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The t-test is a statistical method used to determine if there is a significant difference between the means of two groups. It's particularly useful when dealing with small sample sizes, as it accounts for the variability and distribution of the sample data.

In the context of our problem, we want to see if the average sleep duration of students (6.2 hours) is significantly different from the recommended 8 hours. Essentially, we are checking if this difference is due to random chance or if it truly reflects a lower sleep amount for the population (all students).

The formula to calculate the t-statistic in a one-sample t-test is:
\[ t = \frac{(\bar{x} - \mu)}{(s / \sqrt{n})} \]
Here, \(\bar{x}\) is the sample mean, \(\mu\) is the population mean (hypothesized value), \(s\) is the sample standard deviation, and \(n\) is the sample size.

• \(\bar{x} = 6.2\)
• \(\mu = 8\)
• \(s = 1.70\)
• \(n = 12\)

This statistical tool helps us determine whether the observed sample mean significantly deviates from the expected population mean.

In hypothesis testing, we start by stating two hypotheses; the null hypothesis and the alternative hypothesis.

The **null hypothesis** (H0) generally reflects the status quo or a statement of 'no effect'. It's what we assume to be true until we have evidence to suggest otherwise. In this sleep study, the null hypothesis states that students, on average, are getting 8 hours of sleep per night, expressed as:
\[ H_0: \mu = 8 \]

The **alternative hypothesis** (HA), on the other hand, represents what we want to prove. It suggests that a significant effect exists. Here, it claims that students, on average, are getting less than the stipulated 8 hours:
\[ H_A: \mu < 8 \]

The process is a bit like a courtroom trial, where the null hypothesis is assumed true unless the evidence collected from the data suggests with a high degree of confidence that we should reject it.

Degrees of freedom are an essential concept in statistics, particularly when working with sample data. They are associated with the number of independent values or quantities that can vary in an analysis, without violating any given constraints.

In the context of a t-test, degrees of freedom are important for determining the critical value from a t-distribution table. The degrees of freedom in a one-sample t-test are calculated using the formula:
\[ df = n - 1 \]
where \(n\) is the sample size. In our example of sleep data, \(n = 12\), thus:

• \(df = 12 - 1 = 11\)

The degrees of freedom determine the shape of the t-distribution curve. This, in turn, helps to identify the critical t-value, which is used to decide whether to reject the null hypothesis. A higher number of degrees of freedom results in a t-distribution that closely approximates a normal distribution. This convergence is why larger sample sizes provide more reliable results.