Hypothesis Testing for proportions

The null hypothesis (H0) is a statement that there is no effect or no difference, and it serves as the default assumption that is tested against an alternative hypothesis.

The alternative hypothesis (H1) is a statement that indicates the presence of an effect or a difference, and it is what the researcher aims to support.

A two-tailed test is used when the alternative hypothesis states that the parameter is not equal to the null hypothesis value, allowing for the possibility of an effect in both directions.

The significance level (α) is the probability of rejecting the null hypothesis when it is actually true, commonly set at 0.05.

Rejecting the null hypothesis means that there is sufficient evidence to support the alternative hypothesis.

Failing to reject the null hypothesis means that there is not enough evidence to support the alternative hypothesis.

A p-value is the probability of obtaining test results at least as extreme as the observed results, assuming that the null hypothesis is true.

If the p-value is less than α, the null hypothesis should be rejected.

The test statistic for proportions is calculated using the formula: $$z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}}$$ where $$\hat{p}$$ is the sample proportion, $$p_0$$ is the null hypothesis proportion, and $$n$$ is the sample size.

The critical value is the threshold that determines whether to reject the null hypothesis, based on the significance level and the type of test.