CMA Foundation Paper 3 flashcards

33 cards. Read the term and the definition here. The deck at the top of this page is optional.

Ratio division formula

To split amount N in the ratio a:b:c, the three shares are N·a/(a+b+c), N·b/(a+b+c) and N·c/(a+b+c). The sum of the shares always equals N.

Successive percentage change

Two successive changes of x% and y% produce a net change of x + y + xy/100 percent. A 10% rise then a 10% fall therefore gives a net fall of 1%, not zero.

Compound interest formula

Amount A = P(1 + R/100)^n when interest compounds annually for n years. For half-yearly compounding, halve the rate and double the number of periods.

True versus banker's discount

True discount is simple interest computed on the present value of a bill; banker's discount is simple interest computed on the face value. Banker's discount is therefore always larger.

Quadratic roots and discriminant

For ax² + bx + c = 0, roots sum to −b/a and multiply to c/a. The discriminant b² − 4ac is positive for distinct real roots, zero for equal roots, negative for complex roots.

Sum of an AP

For an arithmetic progression with first term a and common difference d, the nth term is a + (n−1)d and the sum of n terms is n/2[2a + (n−1)d].

Infinite geometric series

The sum of an infinite geometric progression is a/(1 − r), valid only when the common ratio r satisfies |r| < 1. If |r| ≥ 1 the series diverges and has no finite sum.

Combinations versus permutations

nPr = n!/(n−r)! counts ordered arrangements, while nCr = n!/[r!(n−r)!] counts selections where order does not matter. nCr is always nPr divided by r!.

Heron's formula

The area of a triangle with sides a, b and c equals √[s(s−a)(s−b)(s−c)], where s is the semi-perimeter (a+b+c)/2. It needs no height measurement.

Cone volume and slant height

A right cone has volume ⅓πr²h and curved surface area πrl, where slant height l = √(r² + h²). Total surface area adds the base circle, giving πr(l + r).

Sphere and hemisphere formulas

A sphere of radius r has volume (4/3)πr³ and surface area 4πr². A hemisphere has total surface area 3πr², comprising the curved half plus the flat circular base.

Polygon interior angle sum

The interior angles of a polygon with n sides sum to (n − 2) × 180°. For a regular polygon each interior angle is therefore (n − 2) × 180°/n.

Second derivative test

At a stationary point where f′(x) = 0, the function has a minimum if f″(x) > 0 and a maximum if f″(x) < 0. If f″(x) = 0 the test is inconclusive.

Quotient rule

For a function u/v, the derivative is (u′v − uv′)/v². The numerator subtracts in that order, so reversing the terms changes the sign of the answer.

Profit maximisation condition

Profit is maximised where marginal revenue equals marginal cost, that is d(TR)/dQ = d(TC)/dQ. At that output the second derivative of profit should be negative.

Definite integral as area

The definite integral of f(x) from a to b equals F(b) − F(a), where F is any antiderivative of f. It gives the area between the curve and the x-axis over that interval.

Primary versus secondary data

Primary data are collected first-hand by the investigator for the current study. Secondary data already exist, having been gathered earlier by someone else for a different purpose.

Stratified sampling

The population is divided into strata that are internally homogeneous but differ from one another, then random samples are drawn from each stratum. This usually improves precision over simple random sampling.

Ogive intersection

The less-than and more-than cumulative frequency curves, when plotted on the same axes, intersect at a point whose x-coordinate is the median of the distribution.

Components of a time series

A time series is analysed as four components: secular trend (long-term direction), seasonal variation (recurring within a year), cyclical variation (longer waves) and irregular random variation.

Grouped data median formula

Median = L + [(n/2 − cf)/f] × h, where L is the lower limit of the median class, cf the cumulative frequency before it, f its frequency and h the class width.

Coefficient of variation

The coefficient of variation equals (standard deviation / mean) × 100 percent. Being unit-free, it compares consistency between series measured in different units or with different averages.

Empirical mode relation

For moderately skewed distributions, Mode ≈ 3 Median − 2 Mean. The relation fails for highly skewed or J-shaped distributions, where the mean, median and mode diverge sharply.

Quartile deviation

Quartile deviation, also called semi-interquartile range, equals (Q3 − Q1)/2. It measures spread of the middle half of the data and ignores the extreme quarters entirely.

Pearson correlation coefficient

Karl Pearson's r = Σ(x − x̄)(y − ȳ) / √[Σ(x − x̄)²·Σ(y − ȳ)²]. It ranges from −1 to +1, is unit-free, and measures only the strength of linear association.

Spearman rank correlation

Spearman's ρ = 1 − 6Σd²/[n(n² − 1)], where d is the difference between paired ranks. It suits ordinal data and needs a correction when ranks are tied.

Regression coefficients product

The product of the two regression coefficients b_yx × b_xy equals r², the coefficient of determination. Both coefficients share the sign of r, and their geometric mean equals |r|.

Regression lines intersection

The regression line of y on x and the line of x on y both pass through the mean point (x̄, ȳ). They coincide only when the correlation is perfect, r = ±1.

Expected value of a random variable

For a discrete random variable, E(X) = Σ x·P(x), the probability-weighted average of all possible outcomes. It need not equal any value the variable can actually take.

Addition rule of probability

P(A ∪ B) = P(A) + P(B) − P(A ∩ B). For mutually exclusive events the intersection is empty, so the probabilities simply add with no subtraction.

Binomial distribution

With n independent trials each having success probability p, P(X = r) = nCr·p^r·(1 − p)^(n−r). The mean is np and the variance is np(1 − p).

Poisson distribution

P(X = r) = e^(−λ)·λ^r/r! models the count of rare events in a fixed interval. Its single parameter λ is both the mean and the variance of the distribution.

Conditional probability

P(B|A) = P(A ∩ B)/P(A), provided P(A) > 0. It rescales the sample space to event A, giving the chance of B once A is known to have occurred.