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A t-test is a statistical method used to determine if there is a significant difference between the means of two groups, particularly when the sample size is small, and the population standard deviation is unknown. In this particular exercise, the one-sample t-test is used since we're comparing the sample mean of escape times to a known value, that is, 360 seconds. We apply the t-test to check if the observed sample mean of 370.69 seconds significantly deviates from the hypothesized population mean.

To calculate the t-test statistic, use the formula:

• \[ t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}} \]

Here, \( \bar{x} \) is the sample mean, \( \mu_0 \) is the population mean under the null hypothesis, \( s \) is the sample standard deviation, and \( n \) is the sample size. For our scenario, substituting the values gives us a t-value of approximately 2.237, indicating the significance of the difference between the hypothesized and observed mean.

When using a t-test, it is crucial to ensure that the data approximately follows a normal distribution, especially since the sample size is smaller, as it is here with 26 observations.

The significance level, denoted by \( \alpha \), is a threshold used to decide whether to reject the null hypothesis. It is typically set before conducting the test, and it indicates the probability of rejecting the null hypothesis when it is actually true. In hypothesis testing, the significance level provides a balance between sensitivity and the risk of drawing incorrect conclusions.

In this exercise, a significance level of 0.05 was chosen, meaning there is a 5% risk of incorrectly rejecting the null hypothesis. This is a common choice for many researchers as it offers a reasonable balance between too lenient and too strict criteria, ensuring that results are neither too easily dismissed nor too rigorously held.

The chosen significance level directly affects the outcome of the test, guiding whether the calculated t-test statistic warrants rejection of the null hypothesis. The smaller the significance level, the more stringent the test, thereby lowering the risk of Type I error but potentially increasing the chance of a Type II error.

The null and alternative hypotheses are fundamental concepts in statistical hypothesis testing, defining what we aim to test in an experiment. They are the backbone of any hypothesis testing activity, acting as claims about the population.

For this particular exercise:

• The null hypothesis \( H_0: \mu \leq 360 \) seconds posits that the true average escape time for these workers does not exceed 360 seconds (or 6 minutes). It represents the investors' prior belief.
• The alternative hypothesis \( H_a: \mu > 360 \) seconds opposes this, suggesting that the actual average escape time is greater than 360 seconds.

These hypotheses help determine the statistical test's outcome. If evidence strongly supports the null hypothesis, we maintain it. Conversely, if the evidence contradicts it, evident through calculations and critical value comparison, we adopt the alternative hypothesis.

Critical values serve as crucial boundary markers in hypothesis testing. They're derived from the sampling distribution under the null hypothesis and are used to decide the test's outcome.

In our case, we conducted a one-tailed test, significant for testing if the average time is specifically greater than the set threshold (6 minutes). With an \( \alpha = 0.05 \) significance level and 25 degrees of freedom (since \( n-1 \)), we found the critical value using a t-distribution table: approximately 1.708.

The calculated t-statistic (approximately 2.237) is compared to this critical value. Since 2.237 exceeds 1.708, our test statistic falls into the critical region, dictating the rejection of the null hypothesis. Thus, the evidence suggests that the true average escape time indeed surpasses 360 seconds.

In general, if the test statistic is more extreme than the critical value, we reject the null hypothesis, indicating that the observed data contradict the assumption or belief highlighted by the null hypothesis.