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Hypothesis Testing. Central Limit Theorem. Hypotheses and statistics are dependent upon this theorem. Central Limit Theorem. To understand the Central Limit Theorem we must understand the difference between three types of distributions….

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Central Limit Theorem

Hypotheses and statistics are

dependent upon this theorem


Central Limit Theorem

To understand the Central Limit Theorem

we must understand the difference between

three types of distributions…..


A distribution is a type of graph showing the frequency

of outcomes:


Of particular interest is the “normal distribution”


Different populations will create differing frequency

distributions, even for the same variable…



There are three types of distributions:

  • Population distributions


There are three types of distributions:

  • Population distributions


There are three types of distributions:

  • Population distributions


There are three types of distributions:

  • Population distributions


There are three types of distributions:

  • Population distributions

  • Sample distributions


There are three types of distributions:

  • Population distributions

  • Sample distributions


There are three types of distributions:

  • Population distributions

  • Sample distributions

  • Samplingdistributions



2. Sample distributions

The frequency distributions of samples.

The sample distribution should look like

the population distribution…..

Why?


2. Sample distributions

The frequency distributions of samples.


3. Samplingdistributions

The frequency distributions of statistics.


2. Sample distributions

The frequency distributions of samples.

The sampling distribution should NOT look like

the population distribution…..

Why?



Say the mean was equal to 40, if we took

a random sample from this population of a certain

size n… over and over again and calculated the

mean each time……


We could make a distribution of nothing but

those means. This would be a sampling

distribution of means.


Some questions about this samplingdistribution:



2. If the population mean was 40, how many

of the sample means would be larger than 40,

and how many would be less than 40?


Regardless of the shape of the distribution

below, the sampling distribution would be

symmetrical around the population mean of 40.



The means of all the samples will be closer

together (have less variance) if the variance of

the population is smaller.


The means of all the samples will be closer

together (have less variance) if the size of

each sample (n) gets larger.



So the sampling distribution will have a mean

equal to the population mean, and a variance

inversely proportional to the size of the sample (n),

and proportional to the variance of the population.

http://www.khanacademy.org/math/statistics/v/central-limit-theorem

http://www.khanacademy.org/math/statistics/v/sampling-distribution-of-the-sample-mean



Central Limit Theorem

If samples are large, then

the sampling distribution created by those

samples will have a meanequal to the

population mean and a standard deviation

equal to the standard error.




This makes inferential statistics possible

because all the characteristics of a normal curve

are known.


http://www.statisticalengineering.com/central_limit_theorem.htmhttp://www.statisticalengineering.com/central_limit_theorem.htm

A great example of the theorem in action….


https://www.khanacademy.org/math/probability/statistics-inferential/sampling_distribution/v/sampling-distribution-example-problemhttps://www.khanacademy.org/math/probability/statistics-inferential/sampling_distribution/v/sampling-distribution-example-problem

Another great example of the theorem in action….


Hypothesis Testing:https://www.khanacademy.org/math/probability/statistics-inferential/sampling_distribution/v/sampling-distribution-example-problem

A statistic tests a hypothesis: H0


Hypothesis Testing:https://www.khanacademy.org/math/probability/statistics-inferential/sampling_distribution/v/sampling-distribution-example-problem

A statistic tests a hypothesis: H0

The alternative or default hypothesis is: HA


Hypothesis Testing:https://www.khanacademy.org/math/probability/statistics-inferential/sampling_distribution/v/sampling-distribution-example-problem

A statistic tests a hypothesis: H0

The alternative or default hypothesis is: HA

A probability is established to test the

“null” hypothesis.


Hypothesis Testing:https://www.khanacademy.org/math/probability/statistics-inferential/sampling_distribution/v/sampling-distribution-example-problem

95% confidence: would mean that there

would need to be 5% or less probability of

getting the null hypothesis; the null

hypothesis would then be dropped in

favor of the “alternative” hypothesis.


Hypothesis Testing:https://www.khanacademy.org/math/probability/statistics-inferential/sampling_distribution/v/sampling-distribution-example-problem

95% confidence: would mean that there

would need to be 5% or less probability of

getting the null hypothesis; the null

hypothesis would then be dropped in

favor of the “alternative” hypothesis.

1 - confidence level (.95) = alpha


Alphahttps://www.khanacademy.org/math/probability/statistics-inferential/sampling_distribution/v/sampling-distribution-example-problem


Errors:https://www.khanacademy.org/math/probability/statistics-inferential/sampling_distribution/v/sampling-distribution-example-problem


Errors:https://www.khanacademy.org/math/probability/statistics-inferential/sampling_distribution/v/sampling-distribution-example-problem

Type I Error: saying nothing is

happening when something is:

p = alpha


Errors:https://www.khanacademy.org/math/probability/statistics-inferential/sampling_distribution/v/sampling-distribution-example-problem

Type I Error: saying something is

happening when nothing is:

p = alpha

Type II Error: saying nothing is

happening when something is:

p = beta


http://www.youtube.com/watch?v=taEmnrTxuzohttps://www.khanacademy.org/math/probability/statistics-inferential/sampling_distribution/v/sampling-distribution-example-problem

An example from court cases:

http://www.intuitor.com/statistics/T1T2Errors.html


Care must be taken when using hypothesis testing…https://www.khanacademy.org/math/probability/statistics-inferential/sampling_distribution/v/sampling-distribution-example-problem

PROBLEMS

I hypothesize that a barking dog is hungry.


The dog barks, is the dog therefore hungry?https://www.khanacademy.org/math/probability/statistics-inferential/sampling_distribution/v/sampling-distribution-example-problem


To answer that questions, I would have to have somehttps://www.khanacademy.org/math/probability/statistics-inferential/sampling_distribution/v/sampling-distribution-example-problem

prior information.

For example, how often does the dog bark when it is not hungry.


Suppose we flipped a coinhttps://www.khanacademy.org/math/probability/statistics-inferential/sampling_distribution/v/sampling-distribution-example-problem

a hundred times….

It came up heads 60 times.

Is it a fair coin?


No….https://www.khanacademy.org/math/probability/statistics-inferential/sampling_distribution/v/sampling-distribution-example-problem

Because of the Z-test finds that the probability of doing

that is equal to 0.0228.

We would reject the Null Hypothesis!


Suppose we flipped the samehttps://www.khanacademy.org/math/probability/statistics-inferential/sampling_distribution/v/sampling-distribution-example-problem

coin a hundred times again…

It came up tails 60 times.

Is it a fair coin?


But we have now thrown thehttps://www.khanacademy.org/math/probability/statistics-inferential/sampling_distribution/v/sampling-distribution-example-problem

coin two hundred times, and…

It came up tails 100 times.

Is it a fair coin?


Perfectly fairhttps://www.khanacademy.org/math/probability/statistics-inferential/sampling_distribution/v/sampling-distribution-example-problem

The probability of rejecting the null hypothesis is now ZERO!!


Suppose we project ahttps://www.khanacademy.org/math/probability/statistics-inferential/sampling_distribution/v/sampling-distribution-example-problem

Poggendorf figure to one side

of the brain or to the other….

and measure error.


Paired Samples Statisticshttps://www.khanacademy.org/math/probability/statistics-inferential/sampling_distribution/v/sampling-distribution-example-problem

Mean N Std. Error Mean

Pair 1 Right 5.4167 12 .70128

Left 4.9167 12 .62107

t(11) = 2.17, p = 0.053

What do you conclude?


Paired Samples Statisticshttps://www.khanacademy.org/math/probability/statistics-inferential/sampling_distribution/v/sampling-distribution-example-problem

Mean N Std. Error Mean

Pair 1 Right 5.4167 12 .70128

Left 4.9167 12 .62107

t(11) = 2.17, p = 0.053

Now suppose you did this again

with another sample of 12 people.

t(11) = 2.10, p = 0.057


But the probability of independent events is:https://www.khanacademy.org/math/probability/statistics-inferential/sampling_distribution/v/sampling-distribution-example-problem

p(A) X p(B) so that:

The Null hypothesis probability for both studies was:

0.053 X 0.057 = 0.003

What do you conclude now?


But if the brainhttps://www.khanacademy.org/math/probability/statistics-inferential/sampling_distribution/v/sampling-distribution-example-problem

hemispheres are truly

independent….

Then...


Paired Samples Statisticshttps://www.khanacademy.org/math/probability/statistics-inferential/sampling_distribution/v/sampling-distribution-example-problem

Mean N Std. Error Mean

Pair 1 Right 5.4167 12 .70128

Left 4.9167 12 .62107

t(22) = 0.53, p = 0.60

What do you conclude now?


Read the following article….https://www.khanacademy.org/math/probability/statistics-inferential/sampling_distribution/v/sampling-distribution-example-problem

http://commonsenseatheism.com/wp-content/uploads/2011/01/Siegfried-Odds-Are-Its-Wrong.pdf


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