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When we start a hypothesis test, we first have to establish the null hypothesis. The null hypothesis, denoted as \(H_0\), represents a statement of no effect or no difference. It's the hypothesis that there is nothing new or interesting happening, and it serves as a starting point for statistical testing. In the context of hypothesis testing, the null hypothesis often reflects the status quo or the baseline measurement that we want to test against.
For example, in the exercise about smoking proportions, the null hypothesis assumes that the smoking proportion among males equals the smoking proportion among females. This is written as \(H_0 : p_m = p_f\). The null hypothesis is always assumed to be true until evidence suggests otherwise through statistical testing.

The alternative hypothesis is what you want to prove, or at least provide evidence supporting. It's the opposite of the null hypothesis and is usually what you suspect is true before testing. The alternative hypothesis is denoted by \(H_A\) and it posits that there is a significant effect or difference that needs to be investigated. If you find enough evidence to reject the null hypothesis, then you accept the alternative hypothesis.
In the given exercise, the alternative hypothesis aims to find out if a larger proportion of males smoke compared to females. This is formulated as \(H_A : p_m > p_f\). The greater-than sign suggests a directional hypothesis, meaning specifically that males are suspected to smoke more. The formulation of the alternative hypothesis is crucial as it guides the statistical analysis we'll perform.

Statistical parameters are numerical characteristics of a population that we aim to understand or estimate through statistical tests. In hypothesis testing, parameters can include means, proportions, variances, and more. They are the constants that describe the dataset comprehensively.
For this exercise, the parameters of interest are the smoking proportions of males and females, indicated by \(p_m\) and \(p_f\) respectively. These parameters help us set up the context of our hypotheses and provide a framework to carry out statistical tests. By defining and using these parameters, statisticians can perform calculations and draw conclusions about differences between groups. Understanding these parameters is key in making sense of the statistical tests and the hypotheses being evaluated.