#15.2
Be able to identify the probability of success, \(p\), for the binomial distribution.
The binomial distribution as a model for observed data.
Notation:
\(B(n, p), q = 1 - p\)
\(∼\) means ‘has the distribution’.
#15.3
Be able to calculate probabilities using the binomial distribution.
Including use of calculator functions.
#15.4
Understand and use mean \(= np\).
[Excludes: Derivation of mean \(= np\)]
#15.5
Be able to calculate expected frequencies associated with the binomial distribution.
#15.6
Be able to use probability functions, given algebraically or in tables.
Know the term discrete random variable.
Restricted to simple finite distributions.
Notation:
\(X\) for the random variable.
\(x\) or \(r\) for a value of the random variable.
#15.7
Be able to calculate the numerical probabilities for a simple distribution.
Understand the term discrete uniform distribution.
Restricted to simple finite distributions.
Notation:
\(P(X = x)\)
\(P(X \le x)\)
[Excludes: Calculation of \(E(X)\) or \(Var(X)\).]
#15.8
Be able to use the Normal distribution as a model.
Includes recognising when a Normal distribution may not be appropriate.
Understand how and why a continuity correction is used when using a Normal distribution as a model for a distribution of discrete data.
Recognise from the shape of the distribution when a binomial distribution can be approximated by a Normal distribution.
Notation: \(X ∼ N(\mu, \sigma^2)\)
[Excludes: Knowing conditions for Normal approximation to binomial.]
#15.9
Know the shape of the Normal curve and understand that histograms from increasingly large samples from a Normal distribution tend to the Normal curve.
Includes understanding that the area under the Normal curve represents probability.
#15.10
Know that linear transformation of a Normal variable gives another Normal variable and know how the mean and standard deviation are affected. Be able to standardise a Normal variable.
\(y_i = a+bx_i \implies \bar{y} = a+b\bar{x}\), \(s_y^2 = b^2s_x^2\)
Notation:
Standard Normal
\(Z∼N(0,1)\)
\(Z = \dfrac{X - \mu}{\sigma}\)
[Excludes: Proof]
#15.11
Know that the line of symmetry of the Normal curve is located at the mean and the points of inflection are located one standard deviation away from the mean.
#15.12
Be able to calculate and use probabilities from a Normal distribution.
Including use of calculator functions.
#15.13
Be able to model with probability and probability distributions, including recognising when the binomial or Normal model may not be appropriate.
Including critiquing assumptions made and the likely effect of more realistic assumptions.