91Ó°ÊÓ

A z-test is a statistical tool used to determine whether there is a significant difference between the sample mean and the population mean. It is most suitable when the population variance is known, and the sample size is large (typically over 30). The test calculates a z-score, which shows how many standard deviations the sample mean is from the population mean.
This method is applied in hypothesis testing to decide whether to reject the null hypothesis. The z-score is compared against critical values from the z-distribution to make this decision. Whether a one-tailed or two-tailed z-test is used depends on the alternate hypothesis.
Some key points to remember are:

• Applicable when the population standard deviation is known.
• Ideal for testing hypotheses about the mean.
• Uses the standard normal distribution to determine probabilities.

In the context of a z-test, a critical value is the threshold that the calculated z-score must reach or exceed for the null hypothesis to be rejected. This value depends on the significance level, denoted by \(\alpha\), and the nature of the test (one-tailed or two-tailed).
For a given \(\alpha\) level, the critical value can be found using the standard normal distribution (z-table). For example, in a typical two-tailed test with \(\alpha = 0.10\), the critical values are ±1.645. This means if the calculated z-score falls outside this range, the null hypothesis is rejected.
Key aspects of critical values include:

• They set the cutoff points for hypothesis tests.
• Determine the rejection regions for a test.
• Choose them based on the desired level of confidence and type of test.

The standard error (SE) plays a crucial role in hypothesis testing. It represents the variability of the sample mean estimate of a population mean. A smaller standard error indicates that the sample mean is a more accurate reflection of the population mean.
The standard error is calculated as the population standard deviation \(\sigma\) divided by the square root of the sample size \(n\), expressed as \(SE = \sigma / \sqrt{n}\). This formula shows that a larger sample size will result in a smaller standard error, making the sample mean estimate more reliable.
Key points about standard error include:

• Measures the accuracy with which a sample represents a population.
• Decreases with an increase in sample size.
• Used to calculate the test statistic in z-tests.

The null hypothesis (denoted as \(H_0\)) is a foundational concept in hypothesis testing. It represents a statement that there is no effect or no difference, and it serves as the default assumption to be tested against.
In our examples, the null hypothesis generally states that a population parameter (like the mean) is equal to a specific value. Hypothesis testing aims to gather evidence against this null hypothesis.
Its counterpart, the alternative hypothesis \(H_1\), proposes what the research is intended to prove. In the examples, if the test statistic leads us to reject \(H_0\), it suggests the sample provides sufficient evidence that the null hypothesis might not be true.
Crucial aspects of the null hypothesis include:

• Serves as the starting point for hypothesis testing.
• Is formulated such that it can be tested mathematically.
• Rejection of \(H_0\) implies acceptance of \(H_1\), providing evidence for a significant effect or difference.