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The t-test is a fundamental statistical tool often used when the population standard deviation, \( \sigma \), is unknown, and the sample size is small. This test helps determine if there is a significant difference between the mean of a sample and a known or hypothesized population mean. In hypothesis testing, a t-test is chosen if the sample distribution is approximately normally distributed, which becomes especially important with smaller sample sizes (generally if \( n < 30 \)).
One critical element of the t-test is its reliance on the sample standard deviation, \( s \), instead of the population standard deviation, which can be unknown. The formula for the t-statistic is:

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

Here, \( \bar{x} \) is the sample mean, \( \mu \) is the population mean hypothesis, \( s \) is the sample standard deviation, and \( n \) is the sample size. The degrees of freedom used in the t-test is \( n - 1 \).
In hypothesis testing problems, such as exercises (a) and (b) where \( \sigma \) was unknown, the t-test was the appropriate choice due to its adaptability to sample variances and smaller sample sets.

The z-test is another essential method in hypothesis testing, typically used when the population standard deviation is known and the sample size is large (usually \( n \geq 30 \)). It is applicable for comparing a sample mean to a population mean when the data is normally distributed.
In a z-test, the calculated z-score helps us understand how far away our sample mean is from the population mean, in terms of the population standard deviation. The formula for calculating the z-statistic is:

• \[ z = \frac{\bar{x} - \mu}{\sigma/\sqrt{n}} \]

Where \( \bar{x} \) is the sample mean, \( \mu \) is the hypothesized population mean, \( \sigma \) is the population standard deviation, and \( n \) is the sample size. The assumption here is that the population data is normally distributed, which justifies using the z-test.
In exercise problem (c), the z-test was deemed suitable because the population standard deviation \( \sigma \) was known, thus allowing for the direct application of the z-test formula.

The significance level, often denoted by \( \alpha \), is a crucial concept in hypothesis testing. It represents the probability of rejecting a true null hypothesis, essentially indicating the risk we are willing to take of making a Type I error (false positive).
Common significance levels include 0.05, 0.01, and 0.10, and these thresholds are used to determine the critical values for our test statistics. It helps decide whether the findings are meaningful enough to reject the null hypothesis. A smaller \( \alpha \) indicates stronger evidence is needed to reject the null hypothesis.
For example, in exercise problems (a), (b), and (c), significance levels were set at 0.05 and 0.01. These decided how extreme the sample statistic needed to be in order to consider the results statistically significant. More conservatively, a 0.01 level, as used in problems (b) and (c), demands more substantial evidence to reject the hypothesis than a 0.05 level used in problem (a).
Choosing the right significance level depends on the field of study and the potential consequences of making a Type I error, with more critical fields often opting for lower significance level thresholds.