Hypothesis Testing

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Definitions

Null Hypothesis ( H₀ )

Statement of zero or no change. If the original claim includes equality (<=, =, or >=), it is the null hypothesis. If the original claim does not include equality (<, not equal, >) then the null hypothesis is the complement of the original claim. The null hypothesis always includes the equal sign. The decision is based on the null hypothesis.

Alternative Hypothesis ( H₁ or H_a )

Statement which is true if the null hypothesis is false. The type of test (left, right, or two-tail) is based on the alternative hypothesis.

Type I error

Rejecting the null hypothesis when it is true (saying false when true). Usually the more serious error.

Type II error

Failing to reject the null hypothesis when it is false (saying true when false).

alpha

Probability of committing a Type I error.

beta

Probability of committing a Type II error.

Level of Significance (Significance Level)

The probability of committing a type I error, also known as alpha. This is how unusual something must be before we call it too rare to have happened just by chance. The area in the critical region.

Probability Value (P-value)

The probability of getting the results obtained if the null hypothesis is true. If this probability is too small (smaller than the level of significance), then we reject the null hypothesis. If the level of significance is the area beyond the critical values, then the probability value is the area beyond the test statistic.

Test statistic

Sample statistic used to decide whether to reject or fail to reject the null hypothesis.

Critical region

Set of all values which would cause us to reject H₀

Critical value(s)

The value(s) which separate the critical region from the non-critical region. The critical values are determined independently of the sample statistics.

Significance level ( alpha )

The probability of rejecting the null hypothesis when it is true. alpha = 0.05 and alpha = 0.01 are common. If no level of significance is given, use alpha = 0.05. The level of significance is the complement of the level of confidence in estimation.

Decision

A statement based upon the null hypothesis. It is either "reject the null hypothesis" or "fail to reject the null hypothesis". We will never accept the null hypothesis.

Conclusion

A statement which indicates the level of evidence (sufficient or insufficient), at what level of significance, and whether the original claim is rejected (null) or supported (alternative).