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The null hypothesis, often denoted as \( H_0 \), is a foundational concept in significance testing. It serves as a baseline assumption that there is no effect or no difference. For our exercise about the study habits of first-year university students, the null hypothesis assumes that students study exactly 2.5 hours per night on average, which translates to 150 minutes. In practical terms, this means that any observed variation from this average study time in our sample of students is due to random chance rather than an actual deviation from the norm.

When conducting significance tests, the null hypothesis is the statement that will either be rejected or not rejected based on the test results. It is a critical part of the framework that allows us to use statistical methods to infer whether there is enough evidence to support an alternative claim.

The alternative hypothesis, denoted as \( H_1 \) or sometimes \( H_a \), is the statement we are testing against the null hypothesis. Unlike the null hypothesis, the alternative hypothesis suggests that there is a real effect or a significant difference. For the exercise at hand, the alternative hypothesis posits that first-year university students study less than 150 minutes (which is less than 2.5 hours) per night on average.

The alternative hypothesis can be one-tailed or two-tailed, depending on the direction of the effect we expect to find. In this situation, it is a one-tailed hypothesis because we specifically want to see if students study less than 150 minutes, not just a different amount. Establishing a clear alternative hypothesis is essential as it guides the type of statistical test to use and directs how we interpret the results.

A t-test is a statistical method used to determine if there is a significant difference between the mean of a sample and a known value, often the population mean. In this context, we're using a one-sample t-test to investigate whether the average study time of first-year students is significantly less than the assumed mean of 150 minutes. The t-test helps us compute the test statistic, which reflects how far our sample mean deviates from the population mean in units of standard error.

To perform the t-test in our exercise, we calculate the test statistic using the formula:\[ t = \frac{\overline{x} - \mu}{s_x/\sqrt{n}} \]Here, \( \overline{x} \) is the sample mean (137 minutes), \( \mu \) is the population mean (150 minutes), \( s_x \) is the sample standard deviation (45 minutes), and \( n \) is the sample size (30). By calculating the t-statistic, we can assess the likelihood that our sample mean differs from the population mean by random chance alone.

In hypothesis testing, the critical value delineates the boundary or threshold at which we decide whether to reject the null hypothesis. This value comes from the t-distribution, which we refer to when using a t-test. The critical value is determined by the chosen significance level, often denoted as \( \alpha \), and the degrees of freedom of the sample.

For our example, the significance level is 5% or 0.05, and since the sample size is 30, the degrees of freedom would be \( n - 1 = 29 \). Looking up a t-distribution table or using statistical software, the critical value at this significance level for a one-tailed test is approximately -1.699.

If our calculated test statistic falls below this critical value, it suggests the sample data provides enough evidence to reject the null hypothesis. Hence, the critical value acts as the tipping point in our decision-making process regarding the viability of the null hypothesis in presence of the observed data.