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The t-test is a statistical test used to determine if there is a significant difference between the means of two groups. It is particularly useful when dealing with small sample sizes or when the population standard deviation is unknown. In the context of our exercise, the t-test helps determine whether the mean wage of the union employees is statistically different from the reported \$8,000.
To perform a t-test, you start by calculating the t-statistic using the formula:\[ t = \frac{\bar{x} - 渭}{\frac{蟽}{\sqrt{n}}} \] where:

• \(\bar{x}\) is the sample mean
• \(渭\) is the population mean
• \(蟽\) is the sample standard deviation
• \(n\) is the sample size

By interpreting the resulting t-statistic, we assess how far it falls from zero within the t-distribution, allowing us to decide about the hypotheses.

The null hypothesis, often denoted as \(H_0\), is a statement used in statistics that assumes no effect or no difference. It's the baseline assumption that any observed difference in a sample is due to random chance. In our exercise, the null hypothesis is that the mean yearly wage is equal to \\(8,000. This is the claim we are testing against.
A null hypothesis is crucial because it provides a specific statement that can be tested statistically. By assessing the null hypothesis, we can determine whether the sample data provides sufficient evidence to support an alternative hypothesis. The alternative hypothesis, in this context, suggests that the mean wage is not equal to \\)8,000. Rejecting or failing to reject the null hypothesis leads to conclusions about the population parameters based on the sample data.

Critical values are the boundary values that outline regions in the tails of a distribution. These boundaries help in making decisions on whether to reject the null hypothesis. For a two-tailed t-test, like in our example, critical values separate significant results from non-significant ones.
To find the critical values, we use the t-distribution table considering:

• The degrees of freedom, calculated as \(n - 1\)
• The level of significance (alpha), divided by two for a two-tailed test

In our exercise, with 99 degrees of freedom and a 0.05 significance level divided into two tails (0.025 each), the critical values are approximately -1.984 and 1.984. If the calculated t-statistic falls outside these critical values, it indicates a statistically significant difference from the null hypothesis.

The level of significance, denoted by \( \alpha \), represents the probability of rejecting a true null hypothesis, also known as the Type I error rate. It's the threshold set by the researcher to decide whether to accept or reject the null hypothesis. Typically, a common level of significance is 0.05, meaning that there's a 5% risk of concluding that a difference exists when there is none.
In our exercise, we set the level of significance at 0.05, which is customary for many social sciences studies. This means that if the likelihood of observing our sample result is less than 5%, assuming the null hypothesis is true, we would reject the null hypothesis. The choice of this level is somewhat subjective but should reflect the balance between the risk of error and the need for confidence in the results.