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A two-sample t-test is a statistical method used to determine if there is a significant difference between the means of two independent groups. In this exercise, we apply it to compare the average distance traveled by employees from Cincinnati and Pittsburgh.

The main goal is to see if the difference in their travel distances is due to random variation or if it is statistically significant. To perform a two-sample t-test, we compute a test statistic, called the t-statistic, which helps in determining the likelihood that any observed difference is due to chance.

This type of test is appropriate when the data is approximately normally distributed and the sample sizes are relatively similar, although they do not have to be exactly equal. By calculating the t-statistic and comparing it to a critical value, we can make conclusions about our hypotheses.

The significance level is a key concept in hypothesis testing. It represents the probability of making a Type I error, which occurs if we incorrectly reject the null hypothesis. In simpler terms, it is the threshold that tells us how confident we need to be to declare a result as statistically significant.

In our exercise, this level is set at 0.05 (or 5%). This means we are willing to accept a 5% chance of being wrong when concluding that there is a difference in means.

• **Why 0.05?**: This is a commonly used standard in many scientific fields; it's a balance between being too lenient (which might cause us to incorrectly claim significance) and being too strict (which might mean we miss detecting a true effect).
• **Practical implication**: A result is considered statistically significant if the calculated probability (p-value) is less than the significance level, indicating that the observed effect is unlikely to be due to chance alone.

In hypothesis testing, we start by establishing two competing hypotheses: the null hypothesis and the alternative hypothesis.

**Null Hypothesis ( (H_0 ))**: This is a default statement suggesting no effect or difference exists. In our context, it means there is no difference in the mean number of miles traveled by Cincinnati and Pittsburgh employees ( μ_{Cincinnati} = μ_{Pittsburgh} ).

**Alternative Hypothesis ( (H_1 ))**: This proposes a contrasting statement to the null. It suggests that a difference does exist ( μ_{Cincinnati} eq μ_{Pittsburgh} ).

• The null hypothesis is assumed true until evidence suggests otherwise.
• The outcome of the test influences which hypothesis is accepted. If the null hypothesis is not rejected, we do not have sufficient evidence to conclude a difference exists. If rejected, it implies evidence for the alternative hypothesis.

The critical t-value is a threshold value obtained from the t-distribution table. It is used to determine whether the calculated t-statistic falls within a range indicating statistical significance.

For the two-sample t-test, with a given degrees of freedom and a significance level, we find the critical t-value corresponding to a two-tailed test. These are the boundaries for deciding if we reject the null hypothesis.

In this exercise, our degrees of freedom were calculated around 67, providing critical t-values of approximately ±2.00.

• If our calculated t-statistic falls outside this range, we reject the null hypothesis since it indicates a low probability of observing such a difference by chance.
• If the t-statistic falls within the range, as it did here, there is not enough evidence to conclude a significant difference, and we fail to reject the null hypothesis.