Achievement objective S84
In a range of meaningful contexts, students will be engaged in thinking mathematically and statistically. They will solve problems and model situations that require them to:
 Investigate situations that involve elements of chance:
 A. calculating probabilities of independent, combined, and conditional events
 B. calculating and interpreting expected values and standard deviations of discrete random variables
 C. applying distributions such as the Poisson, binomial, and normal.
Indicators
A. Calculating probabilities of independent, combined, and conditional events:
Students learn that some situations involving chance produce discrete numerical variables, that situations involving real data from statistical investigations can be investigated from a probabilistic perspective. These have
probability distributions. They can be investigated by making assumptions about the situation and applying probability rules and/or by doing repeated trials of the situation and collecting frequencies.
 Selects and uses appropriate methods to investigate probability situations including
experiments,
simulations, and
theoretical probability, distinguishing between
deterministic and
probabilistic models.
 Interprets results of probability investigations, demonstrating understanding of the relationship between
true probability (unknown and unique to the situation), model estimates (theoretical probability), and
experimental estimates.
 Selects and uses appropriate tools to solve problems in probability, including
twoway tables, Venn diagrams, and
tree diagrams, including
combined events.
 Solves probability problems involving
conditional probabilities,
randomness,
independence, and
mutually exclusive events.
B. Calculating and interpreting expected values and standard deviations of discrete random variables:
A statistical data set may contain discrete numerical variables. These have
frequency distributions that can be converted to empirical probability distributions. Distributions from both sources have the same set of possible features (centre, spread, clusters, shape, tails, and so on) and we can calculate the same measures (mean, SD, and so on) for them.
C. Applying distributions such as the Poisson, binomial, and normal:
They learn that some situations that satisfy certain conditions can be modelled mathematically. The model may be
Poisson,
binomial,
normal,
uniform,
triangular, or others, or be derived from the situation being investigated.
 Recognises situations in which probability distributions such as
Poisson,
binomial, and
normal are appropriate models, demonstrating understanding of the assumptions that underlie the distributions.
 Selects and uses an appropriate distribution to model a situation in order to solve a problem involving probability.
 Selects and uses an appropriate distribution to solve a problem, demonstrating understanding of the link between probabilities and areas under density functions for continuous outcomes (for example,
normal,
triangular, or
uniform, but nothing requiring integration).
 Selects and uses an appropriate distribution to solve a problem, demonstrating understanding of the way a probability distribution changes as the parameter values change.
 Selects and uses an appropriate distribution to solve a problem involving finding and using estimates of parameters.
 Selects and uses an appropriate distribution to solve a problem, demonstrating understanding of the relationship between true probability (unknown and unique to the situation), model estimates (theoretical probability) and experimental estimates.
 Uses a distribution to estimate and calculate probabilities, including by
simulation.
Progression
S84 links from S74.
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What is new/changed?
Possible context elaborations

Examples of student activities for distributions:
Assessment for qualifications
NCEA achievement standards at levels 1, 2 and 3 have been aligned to the New Zealand Curriculum. Please ensure that you are using the correct version of the standards by going to the
NZQA website.
The NZQA subjectspecific resources pages are very helpful. From there, you can find all the achievement standards and links to assessment resources, both internal and external.
Learn more:
The following achievement standard(s) could assess learning outcomes from this AO:
 AS91585 Mathematics and statistics 3.13 Apply probability concepts in solving problems; External, 4 credits.
 AS91586 Mathematics and statistics 3.14 Apply probability distributions in solving problems; External, 4 credits.
Refer to the
mathematics and statistics matrix.
Last updated September 26, 2013
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