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The null hypothesis, often denoted as \(H_0\), is the starting point in hypothesis testing. It represents the assumption that there is no effect or no difference in a situation or study. In our exercise, the null hypothesis is that the average completion time for orders is 13 minutes. This hypothesis is formulated as \(H_0: \mu = 13\). Essentially, the null hypothesis suggests that whatever the employees claim about their average completion time being 13 minutes is correct. The role of the null hypothesis is to provide a basis for testing the evidence presented by a sample. We will compare the collected data against this hypothesis to determine if it should be rejected, which sets the stage for our conclusions.

In contrast to the null hypothesis, the alternative hypothesis, denoted as \(H_1\) or \(H_a\), is what researchers aim to test. It suggests that the observed effect or difference is genuine. In our example, the alternative hypothesis states that the average completion time is not 13 minutes.This is expressed as \(H_1: \mu eq 13\). The alternative hypothesis is used to challenge the status quo by proposing that the completion time is significantly different from the claimed 13 minutes, whether it's less or more. When we run our hypothesis test, we are essentially looking for evidence that supports \(H_1\). If our collected evidence is strong enough, we can reject the null hypothesis in favor of the alternative hypothesis, suggesting a significant difference in completion times.

The t-test is a statistical test used to compare the sample mean to the known population mean, especially when the sample size is small or the population standard deviation is unknown. In this context, our t-test helps to assess the claim made by the shipping department employees. For this exercise, we utilize a formula: \[ t = \frac{\bar{x} - \mu}{s/\sqrt{n}} \]Where:

• \(\bar{x}\) is the sample mean (14 minutes),
• \(\mu\) is the population mean (13 minutes),
• \(s\) is the sample standard deviation (10 minutes),
• \(n\) is the sample size (400 orders).

We find the t-value to be 2, which tells us how many standard deviations our sample mean is from the population mean. This statistic will be compared against critical values from a t-distribution table to determine the significance of our results.

The level of significance, often represented as \(\alpha\), is the threshold for determining whether the null hypothesis can be rejected. A common level of significance is 0.05, which was used in our exercise. This value indicates a 5% risk of rejecting the null hypothesis when it is actually true. With our two-tailed test, we consider deviations in both directions, meaning the significance is split between the two tails with 0.025 in each.By comparing our calculated t-value with critical values at this significance level, we determine if the observed sample mean significantly deviates from the null hypothesis. Our result, with a t-value exceeding the critical value, allows us to reject the null hypothesis, suggesting the sample provides significant evidence against it.

A sample mean is the average calculated from a sample data set, and it is crucial for statistical analysis and inference. In our exercise, the sample mean is 14 minutes. This value is compared to the population mean of 13 minutes to ascertain whether there鈥檚 enough evidence to suggest a significant difference, which was crucial in the employees' claims about their completion times. The sample mean serves as a primary input in the t-test, allowing us to calculate the t-value and make informed decisions about the null hypothesis. It is a principal component in determining whether there's a statistical basis to view the sample data as significantly different from the population parameter.