Hypothesis Testing

• Branch of statistics concerned with using probability concept to deal with uncertainty in decision making

• It is the process of selecting and using a sample statistic to draw inference about a population parameter based on a sample drawn from the population

• Statistical inference deals with:

1. Hypothesis testing: to test some hypothesis about parent population

2. Estimation: to use the statistics obtained from the sample as an estimate of the unknown parameter of the population

Hypothesis

• It begins with an assumption called Hypothesis, that we make about a population parameter

• It is simply a quantitative statement about a population

Actually, the hypothesis is “an island in the uncharted seas of thought to be used as bases for consolidation and recuperation as we advance into the unknown”

Procedure in hypothesis testing

1. Set up a hypothesis

• The first thing in hypothesis testing is to set up a hypothesis about a population parameter.

• Generally, we are setting up two different hypotheses about the population parameter;

1. Null hypothesis

2. Alternate hypothesis

3. Setting a test criterion

• This involves the selection of an appropriate probability distribution for a particular test eg. Probability distributions that are commonly used in testing procedures are t, F and c²

• Test criteria must employ an appropriate probability distribution example, if only small sample information is available normal distribution can’t be applied.

4. Doing Computation

• We perform various calculative steps from a random sample of size n.

• These calculations involved the testing statistics and standard error associated with it.

5. Decision making

• We may draw a statistical conclusion and make decision.

• The decision will be depending on whether the computed value of the test criteria falls in the region of rejection or acceptance.

Estimation

Interval Estimates

• Statement of two values between which it is estimated that the parameter lies

• Specified by Confidence interval = Upper value – Lower value

• Interval of plausible values, hence named Interval mapping

• If we estimate the average marks obtained by statistics student is 40, it is a point estimate but if we say that the average marks is between 35 to 43, it is an interval estimate

In a scientific investigation many a times it is not necessary to know the exact value of a parameter, but desirable to have some confidence that value lies within certain range

Properties of a good estimator

• The numerical value of the sample means is said to be an estimate of the population mean figure.

• On the other hand, the statistical measure used i.e method of estimation is referred to as an estimator.

• A good estimator, as common sense dictates, is close to the parameter being estimated

Quality of a good estimator

Quality of good estimator is evaluated in terms of,

1. Unbiasedness: An estimator is said to be unbiased if its expected value is identical with the population parameter being estimated i.e. θ ̂≅ θ

2. Consistency: If an estimator (θ ̂) approaches the parameter (θ) closer and closer as sample size n increases, estimator θ ̂ is said to be a consistent estimator of θ. More precisely, the estimator θ ̂ is a consistent estimator of θ, if n approaches infinity.

3. Efficiency: It refers to the sampling variability of an estimator. If two competing estimators are unbiased, the one with smaller variance is said to be relatively more efficient.

4. Sufficiency: An estimator is sufficient if it conveys as much information as is possible about the parameter which is contained in the sample