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Statistical significance is a vital concept in hypothesis testing, used to determine whether the results of an experiment or study are likely to be true or if they may have occurred by random chance. When conducting a hypothesis test, researchers decide in advance how much risk of error they are willing to accept, which is called the significance level, denoted as \( \alpha \).
For example, a significance level of \( \alpha = 0.05 \) means the researcher is willing to accept a 5% probability of rejecting the null hypothesis when it is actually true. This is a common threshold used in many fields.
When a test's result is statistically significant, it means the evidence is strong enough to reject the null hypothesis in favor of the alternative hypothesis. This does not imply absolute certainty, but rather that the observed data is unlikely to have occurred under the assumption that the null hypothesis is correct.
A critical part of this decision is comparing the test statistic to a critical value, which helps to measure how extreme the observed data are. If the test statistic exceeds the critical value, the result is deemed statistically significant.

The Z-Test is a statistical method used to determine if there is a significant difference between sample data points and the population from which the sample was drawn. It is particularly useful when the population standard deviation is known, and the sample size is large (typically \( n \geq 30 \)).
The formula for calculating the Z-Test statistic is:

• \( z = \frac{\bar{x} - \mu}{\sigma/\sqrt{n}} \)

where:

• \( \bar{x} \) is the sample mean,
• \( \mu \) is the population mean under the null hypothesis,
• \( \sigma \) is the population standard deviation,
• \( n \) is the sample size.

The resulting \( z \)-value is then compared against a critical value, which is derived from the significance level \( \alpha \) and the direction of the test (left, right, or two-tailed).
The Z-Test is quite powerful for hypotheses testing because it allows for a more straightforward interpretation when standardized against the standard normal distribution, making the decision-making process clear: if the \( z \)-value transcends the critical threshold, it provides evidence against the null hypothesis.

A right-tailed test, sometimes called a one-tailed test, is a type of hypothesis test where the area of interest is in the right tail of the distribution curve. This test is appropriate when the researcher is testing for the possibility of an effect in only one direction. In the context of hypothesis testing, this means the alternative hypothesis posits that the parameter of interest is greater than a certain value.
In our scenario of testing whether the mean hourly earnings have increased, the null hypothesis \( H_0 \) states that the mean is \( \mu = 14.32 \) dollars, whereas the alternative hypothesis \( H_a \) suggests \( \mu > 14.32 \) dollars. If the test statistic falls into the critical region (extreme right of the curve), we reject \( H_0 \).
The critical region is determined using the significance level \( \alpha \). For instance, with \( \alpha = 0.05 \), the critical value is the \( z \)-score where only 5% of the standard normal distribution lies to its right, typically \( z \approx 1.645 \). If the computed test statistic \( z \) is greater than 1.645, the test results are statistically significant toward the right tail, indicating the mean hourly earnings have indeed increased.