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Hypothesis testing is an essential component of inferential statistics. It helps us decide whether there's enough evidence to support a specific claim about a population. In our problem, the initial step is to set up the hypotheses:

The **null hypothesis** (\(H_0\)) suggests that there's no significant difference or effect. Here, it posits that the mean delay time \(\mu\) equals 45 seconds.

The **alternative hypothesis** (\(H_1\)) claims the opposite, suggesting that the mean delay time is greater than 45 seconds. This hypothesis represents what we want to prove. If we find sufficient evidence against \(H_0\), we accept \(H_1\).

By setting up these hypotheses, we lay the groundwork for making data-driven decisions.

The **t-distribution** is crucial when dealing with small sample sizes or unknown population variances. Unlike the normal distribution, it is a bit flatter and wider, accommodating the additional uncertainty.

In this scenario, we use the t-distribution to calculate the t-score, which helps us determine how many standard deviations the sample mean (50.1 seconds) is away from the hypothesized population mean (45 seconds).

The t-score formula is:\[t = \frac{\bar{x} - \mu}{s/\sqrt{n}}\]where \(\bar{x}\) is the sample mean, \(\mu\) is the population mean, \(s\) is the standard deviation, and \(n\) is the sample size. Calculating this gives us a specific value we can use to compare against t-distribution critical values or determine a p-value.

The **significance level** (\(\alpha\)) is a threshold used to decide whether to reject the null hypothesis. It represents the probability of rejecting \(H_0\) when it is actually true—also known as the Type I error.

In our exercise, \(\alpha = 0.02\) indicates a 2% risk of falsely claiming that the mean delay time is more than 45 seconds. Lower significance levels (like 0.01) reduce the chance of making this type of error but require stronger evidence to reject \(H_0\).

The significance level helps strike a balance between being cautious and decisive. It guides us in making a clear conclusion about the hypothesis test.

The **p-value approach** is a commonly used method for hypothesis testing. It calculates a p-value, which represents the probability of observing the test statistic, or one more extreme, under the null hypothesis.

To find the p-value, consider the calculated t-score from the sample data. Compare this score against a t-distribution table or use statistical software. If the p-value is less than the significance level (\(\alpha = 0.02\)), we reject the null hypothesis.

This method directly tells us how strong the evidence is against \(H_0\). A smaller p-value implies we are more confident in rejecting \(H_0\) in favor of \(H_1\).

The **classical approach** to hypothesis testing is also known as the critical value approach. It involves comparing the t-score to a predefined critical value that corresponds to the chosen significance level and degrees of freedom (\(df\)).

In our example, \(df = n - 1 = 14\), given the 15 system measurements. Using \(\alpha = 0.02\), find the critical value from a t-distribution table.

If the calculated t-score exceeds this critical value, we reject the null hypothesis. This approach gives a straightforward yes-or-no answer based on comparisons, providing an alternative to the p-value method.