For a random sample of size (n) taken from a random variable (X∼N(\mu, \sigma^2)), the sample mean ((\bar{X})) is normally distributed with (\bar{X}∼N\bigg(\mu, \dfrac{\sigma^2}{n}\bigg)).

To perform the test:

• Define the test statistic
• Determine whether a one-tailed or two-tailed test is needed
• Identify the null and alternate hypotheses accordingly
• Calculate (P(\bar{X} < \bar{x})) and/or (P(\bar{X} > \bar{x}))
• Compare with the significance level of the test (usually (5%)) and decide whether or not to reject the null hypothesis
• Conclude with a statement in context of the question

[b]uSample mean of a Normally distributed random variable[/u]/b

(\bar{X}∼N\bigg(\mu, \dfrac{\sigma^2}{n}\bigg))