Final answer:
The null hypothesis in a two-tailed test typically states no difference, while the alternative suggests there is a difference. The test statistic measures how much the sample mean differs from the hypothesized mean. The p-value is the probability of the observed statistic under the null hypothesis, guiding the conclusion.
Step-by-step explanation:
In hypothesis testing, the null hypothesis (H0) and the alternative hypothesis (Ha) are stated in a way that they are mutually exclusive. In a two-tailed test for the mean, the null hypothesis often asserts that there is no difference in the population mean (μ) from a specified value (μ0), while the alternative hypothesis claims that there is a difference.
The test statistic is a standardized value that measures the degree to which the sample statistic differs from the null hypothesis. To calculate the test statistic for the mean, one would typically use the formula: (sample mean - hypothesized mean) / (standard deviation of the sample / sqrt(sample size)). The p-value represents the probability of observing a test statistic as extreme as, or more extreme than, the one observed if the null hypothesis is true. The conclusion about the null hypothesis is made by comparing the p-value with the significance level (α), typically 0.05 or 0.01. If the p-value is less than α, one would reject the null hypothesis, suggesting that the alternative hypothesis may be true.