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A z-test is a statistical method used to determine whether there is a significant difference between sample and population means. This is particularly useful when the population standard deviation is known. To conduct a z-test, the sample mean, population mean, standard deviation, and sample size must be identified.

The z-test is categorized into two types based on the research hypothesis: one-tailed and two-tailed tests. In a two-tailed test, the hypothesis is that the population mean is not equal to a specified value, whereas a one-tailed test considers the mean is greater than or less than the value.

Calculating the z-test involves finding the z-score, which indicates how many standard deviations the sample mean is from the population mean. The formula is:

\[ z = \frac{\bar{X} - \mu}{\sigma / \sqrt{n}} \]

Where \(\bar{X}\) is the sample mean, \(\mu\) is the population mean, \(\sigma\) is the population standard deviation, and \(n\) is the sample size.

• A z-score closer to zero indicates that the sample mean is near the population mean.
• A significant z-score suggests a significant difference between sample and population means, based on the chosen level of significance.

Critical values are thresholds for significance in hypothesis testing. They help to decide whether to reject or not the null hypothesis. You determine these from the standard normal distribution (z-distribution) table.

For different tests and significance levels, the values vary:

• In a two-tailed test with \( \alpha = 0.05 \), the critical values are ±1.96.
• For a one-tailed test with the same alpha, the critical value is either 1.645 or -1.645, depending on whether it is a right or left-tailed test.

These critical values create a boundary for the acceptance region of a hypothesis test. If the test statistic lies outside this range, the null hypothesis is rejected, indicating a significant result.

Observed values, in the context of hypothesis testing, refer to the test statistic result calculated from the sample data. This observed z-value tells us how far and in what direction the sample mean deviates from the population mean, taking into account the sample size and population standard deviation.

To compute the observed value using the given formula, substitute the known values into:

\[ z = \frac{\bar{X} - \mu}{\sigma / \sqrt{n}} \]

Observed values are crucial for decision making in hypothesis testing. You compare them with the critical values to determine whether the observed data fall within the acceptance region or rejection region of the null hypothesis.

• If the observed value lies within the critical region, the null hypothesis is rejected.
• If outside, we do not reject the null hypothesis.

The level of significance, often denoted by \(\alpha\), is a threshold for determining how unlikely a result must be to reject the null hypothesis. Commonly used values are \(0.05\), \(0.01\), and \(0.10\).

A lower alpha level means a stricter criterion for rejecting the null hypothesis. Conversely, a higher alpha level is more lenient.

Choosing an appropriate level of significance depends on the context of the test and the consequences of making a Type I error (incorrectly rejecting a true null hypothesis).

• At \(\alpha = 0.05\), there is a 5% risk of incorrectly rejecting the null hypothesis.
• This level strikes a balance between sensitivity and specificity in many research fields.

In hypothesis testing, the level of significance determines the critical values, which form the boundaries for the acceptance region of the null hypothesis.