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The z-score is an essential concept in statistics that helps us understand where a specific data point stands in relation to the overall data set. It is calculated as the number of standard deviations away from the mean a particular observation is. Simply put, it tells us how far, and in what direction, an observation deviates from the average value of a dataset.

To calculate a z-score, the formula used is:

• \( Z = \frac{X - \mu}{\sigma} \)

Where \(X\) is the value in question, \(\mu\) is the mean of the population, and \(\sigma\) is the standard deviation.

In hypothesis testing, the z-score is critical as it helps in determining the rejection and non-rejection areas of our hypothesis. By comparing the z-score of our sample data to critical z-values, we determine if we should reject or fail to reject the null hypothesis.

Critical values are the threshold values that separate the rejection and acceptance regions in hypothesis testing. They depend on the chosen significance level, often denoted as \(\alpha\), which is the probability of rejecting the null hypothesis when it is actually true. A common significance level is 0.01 or 1%.

For a specific test, whether it is one-tailed or two-tailed, we can find critical values from the standard normal distribution table corresponding to the chosen \(\alpha\).

For example:

• In a two-tailed test with \(\alpha = 0.01\), the critical z-values are usually \(-2.58\) and \(+2.58\).
• In a one-tailed test with \(\alpha = 0.01\), the critical z-value is typically \(+2.33\) for the right tail or \(-2.33\) for the left tail.

These values help us decide whether an observed z-score falls in the rejection region (leading us to reject the null hypothesis) or in the non-rejection region (leading us to fail to reject the null hypothesis).

A two-tailed test is used in hypothesis testing when we are concerned about deviations from the null hypothesis in two potential directions. This test is applied when the alternative hypothesis is expressed as \(H_1: \mu eq \text{some value}\). In other words, the focus is on detecting any significant difference, regardless of it being higher or lower.

In a two-tailed test, the total significance level \(\alpha\) is divided between the two tails of the distribution. So, for a significance level of \(\alpha = 0.01\), each tail would contain 0.005. This division allows us to capture both extremes of the distribution using critical values.

The rejection region would be in both the negative and positive tails, determined by critical values such as \(-2.58\) and \(+2.58\). Observing a z-score outside these values would lead to rejecting the null hypothesis, indicating that the sample provides strong evidence against the null.

A one-tailed test examines the possibility of the relationship or effect in only one direction. It is used when the alternative hypothesis specifies a directional difference, like \(H_1: \mu > \text{some value}\) or \(H_1: \mu < \text{some value}\). A one-tailed test is often used when we are interested in finding an increase or a decrease, but not both.

The entire significance level \(\alpha\) is allocated to one end of the distribution—in the case of a right-tailed test, to the right; for a left-tailed test, to the left. A critical z-value identifies where this tail begins.

For instance:

• In a right-tailed test with \(\alpha = 0.01\), the critical value would be \(+2.33\).
• In a left-tailed test with \(\alpha = 0.01\), the critical value would be \(-2.33\).

If a sample's z-score exceeds \(+2.33\) in a right-tailed test or falls below \(-2.33\) in a left-tailed test, we would reject the null hypothesis. This means the evidence supports the claim that the mean is greater or less than the stated null hypothesis value.