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Hypothesis testing is a fundamental part of statistical inference that allows us to make decisions or draw conclusions about a population based on sample data. In this context, we want to determine whether the difference between two population means is equal, or if one is less than the other.

For the given exercise, our null hypothesis (\(H_{0}\)) states that the difference between the population means is zero: \((\mu_{1}-\mu_{2})=0\). The alternative hypothesis (\(H_{a}\)) proposes that the difference is less than zero: \((\mu_{1}-\mu_{2})<0\).

We calculate a test statistic, which in this case is a t-value, based on our sample data. We then compare this value to a critical t-value from statistical tables, which helps us decide whether to reject or fail to reject the null hypothesis. If our calculated t-value is more extreme than the critical value, we reject \(H_{0}\). In this example, since \(-1.241 > -1.895\), we fail to reject the null hypothesis, meaning there's no strong evidence against it.

This process essentially allows us to weigh the evidence from the data against a pre-specified level of confidence (significance level \(\alpha\)). For a more robust grasp, remember always: hypothesis testing consists of assuming no effect or difference until evidence suggests otherwise.

A confidence interval provides a range of values that likely contain the true difference between the population means. This statistical tool helps us to understand the precision of our sample estimate.

In the exercise, we calculate a 90% confidence interval for the difference \((\mu_{1}-\mu_{2})\). The interval is computed using the sample means, the pooled variance, and a critical t-value, together forming the expression: \[ \left(\bar{X}_{1}-\bar{X}_{2}\right) \pm t_{\alpha/2,df}\cdot \sqrt{\frac{s^2}{n_{1}}+\frac{s^2}{n_{2}}} \]

The calculated interval \((-4.044, 0.844)\) suggests that there is a 90% chance that the true difference falls within this range. This interval includes zero, which corresponds to failing to reject the null hypothesis in our test.

Remember, a confidence interval does not tell us the probability of a parameter being inside the interval; rather, it shows the reliability of the estimate. By consistently considering confidence intervals, we can get a clearer picture of our sample data relative to the population.

Pooled variance is an essential concept when comparing two populations with potentially different samples sizes. It provides a combined estimate of the variance by pooling data from both samples.

In this exercise, pooled variance is applied since we assume that both populations have the same variance. The formula for pooled variance is: \[ s^2 = \frac{(n_{1}-1)s_{1}^{2}+(n_{2}-1)s_{2}^{2}}{n_{1}+n_{2}-2} \]

This formula weighs each sample variance by its degrees of freedom, combining them to form a single, more stable variance estimate. In our case, calculated pooled variance \(s^2=6.514\) aids in computing both the confidence interval and the hypothesis test.

Using pooled variance assumes homogeneity of variance (equal variances) in the two populations. This assumption allows for more efficient inference. Always assess whether this assumption is reasonable in your context. By doing so, you ensure that your statistical conclusions remain valid.