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A tolerance limit helps us determine a range in which we expect a certain percentage of the population to fall. In the exercise, this concept is crucial to verify the union's claim. To do this, a one-sided tolerance limit is calculated. This involves finding the maximum number of hours that 99% of union members are absent, based on the sampled data. The one-sided aspect here means we're only considering an upper bound.The formula used is \( TL = x + (Z \times SEM) \), where \( x \) is the mean of the sample, \( Z \) is the Z-score corresponding to the desired confidence level, and \( SEM \) is the Standard Error of the Mean. In simple terms, if this calculated limit is found to be less than 10 hours, it supports the union's claim that 95% of their members are absent less than 10 hours per month. This tolerance limit calculation acts as a statistical tool to help understand and verify claims about large populations based on smaller samples.

The Standard Error of the Mean (SEM) is essential for understanding how much the sample mean could vary from the actual population mean. It essentially measures the variability of the sample mean, finding how bulletproof it is in estimating the real average.The SEM is calculated using the formula \( SEM = \frac{s}{\sqrt{n}} \), where \( s \) stands for the standard deviation and \( n \) represents the sample size. In our exercise, the sample's standard deviation \( s \) is 2.5, and the sample size \( n \) is 300.A smaller SEM value indicates that the sample mean is a more precise estimate of the population mean. In other words, it assures us of the accuracy and reliability of our sample mean calculations while making tolerance limit decisions.

A confidence interval gives us a range of values in which we expect the true population parameter (like the mean) to lie. It provides a more comprehensive insight than a single estimate, offering a degree of certainty about where the true value likely falls. For the given problem, the Z value for a 99% confidence level is used to find the tolerance limit. This Z value reflects how many standard deviations away a point can lie from the mean with the corresponding level of confidence. By employing the confidence interval concept, we calculate and interpret statistical tolerance limits. Thus, we can make evidence-based statements regarding the overall absenteeism of the union members with a specified level of confidence.

A one-sided test in statistics is applied when the research or claim focuses on a specific direction of effect. In simpler terms, it checks if a population parameter (like the mean) is either greater than or less than a specified value, but not both. In the exercise, a one-sided test is utilized to verify if 99% of union members are absent less than a specified threshold (10 hours). The threshold's one-sided nature means we're only considering it as an upper bound. That differentiates it from a two-sided test, which would check both above and below this threshold. Hence, applying a one-sided test is particularly useful when the research question asks about exceeding or not exceeding a certain value, helping us draw more focused conclusions based on our data and hypothesis.