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  Theory of Hypothesis Testing

Basic Definitions:

Null Hypothesis (H0): It can be stated by the statement of zero or no change. When the original claim comprises equality (<=, =, or >=), it is the null hypothesis. If the original claim doesn’t comprise equality (<, not equal, >) then the null hypothesis is the complement of original claim. The null hypothesis always comprises the equal sign. The decision is mainly based on null hypothesis.

Alternative Hypothesis (H1 or Ha): It is the statement that is true when the null hypothesis is false. The kind of test (left, right or two-tail) is mainly based on alternative hypothesis.

Type I error: Declining the null hypothesis whenever it is true (stating false when true). It is generally the most serious error.

Type II error: Failing to refuse the null hypothesis whenever it is false (stating true when false).

alpha: It is the probability of committing a Type I error.

beta: It is the probability of committing a Type II error.

Test statistic: the sample statistic is employed to decide whether to refuse or fail to refuse the null hypothesis.

Critical region: Set all the values that would cause us to refuse H0

Critical value(s): The value(s) that separate the critical area from the non-critical area. The critical values are recognized independently of the sample statistics.

Significance level (alpha): It is the probability of refusing the null hypothesis whenever it is true. alpha = 0.05 and alpha = 0.01 are very common. When no level of significance is given, use alpha = 0.05. Level of significance is the complement of level of confidence in the estimation.

Decision: It is a statement mainly based on the null hypothesis. It either rejects the null hypothesis or fail to reject the null hypothesis. It is very hard to use the null hypothesis.

Conclusion: It is a statement that points out the level of evidence (that is, sufficient or insufficient), at what level of significance, and whether the original claim is refused (that is, null) or supported alternatively.

Hypothesis Testing:

Be sure to read via the definitions for this part before trying to make sense out of the given.

The very first thing to do whenever given a claim is to write the claim mathematically (whenever possible), and decide whether the claim given is null or alternative hypothesis. If the claim given comprises equality, or a statement of no change from given or accepted condition, then this is the null hypothesis, or else, if it represents change, then this is the alternative hypothesis.

The given illustration is not a mathematical instance, however might help in introducing the concept.

Illustration:

‘He's dead, Jim,’ said Dr. McCoy to the Captain Kirk.

Mr. Spock, as a science officer, is put in charge of statistically determining the accuracy of Bone’s statement and deciding the fate of crew member (that is, to vaporize or try to revive)

His very first step is to arrive at the hypothesis which is to be tested.

Does the statement signify a change in prior condition?

•    Yes, there is change, therefore it is the alternative hypothesis, H1
•    No, there is no change, thus is the null hypothesis, H0

The accurate answer is that there is change. Dead symbolizes a change from the accepted state of alive. The null hypothesis always symbolizes no change. Thus, the hypotheses are:

H0: Patient is alive.
H1: Patient is not alive (that is, dead).

The states of nature are something that you, as a statistician encompass no control over. Either it is, or it is not. This symbolizes the true nature of things.

a) Patient is dead (H0 false - H1 true)
b) Patient is alive (H0 true - H1 false)

Decisions are somewhat which you have control over. You might make an accurate decision or a wrong decision. It mainly depends on the state of nature as to whether your decision is accurate or in error.

Possible states of nature (Based on H0)/Conclusions (based on claim)

a) Reject H0/‘Sufficient evidence to state patient is dead’
b) Fail to reject H0/’Insufficient evidence to state patient is dead’

There are four possibilities which can take place based on the two possible states of nature and the two decisions that we can make.

Statisticians will by no means accept the null hypothesis, we will fail to reject. In another words, we will state that it is not, or that we do not have sufficient evidence to state that it is not, however we will never state that it is, as someone else might come all along with the other sample that shows that it is not and we do not want to be wrong.

Statistically (double) speaking….

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In English....

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Were you right...?

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Out of Type I or Type II, which of the two errors is more serious?

Usually Type I is more serious error and we will mainly concentrate on it. We generally pick alpha to be very small (0.05, 0.01). Note that alpha is not a Type I error. Alpha is a probability of committing a Type I error. Similarly beta is the probability of committing a Type II error.

Conclusion:

These are the sentence answers that comprise whether there is sufficient evidence or not (mainly based on decision), the level of significance and whether the original claim is rejected or supported.

Conclusions are basically based on the original claim, which might be null or alternative hypothesis. Decisions are always based on null hypothesis.

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