A-level Further Mathematics₇₃₆₇

Specification7367

7367

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A-level Further Mathematics Specification for first teaching in 2017

20 Oct 2017

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3.4 Optional application 2 – statistics

3.4.1 SA: Discrete random variables (DRVs) and expectation

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SA1	Understand DRVs with distributions given in the form of a table or function.

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SA2	Evaluate probabilities for a DRV.

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SA3	Evaluate measures of average and spread for a DRV to include mean, variance, standard deviation, mode and median.

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SA4	Understand expectation and know the formulae: $E (X) = \sum x_{i} p_{i}$ ; $E (X^{2}) = \sum x_{i}^{2} p_{i}$ ; $V a r (X) = E (X^{2}) - {(E (X))}^{2}$

	Content
SA5	Understand expectation of linear functions of DRVs and know the formulae: $E (a X + b) = a E (X) + b$ and $V a r (a X + b) = a^{2} V a r (X)$ Know the formula $E (g (X)) = \sum g (x_{i}) p_{i}$ Find the mean, variance and standard deviation for functions of a DRV such as $E (5 X^{3}), E (18 X^{- 3}), V a r (6 X^{- 1})$

Content

SA5

Understand expectation of linear functions of DRVs and know the formulae:

E (a X + b) = a E (X) + b

and

V a r (a X + b) = a^{2} V a r (X)

Know the formula

E (g (X)) = \sum g (x_{i}) p_{i}

Find the mean, variance and standard deviation for functions of a DRV such as

E (5 X^{3}), E (18 X^{- 3}), V a r (6 X^{- 1})

	Content
SA6	Know the discrete uniform distribution defined on the set $\{1,2, \dots, n\}$ . Understand when this distribution can be used as a model.

3.4.2 SB: Poisson distribution

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SB1	Understand conditions for a Poisson distribution to model a situation. Understand terminology $X ~ P o (λ)$ .

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SB2	Know the Poisson formula and calculate Poisson probabilities using the formula or equivalent calculator function.

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SB3	Know mean, variance and standard deviation of a Poisson distribution. Use the result that, if $X ~ P o (λ)$ then the mean and variance of $X$ are equal.

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SB4	Understand the distribution of the sum of independent Poisson distributions.

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SB5	Formulate hypotheses and carry out a hypothesis test of a population mean from a single observation from a Poisson distribution using direct evaluation of Poisson probabilities.

3.4.3 SC: Type I and Type II errors

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SC1	Understand Type $I$ and Type $II$ errors and define in context. Calculate the probability of making a Type $I$ error from tests based on a Poisson or Binomial distribution. Calculate probability of making Type $I$ error from tests based on a normal distribution.

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SC2	Understand the power of a test. Calculations of P(Type $II$ error) and power for a test for tests based on a normal, Binomial or a Poisson distribution.

3.4.4 SD: Continuous random variables (CRVs)

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SD1	Understand and use a probability density function, $f (x)$ , for a continuous distribution and understand the differences between discrete and continuous distributions. Understand and use distributions of random variables that are part discrete and part continuous.

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SD2	Find the probability of an observation lying in a specified interval.

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SD3	Find the median and quartiles for a given probability density function, $f (x)$ .

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SD4	Find the mean, variance and standard deviation for a given pdf, $f (x)$ . Know the formulae $E (X) = \int x f (x) d x$ , $E (X^{2}) = \int x^{2} f (x) d x$ , $V a r (X) = E (X^{2}) - (E (X))^{2}$

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SD5	Understand the expectation and variance of linear functions of CRVs and know the formulae: $E (a X + b) = a E (X) + b$ and $V a r (a X + b) = a^{2} V a r (X)$ Know the formula $E (g (X)) = \int g (x) f (x) d x$ Find the mean, variance and standard deviation of functions of a continuous random variable such as $E (5 X^{3}), E (18 X^{- 3}), V a r (6 X^{- 1})$

Content

SD5

Understand the expectation and variance of linear functions of CRVs and know the formulae:

E (a X + b) = a E (X) + b

and

V a r (a X + b) = a^{2} V a r (X)

Know the formula

E (g (X)) = \int g (x) f (x) d x

Find the mean, variance and standard deviation of functions of a continuous random variable such as

E (5 X^{3}), E (18 X^{- 3}), V a r (6 X^{- 1})

	Content
SD6	Understand and use a cumulative distribution function, $F (x)$ . Know the relationship between $f (x)$ and $F (x)$ . $F (x) = \int_{- \infty}^{x} f (t) d t$ and $f (x) = \frac{d}{d x} F (x)$

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SD7	Understand the rectangular distribution $f (x)$ where $f (x) = {\begin{matrix} \frac{1}{b - a} & a \leq x \leq b \\ 0 & otherwise \end{matrix}$ Know the conditions for the rectangular distribution to be used as a model. Calculate probabilities from a rectangular distribution. Know proofs of mean, variance and standard deviation for a rectangular distribution.

Content

SD7

Understand the rectangular distribution

f (x)

where

f (x) = {\begin{matrix} \frac{1}{b - a} & a \leq x \leq b \\ 0 & otherwise \end{matrix}

Know the conditions for the rectangular distribution to be used as a model.

Calculate probabilities from a rectangular distribution.

Know proofs of mean, variance and standard deviation for a rectangular distribution.

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SD8	Know that if $X$ and $Y$ are independent (discrete or continuous) random variables then $E (X + Y) = E (X) + E (Y)$ and $V a r (X + Y) = V a r (X) + V a r (Y)$

3.4.5 SE: Chi squared tests for association

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SE1	Construction of $n$ $\times$ $m$ contingency tables.

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SE2	Use of $\sum \frac{{(O_{i} - E_{i})}^{2}}{E_{i}}$ as an approximate $χ^{2}$ statistic with appropriate degrees of freedom.

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SE3	Know and use the convention that all $E_{i}$ should be greater than 5.

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SE4	Identification of sources of association in the context of a question.

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SE5	Knowledge of when and how to apply Yates’ correction.

3.4.6 SF: Exponential distribution

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SF1	Know the conditions for an exponential distribution to be used as a model. Know the probability density function, $f (x)$ , and the cumulative distribution function, $F (x)$ , for an exponential distribution.

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SF2	Calculate probabilities for an exponential distribution using $F (x)$ or integration of $f (x)$

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SF3	Know proofs of mean, variance and standard deviation for an exponential distribution.

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SF4	Understand that the lengths of intervals between Poisson events have an exponential distribution.

3.4.7 SG: Inference – one sample t- distribution

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SG1	Test for the mean of a normal distribution with unknown variance using a $t$ with appropriate degrees of freedom.

3.4.8 SH: Confidence Intervals

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SH1	Construct symmetric confidence intervals for the mean of a normal distribution with known variance.

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SH2	Construct symmetric confidence intervals from large samples, for the mean of a normal distribution with unknown variance.

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SH3	Make inferences from constructed or given confidence intervals.

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SH4	Construct symmetric confidence intervals from small samples, for the mean of a normal distribution with unknown variance using the $t$ -distribution.

3.3 Optional application 1 – mechanics

3.5 Optional application 3 – discrete mathematics