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The Z-test for proportions is a statistical method used to determine if there is a significant difference between an observed sample proportion and a known population proportion.
It helps us understand if any observed deviation could be attributed to random chance or if it suggests a real effect.

In the context of the exercise, we want to see if the proportion of cars with two or more occupants on the New Jersey Turnpike is different from the stated 10%.

• To perform this test, we set up our sample proportion, \(\hat{p}\), which is calculated by dividing the number of successes (here, cars with two or more occupants) by the total sample size (in this exercise, it’s 63 out of 300).
• We then use the population proportion \(p_0\) (0.10 in this exercise) and compare \(\hat{p}\) to \(p_0\).
• The Z-test formula measures how many standard deviations our sample proportion \(\hat{p}\) is from the population proportion \(p_0\).

Paying attention to these calculations ensures we correctly interpret whether the observed difference is statistically significant.

In hypothesis testing, we always start with two opposing hypotheses.
The null hypothesis is a statement assuming there is no change or difference, while the alternative hypothesis is what we want to gather evidence for.

• Null Hypothesis (\(H_0\)): This is a commonly accepted fact unless proven otherwise. In this exercise, \(H_0\) states that 10% of cars have two or more occupants: \(p = 0.10\).
• Alternative Hypothesis (\(H_a\)): This suggests that there is a significant effect or difference. Here, \(H_a\) posits that the actual proportion of cars is different from 10%: \(p eq 0.10\).

By testing these hypotheses, we seek to either reject the null hypothesis in favor of the alternative or fail to provide enough evidence against the null hypothesis. This process involves calculating test statistics and comparing them to critical values derived from significance levels.

The significance level, denoted by \( \alpha \), is a threshold used to determine whether the results are statistically significant.
It is the probability of rejecting the null hypothesis when it is actually true, also known as the Type I error.

In most hypothesis tests, a common significance level used is 0.05; however, in this exercise, we use a stricter level of 0.01.

• A lower \( \alpha \) level, like 0.01, necessitates stronger evidence to reject the null hypothesis. This means the test is more stringent and less likely to yield false positives.
• When the p-value (calculated probability) is below the significance level, we reject the null hypothesis.

In the exercise, since our calculated \(z\)-value is beyond the threshold of ±2.576 (which corresponds to a 0.01 significance level in a two-tailed test), it means our findings are statistically significant, and hence, we reject the null hypothesis, indicating a different proportion of cars with two or more occupants.