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In hypothesis testing, the initial step involves defining the null and alternative hypotheses. The null hypothesis (\( H_0 \)), represents a statement of no effect or no difference—in this case, it's the assumption that the proportion of smokers in the population is 20%, written as \( P = 0.20 \).

The alternative hypothesis (\( H_a \)) represents the opposite claim which we are trying to find evidence for. For the example at hand, it posits that the proportion of smokers is different from 20%, expressed as \( P eq 0.20 \). The task of hypothesis testing is essentially to determine whether there's enough statistical evidence to reject the null hypothesis in favor of the alternative.

When investigating the proportion of smokers within a population, we analyze a sample to make inferences about the population as a whole. The proportion from the sample, denoted \( P_{\text{sample}} \), helps us estimate the population proportion. Our goal is to check whether there is a significant difference between the sample proportion and an assumed population proportion (20% in our example).

The accuracy of our estimation can be affected by the size of the sample. Larger samples tend to give more reliable estimates of the population proportion, as they reduce the margin of error and the impact of outliers or anomalies.

The z-score is a statistical measure that describes the position of a raw score in terms of how many standard deviations it is from the mean. In the context of hypothesis testing, it's used as a test statistic to decide whether to reject the null hypothesis. To calculate it, we use the formula:
\[Z = \frac{(P_{\text{sample}} - P_{H_0})}{\sqrt{\frac{P_{H_0}(1 - P_{H_0})}{n}}}\]

where \( P_{H_0} \) is the hypothesized population proportion and \( n \) is the sample size. The resulting z-score is then compared against critical values from the standard normal distribution to determine whether the sample proportion differs significantly from the hypothesized proportion.

The significance level, commonly denoted by \( \alpha \), is a threshold used to determine when a result is statistically significant. A typical significance level used in many studies is 0.05 or 5%, which means that there is a 5% risk of concluding that a difference exists when there is no actual difference (false positive).

In hypothesis testing, comparing the z-score to a significance level helps us decide whether to reject the null hypothesis. If the calculated z-score is beyond the critical values associated with the chosen significance level (±1.96 for a 95% confidence level in a two-tailed test), the null hypothesis is rejected, indicating that the sample provides enough evidence that the population proportion of smokers is different from 20%.