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Hypothesis Testing

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This modules discusses the concepts of hypothesis testing, including α-level, p-values, and statistical power.

A random sample of n = 25 measurements of chest circumferences from a population of newborns having
= 0.7 inches provides a sample mean of = 12.6 in. Is it likely that the population mean has the value µ = 13.0 in.?


The Process of Testing Hypotheses


The null hypothesis is never proved or established, but is possibly disproved in the course of experimentation.”

“Every experiment may be said to exist only to give the facts a chance of disproving the null hypothesis.”
R.A.Fisher
Design of experiments


Hypothesis Testing Key Concepts


p-value: For a specific test of a hypothesis, the likelihood or probability of observing, under the assumption that the null hypothesis is true, an outcome as far away or further from the null hypothesis than the one observed.
The p-value measures the rareness of an observed outcome, under the assumption that the null hypothesis is true. If the p-value is small, typically p < 0.05, then it is often judged that the null hypothesis is unlikely to be true because, if it were, one would not expect to have observed so unlikely an outcome.

Hypothesis Testing

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Example 1

Mango farms in Ratnagiri District produce average 500mangoes (per farm)with a standard deviation 96
(Information source-Food Report journal)
After Inclusion of special fertilizer, out put was measured from 50farms, The average output was 535mangoes per farm

Exercise

The maker of a certain model car claimed that his car averaged at least 31 milesper gallon of gasoline. A sample of 36cars was selected and each car was driven with one gallon of regular gasoline. The sample showed a mean of 29.43miles with a standard deviation of 3 miles. With 95 % Confidence level, what do you conclude about the manufacturers claim?

Errors in Hypothesis Testing

A type I error consists of rejecting the
null hypothesis H0 when it was true.
A type II error consists of not rejecting
H0 when H0 is false.
 and  are the probabilities of type
I and type II error, respectively (The so
called Alfa, Beta Error)

P-Value

The P-valueis the probability, calculated assuming H0is true, of obtaining a test statistic value at least as contradictory to H0as the value that actually resulted. The smaller the P-value, the more contradictory is the data to H0.

HYPOTHESIS


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An hypothesis is a preliminary or tentative explanation or postulate by the researcher of what the researcher considers the outcome of an investigation will be.
It is an informed/educated guess.

TYPES OF HYPOTHESIS

Null hypothesis
Alternative hypothesis
Crude hypothesis
Refined hypothesis
Non Directional hypothesis
Directional hypothesis

ALTERNATIVE HYPOTHESIS

In testing hypothesis null hypothesis is either accepted or rejected
When null hypothesis is rejected we accept another hypothesis, then that another hypothesis is called alternative hypothesis

DIRECTIONAL HYPOTHESIS

This is a type of alternative hypothesis that specifies the direction of expected findings.·
Eg: Children with high IQ will exhibit more anxiety than children with low IQ”

Hypothesis Testing

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Decision Making Under Uncertainty

You have to make decisions even when you are unsure. School, marriage, therapy, jobs, whatever.
Statistics provides an approach to decision making under uncertainty. Sort of decision making by choosing the same way you would bet. Maximize expected utility (subjective value).
Comes from agronomy, where they were trying to decide what strain to plant.

Statistical Hypotheses

Statements about characteristics of populations, denoted H:
H: normal distribution,
H: N(28,13)
The hypothesis actually tested is called the null hypothesis, H0
E.g.,
The other hypothesis, assumed true if the null is false, is the alternative hypothesis, H1
E.g.,

Testing Statistical Hypotheses - steps

State the null and alternative hypotheses
Assume whatever is required to specify the sampling distribution of the statistic (e.g., SD, normal distribution, etc.)
Find rejection region of sampling distribution –that place which is not likely if null is true
Collect sample data. Find whether statistic falls inside or outside the rejection region. If statistic falls in the rejection region, result is said to be statistically significant.

Conventional Rules

Set alpha to .05 or .01 (some small value). Alpha sets Type I error rate.
Choose rejection region that has a probability of alpha if null is true but some bigger (unknown) probability if alternative is true.
Call the result significant beyond the alpha level (e.g., p < .05) if the statistic falls in the rejection region.

Power (1)

Alpha ( ) sets Type I error rate. We say different, but really same.
Also have Type II errors. We say same, but really different. Power is 1- or 1-p(Type II).
It is desirable to have both a small alpha (few Type I errors) and good power (few Type II errors), but usually is a trade-off.
Need a specific H1 to figure power.

Summary

Conventional statistics provides a means of making decisions under uncertainty
Inferential stats are used to make decisions about population values (statistical hypotheses)
We make mistakes (alpha and beta)
Study power (correct rejections of the null, the substantive interest) is partially under our control. You should have some idea of the power of your study before you commit to it.