421)

– Original x-bar was 1.329

– Original s was .484

– Original CI (1.185, 1.473)

Can we do this?

• Not really.

• The outlier suggests that the distribution is

not Normal.

• A single outlier can have a drastic effect on

the results of t procedures.

• t procedures are quite robust against non-

Normality when there are no outliers.

Always make a plot to check for skewness and outliers before using t

procedures for small samples.

Usually it is safe to use one-sample t procedures when n ≥ 15 unless an

outlier or strong skewness is present.

We have the data for the whole population, so formal inference makes no sense.

We can calculate the exact mean of the population.

Here is a histogram of the percentage of every state’s population

over the age of 65

We have 70 observations with a symmetric distribution.

We can use t procedures to draw conclusions about the mean time

of a day’s first lightening strike with complete confidence.

This is a histogram that shows the time of day of the first lightning

strike in a region of Colorado.

This is skewed to the right and we aren’t told how large the sample is.

We can use t procedures for a distribution like this if the sample size is 30

or larger.

This is a histogram of the word lengths in Shakespeare’s plays.

Assignment

• Exercises 10.34, 10.35, 10.36

• Read Pages 661 – 668