applied maths

define census, and give pros and cons

census: measures every member of a population

😊: accurate result

☹: expensive/testing may destroy

define sampling units and sampling frame

sampling units: individuals of a population

sampling frame: list of sampling units

define simple random sampling and how to do it, and give pros and cons

it means every unit has the same chance of being selected, you can use random number/lottery sampling - out of a hat

😊 bias free

☹ need sampling frame

define systematic sampling and how to do it, and give pros and cons

type of random sampling, take every kth unit, k = pop/sam, pick a random number been 1 and k for starting

😊 quick to use

☹ need sampling frame

define systematic sampling and how to do it, and give pros and cons

type of random sampling, sample represents groups (strata) of a population. sam/pop x strate for each strate, pick randomly.

😊 reflects population

☹ pop must be classified in strata

define quota sampling and how to do it, and give pros and cons

type of non random sampling, like stratified, but strata filled by interviewer/researcher

😊 no sampling frame required

☹non-random, potential bias

define opportunity sampling and how to do it, and give pros and cons

type of non random sampling, quota filled by those available at the time

😊 easy, cheap

☹unlikely to be representative

define the types of data

qualitative: non-numerical

quantitative: numerical

discrete: can only take certain values in a range

continuous: can take any value in a range

where are the uk weather stations

(alphabetical upwards apart from heathrow and hurn)

general rules for weather at uk stations

coastal: windier, wetter

south: warmer, more hours of sunshine

when is large data set recorded from?

may-oct ‘87 and ‘15 (only 6 months of year)

name the international stations and general weather rules there

1. perth, australia, summer and winter switched, very hot

2. beijing, china, very hot and rainy in summer, very cold in winter

3. jacksonville, florida usa, very hot, prone to hurricanes, hurricane in oct 87 and oct 15

what does tr mean in the rainfall data

trace, < 0.05mm, treat as 0 in calculations

what does n/a mean in lds

means reading is not available, so can’t use in sample

what is cloud cover measured in?

okras, discrete values integers 0-8 (9 values it can take)

what is max gust measured in?

knots, 1kn = 1.15mph, great storm oct 15th/16th ‘87.

formula for mean!

how to find LQ median and UQ in listed data

LQ = n/4

median = n/2

UQ = 3n/4

if a decimal, round up

if whole, use midpoint between that value and one above

how to find percentiles and deciles e.g

P(57) = 0.57 x n

D(3) = P(30) = 0.3 x n

do not round, use linear interpolation.

how to find IQR and interpretive range, and an advantage of these measures

IQR = Q3 - Q1

e.g 10th to 90th IPR = p(90) - p(10)

😊 ignores extremes

how to find variance and standard deviation

coding?

cumulative frequency diagrams/ box plots, and what are the outlier boundaries?

why should you use histograms, what is frequency density and what is the area?

area = frequency x k

what two things do you need to compare if asked to compare two diagrams

1) measure of location

2) measure of spread

put in context

what does the PMCC measure and what values can it take

measures strength and +/- of correlation

-1 <= r =< 1

what is a regression line, what form does it often take and what does it mean, and what to do if model is non linear

line of best fit, y = a + bx

a is y when x = 9

b is how much y changes when x increases by one

if y = ab^x or y= ax^n: take logs to make linear

what is interpolation and extrapolation, and are they reliable

interpolation: estimating inside data range, reliable

extrapolation: estimating outside data range, unreliable

what are unions and intersections

tree diagrams

if a and b are mutually exclusive?

if a and b are independent?

conditional probability formula?

addition law for probabilities?

what does it mean for a distribution to be discrete and uniform

probabilities of outcomes are all equal

when should you use binomial distribution

FFIT:

Fixed number of trials, n

Fixed probability of success, p

Independent trials

Two outcomes, success/failure

normal distribution? where is p.o.i and where does the data lie

standard normal distribution? what is the coding?

what conditions needed to approximate binomial as normal

continuity correction for approximating binomial as normal

going from discrete to continuous, increase range by 0.5

define null hypothesis

what we assume to be true, H0

define alternative hypothesis

H1, what would be true if H0 is wrong

define significance level

the given threshold of likeness, alpha

when do you do one-tailed test/ two-tailed test

one-tailed: when H1: p > k or p < k

two-tailed test: when H1: p ≠ k → halve s.l for each tail

how do you do correlation testing

how to do binomial testing

how to do normal hypothesis testing of a sample mean

what formula to use when a = 0

distance = speed x time

suvats

v = u + at

s = ut + 1/2at²

s = vt - 1/2at²

v² = u² + 2as

s = (u + v)t

variable acceleration

can tell it’s variable acceleration when expressed in terms of t

s differentiates to v which differentiates to a (with respect to time) integration opposite

speed-time graphs

distance travelled = area under the curve

vector motion when

a. no acceleration

b. constant acceleration

c. variable acceleration

a. r = r(0) + vt

b. v = u + at,

r = r(0) + ut + 1/2at²

r = r(0) + vt - 1/2at²

c. r differentiates to v differentiates to a, differentiate i and j components separately

projectiles are a and b. connected by t.

what does it mean for a vector to be ‘moving in the direction’

parallel to it, e.g moving in the north direction = k(0,1)

what effect do each of these modelling assumptions have on calculations

smooth pulley:

light string:

inextensible string:

particle:

rod:

smooth pulley: tension on either side of the pulley is equal

light string: tension is equal

inextensible string: both particles have same acceleration

particle: ignore air resistance and ignore rotational effects

rod: rigid, so it doesn’t bend, it has no thickness.

what does it mean when described as:

• static/at rest

• on the point of slipping/limiting equilibrium

• constant speed

• vector forces f1 + f2 + f3 = 0

in equilibrium, forces are balanced: down forces = up forces, left forces = right forces.

what does it mean and what to do when forces not in equilibrium

it is accelerating/decelerating, resolve using F=ma, in the direction of motion.

Vectors: use F=ma where F is the resultants R

friction

always opposes the direction of motion/it’s about to move in

if static, Fr <= muR

if limiting equilibrium/dynamic Fr = muR

0 <= mu (mu is generally < 1, < 4)

resultant force on pulley?

method for connected particles

1) F = ma

2) F= ma

3) Sim Equations

4) SUVAT, find v → u after string breaks after particle hits floor

5) New acceleration, T = 0

6) More SUVAT, use new a and u will be v from before

what do these terms mean for calculations:

uniform:

non-uniform:

on the point of tilting about A:

uniform: weight acts at centre

non-uniform: weight doesn’t have to act at centre

on the point of tilting about A: all other R = 0

rod suspended by string

y can be up or down

rod resting on peg

use pythagoras to find R

formula for moment

moment = force x perpendicular distance

or

moment = perpendicular force x distance

if moment is in equilibrium

anticlockwise moment = clockwise moment

general method for moments

1) resolve up/down

2) resolve left/right

3/ take moments