## 6.7 **Optional questions**

These questions are **optional**; e.g., if you need more practice, or you are studying for the exam.
(Answers are available in Sect. A.6.)

### 6.7.1 **Optional**: Working with normal distributions

This question has a video solution in the online book, so you can hear and see the solution.

The Southern Oscillation Index (SOI) is a unitless^{2} climatological measure that is easily computed, and has been shown to be related to the weather conditions in eastern Australia (Stone and Auliciems 1992; Dunn 2001).

The daily SOI has an approximate normal distribution, and is designed to have a mean of \(0\) and a standard deviation of \(10\).

- Draw a rough, labelled sketch of the theoretical distribution of the SOI values.
- What is the probability that the daily SOI exceeds \(20\)?
- What are the
*odds*that the daily SOI exceeds \(20\)? - What is the probability that the daily SOI is less than \(-25\)?
- What is the probability that the daily SOI is greater than \(-12\)?
- What is the probability that the daily SOI is between \(-10\) and \(20\)?
- The SOI is
*less than*what value about 80% of the time? - The SOI is
*greater*than what value about 35% of the time?

### References

Dunn PK. Bootstrap confidence intervals for predicted rainfall quantiles. International Journal of Climatology. 2001;21(1):89–94.

Stone RC, Auliciems A. soi phase relationships with rainfall in eastern Australia. International Journal of Climatology. 1992;12:625–36.

That is, it is not measured in kilograms or seconds etc.It is just a number; in fact, it is a bit like a \(z\)-score.↩︎