15.4 Statistical test using the Normal distribution

AQA Edexcel OCR A OCR B (MEI)
For a random sample of size nn taken from a random variable XN(μ,σ2)X∼N(\mu, \sigma^2), the sample mean (Xˉ\bar{X}) is normally distributed with XˉN(μ,σ2n)\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(Xˉ<xˉ)P(\bar{X} < \bar{x}) and/or P(Xˉ>xˉ)P(\bar{X} > \bar{x})
- Compare with the significance level of the test (usually 5%5\%) and decide whether or not to reject the null hypothesis
- Conclude with a statement in context of the question
Important
Sample mean of a Normally distributed random variable

XˉN(μ,σ2n)\bar{X}∼N\bigg(\mu, \dfrac{\sigma^2}{n}\bigg)
3