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Where we have been

Making predictions about likelihood of true population

mean falling within specific range around sample mean

Want to know how closely sample statistic approximates

population parameter

In other words, we interested drawing conclusions about

characteristic of SINGLE VARIABLE (ie. test scores or time

spent studying) from sample data

mean falling within specific range around sample mean

Want to know how closely sample statistic approximates

population parameter

In other words, we interested drawing conclusions about

characteristic of SINGLE VARIABLE (ie. test scores or time

spent studying) from sample data

Where we are going

Test whether mean value of SINGLE variable is greater

or less than predetermined value

AND

Test whether mean values of SINGLE variable from

TWO DIFFERENT samples are equivalent

or less than predetermined value

AND

Test whether mean values of SINGLE variable from

TWO DIFFERENT samples are equivalent

TEST HYPOTHESES about

Single variable in relation to specific value OR

Single variable from one sample in relation to same

variable from another sample

Single variable from one sample in relation to same

variable from another sample

Two Types of Hypothesis Tests

single means/two means test

single means test

Test whether population

mean is greater to or lesser than predetermined

value

mean is greater to or lesser than predetermined

value

two means test

Test whether population

means from two distinct groups actually differ

means from two distinct groups actually differ

single means test example

New 5th grade reading

program increases standardized test scores by

more than 1.25 grade levels

program increases standardized test scores by

more than 1.25 grade levels

two means test example

Reading Program X increases

standardized test scores more than Reading

Program Y

standardized test scores more than Reading

Program Y

null hypothesis (Ho)

States that in population there is

no association, no change, or no difference between two

variables or conditions. Indicates statistical

INDEPENDENCE

no association, no change, or no difference between two

variables or conditions. Indicates statistical

INDEPENDENCE

alternative hypothesis (H1)

States that in population

there is an association, change, or difference between two

variables or conditions. Indicates statistical

DEPENDENCE.

there is an association, change, or difference between two

variables or conditions. Indicates statistical

DEPENDENCE.

null hypothesis example

New 5th grade

reading program does NOT increase

standardized test scores by more than 1.25

grade levels per year.

reading program does NOT increase

standardized test scores by more than 1.25

grade levels per year.

alternative hypothesis example

New 5th

grade reading program increases

standardized test scores by more than 1.25

grade levels per year.

grade reading program increases

standardized test scores by more than 1.25

grade levels per year.

What's up with this null hypothesis nonsense?

Because of basic assumptions about epistemology & philosophy

of science, can only directly test null hypothesis

Can only REJECT or FAIL TO REJECT null hypothesis. Cannot

prove that alternative hypothesis is true.

of science, can only directly test null hypothesis

Can only REJECT or FAIL TO REJECT null hypothesis. Cannot

prove that alternative hypothesis is true.

Central principle of inductive reasoning (null)

single study can

never PROVE something to be true. We can only FAIL TO

PROVE that it is false (thanks Karl Popper).

This is what is meant by "falsifiability" in science. In order for

hypothesis to be testable, it has to be possible to prove it to be

false.

It is basically science's way of being VERY CONSERVATIVE

about conclusions we draw

never PROVE something to be true. We can only FAIL TO

PROVE that it is false (thanks Karl Popper).

This is what is meant by "falsifiability" in science. In order for

hypothesis to be testable, it has to be possible to prove it to be

false.

It is basically science's way of being VERY CONSERVATIVE

about conclusions we draw

Type I Error

We reject null hypothesis when, in fact, null hypothesis is

really true

Conclude that treatment had effect when it actually was

not effective (less conservative conclusion)

Occurs when information from sample is misleading.

Cannot make "correct" estimates about population

parameters from sample statistics

Probability of making Type I error is alpha

really true

Conclude that treatment had effect when it actually was

not effective (less conservative conclusion)

Occurs when information from sample is misleading.

Cannot make "correct" estimates about population

parameters from sample statistics

Probability of making Type I error is alpha

Type II Error

We fail to reject null hypothesis when, in fact, null

hypothesis is really false

Conclude that treatment had no effect when it actually

was effective (more conservative conclusion)

Occurs when hypothesis test fails to detect statistical

dependence

Probability of making Type II error is beta (b)

hypothesis is really false

Conclude that treatment had no effect when it actually

was effective (more conservative conclusion)

Occurs when hypothesis test fails to detect statistical

dependence

Probability of making Type II error is beta (b)

Which Error to Minimize?

Need to carefully examine specific research question

What if you want to determine if sexual contact is related

to particular viral infection

Want to use this information to decide whether or not to

inform public about potential risk.

Ho: Sexual contact is not related to viral infection (do not

inform patients)

H1: Sexual contact is related to viral infection (inform

patients)

What if you want to determine if sexual contact is related

to particular viral infection

Want to use this information to decide whether or not to

inform public about potential risk.

Ho: Sexual contact is not related to viral infection (do not

inform patients)

H1: Sexual contact is related to viral infection (inform

patients)

Which Error to Minimize?

Fail to reject null hypothesis & say that sexual contact is NOT

related to virus when, in fact, it is (Type II error). Therefore, you

do not inform public of risks.

Implications: jeopardize health of sexually active

individuals

OR

Reject null hypothesis & conclude that sexual contact IS related

to virus when, in fact, it is not (Type I error). Therefore, you tell

public there are risks which really do not exist.

Implications: people have safer sex when they don't really

need to, as least as far as this virus is concerned

related to virus when, in fact, it is (Type II error). Therefore, you

do not inform public of risks.

Implications: jeopardize health of sexually active

individuals

OR

Reject null hypothesis & conclude that sexual contact IS related

to virus when, in fact, it is not (Type I error). Therefore, you tell

public there are risks which really do not exist.

Implications: people have safer sex when they don't really

need to, as least as far as this virus is concerned

What is the point?

Must carefully consider your research question(s) when

deciding what type of error to minimize

However, we will largely focus on decreasing Type I error

because it is more common in social sciences AND

Type I error typically can result in potentially serious

consequences

deciding what type of error to minimize

However, we will largely focus on decreasing Type I error

because it is more common in social sciences AND

Type I error typically can result in potentially serious

consequences

How to Minimize Errors

increase sample size/replicating study by selecting new sample

increase sample size

Reduces error because samples are never identical

to population from which drawn

to population from which drawn

Replicate study by selecting new sample

Reduces error because samples are never identical

to one another

to one another

Steps to follow

1. State null & alternative hypotheses

2. Set significance level

3. Determine critical region

4. Collect data & compute test statistic(s)

5. Make decision to either reject null hypothesis or fail

to reject null hypothesis

2. Set significance level

3. Determine critical region

4. Collect data & compute test statistic(s)

5. Make decision to either reject null hypothesis or fail

to reject null hypothesis

Take Home Message (of example)

By doubling sample size, we can now reject null hypothesis

and conclude, with 99% certainty, that the Cooper gets

less than 35 miles to the gallon...

Even though sample mean & population SD did not change

The bigger sample size, more information we have about

given population so...

Our sample statistics will more accurately approximate

population parameters

and conclude, with 99% certainty, that the Cooper gets

less than 35 miles to the gallon...

Even though sample mean & population SD did not change

The bigger sample size, more information we have about

given population so...

Our sample statistics will more accurately approximate

population parameters

Two Tailed Single Means Test

Tests whether population mean is equal to or not

equal to predetermined value

Previously (with one-tailed test) we were testing

whether population mean was greater than or less

than predetermined value

equal to predetermined value

Previously (with one-tailed test) we were testing

whether population mean was greater than or less

than predetermined value

Error in TwoTailed Test

Alpha level that we choose (probability of Type I error) is

now distributed in BOTH tails of distribution (rather than

in just one tail)

You can make a Type I error in two ways:

(a) by rejecting H0 because you think μ is greater than

value of interest when it is not OR

(b) by rejecting H0 because you think μ is less than value

of interest when it is not

now distributed in BOTH tails of distribution (rather than

in just one tail)

You can make a Type I error in two ways:

(a) by rejecting H0 because you think μ is greater than

value of interest when it is not OR

(b) by rejecting H0 because you think μ is less than value

of interest when it is not

Relationship between One & Two

Tailed Critical Value

Tailed Critical Value

For a given a, one-tailed CV will be smaller than

two-tailed CV (ie. closer to zero)

For a = 0.05, one-tailed CV is 1.65 while two-tailed

CV is 1.96

This is because we are dividing alpha by two

two-tailed CV (ie. closer to zero)

For a = 0.05, one-tailed CV is 1.65 while two-tailed

CV is 1.96

This is because we are dividing alpha by two

Interpret Results

Based on our findings, we reject null hypothesis &

conclude that educational attainment was significantly

different in 2007 than in 2000

conclude that educational attainment was significantly

different in 2007 than in 2000

A Reminder

We expect sample mean to approximate population mean

Standard error provides simple measure of degree to which

sample mean differs from population mean

Based on mean & SD we can calculate Z-score

This Z-score indicates whether observed difference is

significantly greater than would be expected by

chance alone

Standard error provides simple measure of degree to which

sample mean differs from population mean

Based on mean & SD we can calculate Z-score

This Z-score indicates whether observed difference is

significantly greater than would be expected by

chance alone