Statistical Analysis (ME-S1)

6 dot points across 6 inquiry questions, in syllabus order. Each dot point has a focused answer, with exam-style questions and worked answers where available.

What is a Bernoulli trial, and what are its mean and variance?

Define a Bernoulli random variable, compute its mean and variance, and recognise scenarios that fit the model

Bernoulli trials. The definition, mean $p$ , variance $p (1 - p)$ , and the role of Bernoulli trials as the building block of the binomial distribution.

10 min read →

What is the binomial distribution, and what are its mean and variance?

Define the binomial distribution

B(n, p)

, state its probability mass function, and find its mean

n p

and variance

n p (1 - p)

Binomial distribution. The pmf $P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k}$ , mean $n p$ , variance $n p (1 - p)$ , and standard situations that fit the model.

11 min read →

When and how do we use the normal distribution to approximate binomial probabilities?

Use the normal approximation

X \sim N(n p, n p (1 - p))

to approximate binomial probabilities for large

n

Normal approximation of the binomial. The rule of thumb $n p \ge 5$ and $n(1 - p) \ge 5$ , continuity correction, standardising and computing approximate probabilities, with worked examples.

12 min read →

How does the sample proportion behave as a random variable, and how do we use its normal approximation?

Use the sample proportion

\hat{p} = X/n

as a random variable, with mean

p

, variance

pq/n

and standard deviation

\sqrt{pq/n}

, and apply its normal approximation

\hat{p} \sim N(p,\, pq/n)

Sample proportions. What $\hat{p} = X/n$ is, why it is a random variable, its mean $p$ , variance $pq/n$ and standard deviation $\sqrt{pq/n}$ , the normal approximation $\hat{p} \sim N(p, pq/n)$ , computing probabilities about $\hat{p}$ , the effect of sample size, and expected-range reasoning for polling and quality control.

14 min read →

How much do sample means vary from sample to sample, and how does the central limit theorem let us find the probability that a sample mean lies within given bounds?

Apply the central limit theorem to estimate the probability that the sample mean lies within given bounds

The Year 12 Extension 1 answer on the sampling distribution of the mean, new in the 2024 syllabus: why sample means vary, E of X-bar = mu and Var of X-bar = sigma squared on n, the central limit theorem for n at least 30 whatever the population's shape, and finding probabilities for a sample mean with z-scores, including NESA's sample exam question.

13 min read →

How do we calculate probabilities involving the binomial distribution, including ranges and complements?

Compute exact probabilities for the binomial distribution including

P(X = k)

P(X \le k)

P(X \ge k)

, and use complementary counting

Computing binomial probabilities. Exact pmf values, cumulative sums, complements (at least, at most), and standard problem patterns, with worked examples.

11 min read →