Hypothesis Testing. Central Limit Theorem Hypotheses and statistics are dependent upon this theorem.

Hypothesis Testing

Central Limit Theorem

Hypotheses and statistics are dependent upon this theorem


To understand the Central Limit Theoremwe must understand the difference between

three types of distributions…..

A distribution is a type of graph showing the frequencyof outcomes:

Of particular interest is the “normal distribution”

Different populations will create differing frequencydistributions, even for the same variable…

There are three fundamental types of distributions:1. Population distributions

There are three types of distributions:1. Population distributions




There are three types of distributions:1. Population distributions2. Sample distributions



3. Sampling distributions

1. Population distributionsThe frequency distributions of a population.

2. Sample distributionsThe frequency distributions of samples.

The sample distribution should look likethe population distribution…..

Why?


3. Sampling distributionsThe frequency distributions of statistics.


The sampling distribution should NOT look likethe population distribution…..

Why?

Suppose we had population distributions that looked like these:

Say the mean was equal to 40, if we tooka random sample from this population of a certainsize n… over and over again and calculated themean each time……

We could make a distribution of nothing butthose means. This would be a samplingdistribution of means.

Some questions about this sampling distribution:

1. What would be the mean of all those means?

2. If the population mean was 40, how manyof the sample means would be larger than 40,and how many would be less than 40?

Regardless of the shape of the distributionbelow, the sampling distribution would be symmetrical around the population mean of 40.

3. What will be the variance of the sampling distribution?

The means of all the samples will be closertogether (have less variance) if the variance of

the population is smaller.

The means of all the samples will be closertogether (have less variance) if the size of

each sample (n) gets larger.

Sample

n = number of samples

So the sampling distribution will have a mean equal to the population mean, and a varianceinversely 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



If samples are large, thenthe sampling distribution created by those

samples will have a mean equal to thepopulation mean and a standard deviation

equal to the standard error.

Sampling Error = Standard Error

The sampling distribution will be a normal curve with:

x o and Snxo

This makes inferential statistics possiblebecause all the characteristics of a normal curveare known.

http://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-problem

Another great example of the theorem in action….

Hypothesis Testing:

A statistic tests a hypothesis: H0

100:0 H

Hypothesis Testing:


The alternative or default hypothesis is: HA

100: AH

Hypothesis Testing:


The alternative or default hypothesis is: HA

A probability is established to test the “null” hypothesis.

Hypothesis Testing:

95% confidence: would mean that therewould need to be 5% or less probability ofgetting the null hypothesis; the nullhypothesis would then be dropped infavor of the “alternative” hypothesis.

Hypothesis Testing:

95% confidence: would mean that therewould need to be 5% or less probability ofgetting the null hypothesis; the nullhypothesis would then be dropped infavor of the “alternative” hypothesis.

1 - confidence level (.95) = alpha

Alpha

Errors:

Errors:

Type I Error: saying nothing ishappening when something is: p = alpha

Errors:

Type I Error: saying something ishappening when nothing is: p = alpha

Type II Error: saying nothing is happening when something is: p = beta

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

An example from court cases:

http://www.youtube.com/watch?v=taEmnrTxuzo

Care must be taken when using hypothesis testing…

PROBLEMS

I hypothesize that a barking dog is hungry.

The dog barks, is the dog therefore hungry?

To answer that questions, I would have to have someprior information.

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

Suppose we flipped a coina hundred times….

It came up heads 60 times.Is it a fair coin?

No….

Because of the Z-test finds that the probability of doingthat is equal to 0.0228.

We would reject the Null Hypothesis!

Suppose we flipped the samecoin a hundred times again…

It came up tails 60 times.Is it a fair coin?

But we have now thrown thecoin two hundred times, and…

It came up tails 100 times.

Is it a fair coin?

Perfectly fair

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

Suppose we project aPoggendorf figure to one sideof the brain or to the other….

and measure error.

Paired Samples Statistics

Mean N Std. Error MeanPair 1 Right 5.4167 12 .70128 Left 4.9167 12 .62107

t(11) = 2.17, p = 0.053

What do you conclude?



t(11) = 2.17, p = 0.053

Now suppose you did this againwith another sample of 12 people.

t(11) = 2.10, p = 0.057

But the probability of independent events is: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 brainhemispheres are truly

independent….Then...



t(22) = 0.53, p = 0.60

What do you conclude now?

Read the following article….

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

Hypothesis Testing. Central Limit Theorem Hypotheses and statistics are dependent upon this theorem.

Documents

population distributions

sample distributions

theorem slide

sampl ing distributions

variable slide

time slide

normal distribution

population mean