91Ó°ÊÓ

Understanding proportion tests is fundamental in hypothesis testing, particularly when you want to compare a sample proportion to a known population proportion. In the context of the garage example, we compare the sample proportion of repeat patrons against the known population proportion of 68%.
Proportion tests come into play when:

• You have categorical data, often as a count or a binary outcome (e.g., repeat vs. new customer).
• You need to evaluate claims about the proportion of categories in a population (like testing if it is less than a certain value).

Proportion tests are a subset of hypothesis tests designed for cases where the data are in proportions or percentages, making them suitable for the garage's repeat patron analysis.

The z-statistic calculation is a critical step when conducting a proportion test. It helps quantify how much the sample proportion deviates from the hypothesized population proportion under the null hypothesis. In our example:
The formula for calculating the z-statistic is:\[ z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}} \]Here:

• \( \hat{p} = 0.57 \) represents the sample proportion of repeat patrons.
• \( p_0 = 0.68 \) is the hypothesized population proportion.
• \( n = 200 \) is the sample size.

Plug these into the formula to get \( z \approx -2.36 \), indicating that the sample proportion is significantly lower than the hypothesized proportion. This tells us that our sample differs from the hypothesized population value substantially.

The significance level, often denoted as \( \alpha \), is the threshold used to decide whether to reject the null hypothesis. In hypothesis tests, it defines the probability of making a Type I error, which is rejecting a true null hypothesis.
In this exercise, a significance level of 1% (or 0.01) is used. This low \( \alpha \) level means that we are being conservative, allowing only a 1% chance of falsely declaring a significant effect. In the context of the garage example:

• The critical z-value at the 1% level for a one-tailed test is approximately -2.33.
• We reject the null hypothesis if the calculated z-statistic is less than this critical value, indicating strong evidence against the null hypothesis.

Here, since our test statistic \( z = -2.36 \) is less than \( -2.33 \), we conclude that the evidence is strong enough to reject the null hypothesis with high confidence.

P-value computation is an essential part of hypothesis testing. It provides the probability of observing a test statistic as extreme as, or more extreme than, the one calculated, assuming the null hypothesis is true. In practical terms:
- A smaller p-value indicates stronger evidence against the null hypothesis.- The computed p-value should always be compared with the significance level \( \alpha \) to accept or reject the null hypothesis.
For our scenario:

• We found a test statistic of \( z = -2.36 \).
• The corresponding p-value is approximately 0.0091 based on z-tables or computational tools.

This p-value is less than the 1% significance level, meaning the result is statistically significant. In essence, there is less than a 1% probability that the observed or more extreme result would occur if the population proportion were truly 68%, supporting the decision to reject the null hypothesis.