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Review Chaps 1-2

Chapter 1

Inference: When we make an inference we are drawing a conclusion from incompleteinformation. The science of statistics is to formulate rules about the validity of these

inferences. The realm of valid inferences is called thescope of inference.

Causal Inference can be justified only from arandomized experiment. A

randomized experiment is defined as an experiment where the investigator uses some

randomization mechanism to assign experimental units to groups.Another kind of experiment is theobservational study. While we can obtain

much interesting and important information from an observational study, we can not infer

cause and effect and must be very careful as to what conclusions we draw.Randomization in an experiment helps us control forconfounding variables. We cannot

be 100% sure that we have accounted for all confounding variables in an experiment.However, randomization gives us the best chance that what we observe is caused by thetreatments and not by unobserved variables that are related to both group membership

and the response variable.Example: Creativity Experiment

The investigator randomly assigns students in creative writing at her university to

one of two groupsthe intrinsic group, who completes a questionnaire focusing on

intrinsic rewards for creativity, and an extrinsic group, who completes a questionnaire

focusing on extrinsic rewards for creativity. This is a randomized experiment, even

though the students are not a random sample from the population of all creative writers.

We can infer cause and effect from the results, however, we can not extrapolate to a

broader population than that of the students we are studying.

Population Inference can be justified only from a random sample. Only whenwe take a random sample from the population do we have mathematical models for

quantifying the behavior of our population estimates. Randomization gives us the highest

probability that what we sample is representative of the population proportions anddistribution.

Example: Lead content in teeth.

Researchers measured the lead content in teeth and the IQ score for 3,229

children attending first and second grade in the period 1975 and 1978.

Is this a randomized experiment or an observational study?

What is the scope of inference?

What is the value of this study?

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Standard error: of a statistic is an estimate of the standard deviation of its

sampling distribution.

Example: the standard error of the sample average is ( )s

SE Yn

= . Where

( )2

2 1

1

n

i

i

y y

sn

=

=

Z and t - ratios: the ratio of an estimates error to its standard error is a convenienttest statistic. When the standard deviation is known, we use theZ-ratio. When the

standard deviation is estimated, we use thet- ratio. Most often we will need to use an

estimate of the standard error and hence we will most often use the t-ratio.

Example: 0( )

ytSE y

= is the t-ratio used to test the hypothesis that the true mean

is 0 in a one sample t-test.

Example:( )

1 2

1 2

0y yt

SE y y

=

is the t-ratio to test the hypothesis that the true

difference between paired observations is zero.

Example: Schizophrenia Study

Researchers examined 15 pairs of monozygotic (identical) twins where one twin

had schizophrenia and the other did not. They measured the volumes of several regions

inside the twins brains using MRI. The observed average difference in the means was

0.199 cm3 and the sample of differences had a standard deviation of 0.238 cm3.

What is the t-ratio?

What is the p-value?

What is the confidence interval?

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Can the difference be attributed to chance alone?

Two sample t tests: In addition to the normal or near-normal assumption, for the two

sample t test we also require the assumption of independent samples. The observationswithin a sample must be both independent of each other and of all observations in the

other sample. The two-sample test statistic is given by

( )( )

1 2 1 2

1 2

( )y yt

SE y y

=

Where

( )2 2

1 21 2

1 2

s sSE y y

n n = + when equal variances cannot be assumed. And is given by

( )2 2

1 2

1 2 1 2

1 1p pp

s sSE y y s

n n n n = + = + where

( ) ( )2 21 1 2 221 2

1 1

2p

n s n ss

n n

+ =

+ when equal

variances can be assumed.

Note: The course text uses the pooled estimate of the standard error because this is the

estimate used in ANOVA, an extension of the t-test. As mentioned in 444, the unpooledestimate is considered superior for most tests. However, we cannot use the unpooled in

the ANOVA situation and so we review the pooled estimate here.

Example: Sparrow humerus length and survival

Investigators wanted to test whether selection pressure is acting on sparrow humerus

length. After a severe cold snap, they measured humerus length of 24 sparrows that died

and 35 that survived.

Group N Average (in.) SD(in)

Died 24 0.72792 0.02355

Lived 35 0.73800 0.01984

What is the estimate of the difference?

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What are the degrees of freedom?

What is the confidence interval?

What is the t-ratio?

What is the p-value?

What is the conclusion?