Final answer:
To test the null hypothesis of the population mean using a two-tailed test, establish the null and alternative hypotheses, calculate the test statistic, and compare it to critical values to decide whether to reject the null hypothesis.
Step-by-step explanation:
To test the null hypothesis of the population mean using a two-tailed test, one must first establish the null hypothesis (H0) and the alternative hypothesis (Ha). For instance, the null hypothesis might be that the population mean (μ) is equal to a specific value, H0: μ = 50. The alternative hypothesis (Ha) for a two-tailed test would be that the mean is not equal to this value, Ha: μ ≠ 50.
After collecting sample data, you would calculate the test statistic, typically a t-score or z-score, based on the sample mean, sample size, population mean, and standard deviation. Compare the test statistic to the critical values from the t-distribution (or normal distribution if the sample size is large enough). For example, if the critical value for α = 0.05 for a two-tailed test using the t-distribution is 2.045 and your calculated test statistic is less than this, you would fail to reject the null hypothesis, suggesting no significant deviation from the hypothesized population mean.
If the two-tailed test was for comparing two population means, such as whether science students spend more on textbooks than humanities students, you might use a similar process with a t-test for independent samples, again concluding based on the critical values associated with your significance level.