Consider if we are simply given that $X ~\sim~ N(\mu_0, \sigma^2) and$latex \mathrm{P}(X < 4.2) = 0.2\$ and $\mathrm{P}(X > 8) = 0.1$. We will be able to solve this question comfortably by standardising the two given probability equations and then finding the z-value, etc.

However, what if the question throw in more unknowns? Like $\alpha \mathrm{P}(X < \mu_0 - \alpha) = \mathrm{P}(X < \mu_0 + \alpha)$. In this case we cannot really doing our usual approach of doing standardising and inverse normal since we lack the area.

Considering the symmetrical properties of the graph above, we can derive some results.

Many students tend to confuse and think that normal distribution is always centred about zero, this is inaccurate as only the standard normal distribution is centred about zero. In general, we can say that normal distribution is always centred about its mean, $\mu$. And it so happens that the mean of the standard normal distribution is zero.

Students may want to reference a 2014 ACJC Prelim Question here to see if they can do it.

• […] Type I vs Type II Errors 9. A little about Central Limit Theorem 10. What is a good estimate? 11. Using symmetrical properties of Normal Curve to solve questions 12. Bayesian vs Frequentist […]

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