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In hypothesis testing, we begin by stating two opposing statements known as the null and alternative hypotheses. The null hypothesis, denoted as \(H_0\), represents a presumption of no effect or no difference; it's the scenario we assume to be true before collecting any data. In our exercise, the null hypothesis is that the mean number of hours worked by assembly workers on Mondays \(\mu\) is equal to 7 hours (\(H_0: \mu = 7\)).

On the flip side, the alternative hypothesis, denoted as \(H_1\) or \(H_a\), posits that there's a meaningful effect, difference, or relationship. In this scenario, the alternative hypothesis is that the mean number of hours \(\mu\) is less than 7 hours (\(H_1: \mu < 7\)). The direction we specify in \(H_1\) indicates we are conducting a one-tailed test, looking for evidence of a decrease in working hours after implementing a flextime schedule.

Understanding and clearly stating these hypotheses is crucial, as they lay the groundwork for the subsequent steps in our hypothesis testing process.

The t-score is a standardized statistic that measures the number of standard deviations a sample mean is away from the population mean under the null hypothesis. It is calculated using the formula:\[t = \frac{\bar{x} - \mu_0}{\frac{s}{\sqrt{n}}}\] In this formula:\[\begin{align*}&\bar{x} \text{ is the sample mean},\&\mu_0 \text{ is the hypothesized population mean},\&s \text{ is the sample standard deviation},\&n \text{ is the sample size}.\end{align*}\] For the given problem, the t-score was found to be \(t = -1.5661\), suggesting the sample mean is approximately 1.5661 standard errors below the hypothesized mean. This t-score will be compared against a critical value to decide whether there's enough statistical evidence to reject the null hypothesis.

Determining the critical value in hypothesis testing is like setting the 'goalposts' for a significant result. This value is based on the chosen significance level \(\alpha\), which represents the probability of mistakenly rejecting the null hypothesis when it's actually true (a type I error). Since we are conducting a one-tailed t-test at \(\alpha = 0.05\), we're accepting a 5% chance of a type I error.

To find the critical value, we look at a t-distribution table or use an online calculator, checking for the value that corresponds to the particular \(\alpha\) level and the degrees of freedom \(df\), which is the sample size minus one (79 in this case). The critical t-value obtained is \(t_{\alpha} = -1.664\). By comparing our calculated t-score to this critical value, we determine whether the observed sample mean would be considered statistically unusual if the null hypothesis were true.

The one-tailed t-test is a statistical method that tests for a difference in one specific direction. Unlike its two-tailed counterpart which tests for any significant difference, a one-tailed test only considers an extreme outcome in the direction specified by the alternative hypothesis.

In our example, we use a one-tailed t-test because we only want to know if there's evidence of the mean hours worked being less than 7; we're not testing for the mean being greater than 7. When we compared the t-score with the critical value, we found that \(t > t_{\alpha}\), meaning the calculated t-score didn't cross the significance 'goalposts' set by the critical value. Therefore, we do not reject the null hypothesis and conclude there's no statistically significant evidence to suggest the mean work hours on Mondays will be less than 7 hours with the implementation of flextime. This decision process highlights the importance of understanding the type of t-test to apply based on the research question at hand.