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When conducting a two-tailed test, we're interested in determining whether there is a significant difference in either direction between two sample means. The null hypothesis here is that there is no difference between the population means, denoted as \(H_{0}: \mu_{1}-\mu_{2}=0\). In contrast, the alternative hypothesis suggests that the two means are not equal, represented as \(H_{a}: \mu_{1}-\mu_{2} eq 0\). This type of test considers variability on both sides of the distribution, looking for evidence that the effect is present in either direction.

A two-tailed test thus involves evaluating whether your observed test statistic falls into the lower or upper extreme of a statistical distribution. The critical values mark the boundaries of this extreme region, typically based on the chosen significance level, \(\alpha\). If the test statistic lands beyond these critical values, the null hypothesis is rejected. Otherwise, it is not rejected. This testing is crucial when outcomes could differ in both directions substantially and we do not have information to expect deviation in one particular direction.

In hypothesis testing, the test statistic is a standardized value used to determine whether to reject the null hypothesis. For comparing two independent samples, the test statistic formula is critical because it helps quantify the difference between sample means in standardized units.

Here's the formula for the test statistic in a two-sample z-test:

• \[ z = \frac{(\bar{x}_1 - \bar{x}_2) - (\mu_1 - \mu_2)}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}} \]

In this formula:

• \( \bar{x}_1 \) and \( \bar{x}_2 \) are the sample means.
• \( \sigma_1 \) and \( \sigma_2 \) are the population standard deviations.
• \( n_1 \) and \( n_2 \) are the sample sizes.
• \( \mu_1 \), \( \mu_2 \) are the population means under the null hypothesis.

This equation takes into account the variability within both samples and adjusts for the sample size to provide a comprehensive picture of how different the two sample means are from each other. Once calculated, this z-score helps determine how likely it is to observe such a result under the assumption that the null hypothesis is true.

The p-value in hypothesis testing is a measure of the strength of the evidence against the null hypothesis. It tells us the probability of obtaining a test statistic as extreme as, or more extreme than, the observed result, assuming the null hypothesis is true.

In a two-tailed test, because we are examining the potential for extreme outcomes in both directions, the p-value represents the combined area in both tails of the distribution beyond the calculated test statistic. For example, when a calculated z-score is \(-1.65\), the p-value is found by determining the probability of z being less than \(-1.65\) (for the left tail) and more than \(1.65\) (for the right tail), and then summing these probabilities. Using the standard normal distribution:

• The left tail probability for \(-1.65\) is approximately 0.0492.
• The right tail probability for \(1.65\) is also approximately 0.0492.
• Adding these gives a p-value of about 0.0984.

If this p-value is greater than the pre-set significance level \( \alpha \), the evidence is not strong enough to reject the null hypothesis. In practical terms, a larger p-value indicates more support for the null hypothesis, suggesting that the observed data could occur by random chance alone.