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The null hypothesis is the starting point for any hypothesis test. It represents a statement of no effect or no difference, that you want to challenge. In this exercise, the null hypothesis, denoted as \(H_0\), is that 20% or fewer of the cars in the fleet fail to meet pollution control guidelines. This translates to the mathematical expression \(p = 0.20\). The null hypothesis is essentially a benchmark or baseline from which we measure deviation. It remains our assumption until evidence suggests otherwise. We usually retain the null hypothesis unless statistical analysis suggests significant evidence to the contrary.

In contrast to the null hypothesis, the alternative hypothesis proposes a different outcome from the status quo. Here, our alternative hypothesis, represented as \(H_A\), is the claim that more than 20% of the cars might be non-compliant with pollution control requirements. Mathematically, it is expressed as \(p > 0.20\). The alternative hypothesis is what you are trying to provide evidence for through testing. When performing hypothesis testing, the goal is often to find evidence that supports this alternative hypothesis by rejecting the null hypothesis.

The Z-score is a statistical measure that shows how far away our sample proportion is from the null hypothesis proportion, expressed in standard deviation units. To compute the Z-score, we use the formula:
\[ Z = \frac{ \hat{p} - p }{ \sqrt{\frac{p(1 - p)}{n}} } \]
Where:

• \(\hat{p}\) is the sample proportion (i.e., the observed proportion of cars failing, which is \(\frac{7}{22}\)).
• \(p\) is the null hypothesis proportion (0.20 in this case).
• \(n\) is the sample size (22).

The computed Z-score helps us understand how extreme our sample result is. The larger the Z-score, the farther our observation is from the null hypothesis, indicating more substantial evidence to reject it.

The p-value is a crucial measure in hypothesis testing. It quantifies the probability of obtaining a test statistic as extreme as the one observed, given that the null hypothesis is true. In our scenario, a high p-value means that the observed sample result (or one more extreme) might occur under the null hypothesis. A low p-value, typically less than 0.05, suggests that the observed result is quite unlikely under the null hypothesis.
To find the p-value, one usually uses statistical tables or software to see where the observed Z-score lies on the standard normal distribution. A p-value below the chosen significance level (0.05 commonly) leads to rejecting the null hypothesis, indicating stronger evidence for the alternative hypothesis that more than 20% of the fleet fails to meet the guidelines.