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A proportion test is a statistical method used to determine if there is a significant difference between an observed proportion and a theoretical one. It is particularly useful in situations where you want to compare a sample proportion to a known population proportion.
In our exercise, we are testing if the proportion of mechanics who can correctly identify an engine problem is less than 0.75. This involves determining whether the observed sample proportion (42 out of 72 mechanics) significantly deviates from the hypothesized population proportion, which is 0.75 in this case.
The proportion test helps ensure that observed differences aren't due to random chance but rather reflect a real deviation from the expectation. It is one of several types of hypothesis tests used in statistics.

The Z-score is a crucial component in hypothesis testing, measuring how far an observed data point is from the expected value under the null hypothesis, in terms of standard deviations.
To calculate the Z-score for a proportion test, you use the formula:

• \[ Z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}} \]

Where:

• \( \hat{p} \) is the observed sample proportion,
• \( p_0 \) is the hypothesized proportion, and
• \( n \) is the sample size.

In this exercise, the Z-score helps us quantify how unusual the observed sample proportion (0.5833) is compared to the hypothesized value (0.75). The resulting Z-score of -3.24 suggests that the observed proportion is unusually low if the true proportion were indeed 0.75.

The P-value in hypothesis testing helps us determine the significance of our results. It is the probability of obtaining a result as extreme as, or more extreme than, the observed data, assuming the null hypothesis is true.
A low P-value suggests that the observed data is unlikely under the null hypothesis, thus providing evidence against it.

• In our scenario, the computed P-value is approximately 0.0006.

Given that this P-value is considerably lower than a typical significance level, like 0.05, it tells us that the observed result (that only 42 out of 72 mechanics could identify the problem) is highly unlikely if the true proportion were 0.75. Hence, we have strong evidence to reject the null hypothesis.

The alternative hypothesis represents what we seek to prove or disprove with our test. It is the statement employed when the null hypothesis is rejected.
In our exercise, the alternative hypothesis is that the true proportion of mechanics who can identify the problem is less than 0.75. This is articulated as \( H_1: p < 0.75 \).
The direction of the alternative hypothesis ("less than") indicates a left-tailed test. This corresponds to checking if the observed sample proportion is significantly less than the hypothesized proportion.

• The decision to accept or reject depends on the P-value relative to a chosen level of significance (usually 0.05).

Ultimately, the goal of setting an alternative hypothesis is to challenge the status quo (null hypothesis) and test if there is enough statistical evidence to support a new claim about the population.