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The Z-test for proportions is a statistical method used to determine whether there is a significant difference between an observed sample proportion and a hypothesized population proportion. This test is particularly useful when dealing with categorical data, such as the proportion of items in specific categories. In this context, we are interested in testing whether the proportion of misshelved books is below a certain threshold.

The calculation involves determining a Z-score with 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 population proportion, and \( n \) is the sample size. This formula helps us measure how far the sample proportion is from the hypothesized one, adjusted for the number of observations.

If the calculated Z-score falls within the critical region, defined by the significance level, we reject the null hypothesis in favor of the alternative. Essentially, this method gives a statistical basis for deciding on actions based on sample data.

In hypothesis testing, Type I and Type II errors are potential mistakes that can occur. Understanding these errors is crucial when making decisions based on statistical tests.

• **Type I Error:** Occurs if we reject the null hypothesis when it is actually true. In this exercise, it would mean deciding not to conduct the inventory when the proportion of misshelved books is actually above the acceptable threshold. The probability of this error is denoted by \( \alpha \), known as the significance level, which is set at 0.05.
• **Type II Error:** Happens if we fail to reject the null hypothesis when the alternative hypothesis is true. This would be the case if we went ahead with the inventory when the true proportion was below 0.02. The chance of this error, denoted by \( \beta \), is related to the power of the test.

Effective hypothesis testing requires balancing these risks, often by selecting an appropriate \( \alpha \) level that minimizes the chance of a Type I error while being mindful of the implications of a Type II error.

Critical values are thresholds in hypothesis testing that determine the boundary of rejecting the null hypothesis. They mark the point where the test statistic's value leads us to accept or reject our hypothesis.

For a Z-test, critical values are derived from the standard normal distribution. They correspond to the significance level \( \alpha \), which is the probability of committing a Type I error. In this case, for a one-tailed test with \( \alpha = 0.05 \), the critical value is approximately -1.645.

This means that if the calculated Z-statistic is smaller than -1.645, it falls into the critical region, leading us to reject the null hypothesis. However, if it is greater, as in this exercise (with a Z of -1.1294), we do not have enough evidence to reject it, thus maintaining the original hypothesis.

Setting up null and alternative hypotheses is the first step in hypothesis testing. These hypotheses are foundational elements that help clarify the research question and the statistical test's direction.

• **Null Hypothesis (\(H_0\)):** Represents the status quo or a statement of no effect. In this scenario, \( H_0 \) asserts that the proportion of misshelved books is at least 0.02, suggesting no need to postpone the inventory.
• **Alternative Hypothesis (\(H_a\)):** This is what we aim to support through our test; it indicates a new effect or difference. Here, \( H_a \) posits that the proportion of misshelved books is less than 0.02, supporting the potential postponement of inventory.

The choice between these hypotheses directs the statistical analysis. It sets the stage for evaluating evidence from the data, testing if the observed sample supports a shift from the null towards the alternative.