Formulatoin of a hypothesis

Formulating a Hypothesis

A hypothesis is a prediction or claim about a population parameter which is to be tested. Formulating hypotheses is a fundamental step before conducting statistical hypothesis testing.
Two complementary hypotheses are formulated, the null hypothesis (H0) and the alternative hypothesis (H1 or Ha).

Null Hypothesis (H0)

The null hypothesis is an assumption of no effect or no difference in the population. It’s a statement of status quo.
It’s usually formulated with an equality sign, i.e., H0: µ = µ0 or H0: µ1 = µ2, where µ is the population parameter, often the population mean.

Alternative Hypothesis (H1 or Ha)

The alternative hypothesis asserts that there is an effect or difference in the population contradicting the null hypothesis.
It can be formulated as one-sided (e.g. Ha: µ > µ0 or Ha: µ1 < µ2) or two-sided (e.g. Ha: µ ≠ µ0 or Ha: µ1 ≠ µ2)

Significance Level

The significance level (alpha), usually 0.05 or 0.01, is set before testing. It threshold for deciding when to reject the null hypothesis.
Lower alpha values indicate a higher level of evidence to reject the null hypothesis.

Hypothesis Test Decision

A test statistic is computed from the sample data and this statistic (or its p-value) is compared to the significance level to decide whether to reject the null hypothesis.
If the test statistic is more extreme than what is likely under the null hypothesis (i.e., p-value <= alpha), reject the null hypothesis favoring the alternative. Else, fail to reject the null hypothesis.

Considerations

Type I Error: Rejecting a true null hypothesis. Its probability equals the significance level.
Type II Error: Failing to reject a false null hypothesis. Its probability (beta) should also be considered.
Power: The capacity of a test to find an effect when one actually exists. It’s calculated as 1- beta. A powerful test can detect smaller effects.

Hypothesis Testing using t-distribution

With small sample sizes and/or unknown population variance, hypothesis testing often involves the t-distribution. The t-test statistics follow a t-distribution under the null hypothesis.
The degrees of freedom for the t-test are often determined by the sample size. For a simple one-sample or paired t-test, df=n-1, where n is the sample size. For a two-sample t-test, df is usually approximated to the smaller of n1-1 and n2-1, where n1 and n2 are the sizes of the two samples.