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Hypothesis testing is a statistical method used to make decisions about a population based on sample data. This method helps determine if there is enough evidence to support a specific hypothesis about the population.

In hypothesis testing, we start by stating two hypotheses:

• The null hypothesis (\(H_{0}\)) represents a statement of no effect or no difference, often suggesting that any observed difference is due to random chance. In the given exercise, the null hypothesis \(H_{0}: \mu_{1}=\mu_{2}\) assumes that there is no difference between the two population means.
• The alternative hypothesis (\(H_{a}\)) is what we aim to support, indicating a change or difference. In our exercise, it suggests that the population means are unequal: \(H_{a}: \mu_{1} eq \mu_{2}\).

Once the hypotheses are set, we use sample data to test them. By calculating a test statistic and comparing it to a critical value, we either "reject" or "fail to reject" the null hypothesis. If the test statistic falls into the critical region determined by the critical value, we have enough evidence to reject \(H_{0}\). If not, any difference found may just be due to chance.

The t-distribution is crucial for hypothesis testing, especially when dealing with small sample sizes or when the population standard deviation is unknown. It is similar to the normal distribution but has thicker tails. This design accounts for the additional uncertainty present in small samples.

The shape of the t-distribution depends on the degrees of freedom, which relates to the sample sizes. The degrees of freedom are typically calculated as the total number of observations minus the number of groups being compared. For our exercise, with two groups (samples) of size 100 and 80, we have \(178\) degrees of freedom (\(n_{1} + n_{2} - 2 = 100 + 80 - 2\)).

The thicker tails in the t-distribution mean that extreme values are more probable under this distribution compared to the normal distribution. As the sample size increases, the t-distribution approaches the shape of the normal distribution. The t-distribution is used to find the critical value needed in hypothesis tests, like the one in our exercise, and helps provide the threshold for deciding whether to reject the null hypothesis.

Pooled standard deviation is a method to estimate the shared standard deviation for two independent samples. It provides a single measure that accounts for variation in both samples. This estimation is particularly useful when comparing means in hypothesis testing, as seen in our exercise.

To calculate the pooled standard deviation, we use the formula: \[\sqrt{\frac{(n_1 - 1)s_1^2 +(n_2 - 1)s_2^2 }{n_1 + n_2 - 2}}\] Here, \(n_1\) and \(n_2\) are the sample sizes, while \(s_1\) and \(s_2\) are the sample standard deviations. The numerator sums the squared deviations of each sample, weighted by their degrees of freedom. Dividing this by the combined degrees of freedom gives us the pooled variance, and taking its square root provides the pooled standard deviation.

Using the values \(n_1 = 100, s_1 = 11.6\) and \(n_2 = 80, s_2 = 14.3\), we compute the pooled standard deviation to gauge the overall variation. This value plays a pivotal role in calculating the t statistic, vital for determining whether the observed difference between sample means is statistically significant.