paper 1

population

all of the possible data

sample

part of the possible data

parameter

a numerical property of the population

statistic

a numerical property of a sample

sampling frame

a list of all the members of a population

outliers

a value less than: Q1 - 1.5 x IQR

a value more than: Q3 + 1.5 x IQR

a value less than 2 standard deviations below the mean

a value more than 2 standard deviations above the mean

frequency density formula

frequency / class width

simple random sample

1. number your sampling frame from 1 - N

2. use a random number generator to generate numbers between 1 - N

3. generate n numbers within the range 1 - N

4. ignore any repeats (restricted) or include repeats (unrestricted)

5. choose the members of the population that match the numbers

Adv and disadv of simple random sample

Adv - not bias, cheap, equal chance of selection

Disadv- sampling frame needed, not suitable for large population

proportional stratified sampling

1. block the population in strata (groups) by a chosen characteristic

2. select sample sizes for each strata in the same proportion as the population

3. use a simple random sample to select members of each strata

non proportional stratified sample

1. block the population in strata by a chosen characteristic

2. select sample sizes for each strata as decided by the person administering the experiment

3. use a simple random sample to select members of each strata

Adv and disadv of stratified sampling

Adv- representative of population, proportional representation

Disadv - population must be classified into strata

cluster sample

1. randomly select representative clusters within a population

2. select your sample from these clusters (either all members or a simple random sample from each cluster)

Adv and disadv of cluster sampling

Adv - easy to carry out, cheap

Disadv - bias, probability of number being selected depends on size of cluster

systematic sample

1. number your sampling frame

2. divide population by sample size = k

3. generate one random number between 1 and k to use as starting point

4. your sample size is selected by choosing the starting point followed by every kth person in the list after

Adv and disadv of systematic sampling

Adv- simple& quick, suitable for large populations

Disadv- sampling frame needed, bias if sampling frame not random

Adv and disadv of judgemental sampling

Adv- only selects most suitable candidates, saves money and effort on collecting unnecessary data, representative of the population

Disadv - selection criteria are subjective, sample can be influenced by the researchers viewpoint, dependant on skill of researcher

adv & disadv of snowball sampling

adv- allows researchers to reach populations that are difficult to access, cost effective, requires little planning

disadv- may not be representative, can lead to sampling bias, subjects share similar traits, may be difficult/ time consuming to get a large enough sample

experimental design

1. select a large random sample of participants (randomisation)

2. group participants by a particular attribute (blocking)

3. select a control group receives the placebo

4. run an experiment with the control group and experiment group (paired comparison

1. blind trial

   • participants dont know if they are receiving treatment or placebo

2. double blind trial

   • neither participants or experimenter know who is receiving treatment or placebo

probability notation

P(A n B) = probability that A and B both occur (intersection)

P(A u B) = probability that either A or B or both occur (union)

probability laws

(addition law) P(A u B) = P(A) + P(B) - P(A n B)

(multiplication law) P(A n B) = P(A) x P(B | A) or P(A n B) = P(B) x P(A | B)

(if A and B are mutually exclusive) P(A u B) = P(A) + P(B) or P(A n B) = 0

(if A and B are independent) P(A n B) = P(A) x P(B)

position from a list of data or frequency table

• position of median: n+1/2

• position of Q1: n+1/4 ( half of position of median )

• position of Q3: 3(n+1)/4 (position of Q1 + position of median)

position from a grouped data table

• position of median: n/2

• pos of Q1: n/4

• pos of Q3: 3n/4

• pos of xth percentile: x/100 x n

• pos of xth decile: x/10 x n

linear interpolation of the median from a grouped data table

1/2n - f / fc x c + b

b = lower class boundary of the median

f = sum of frequencies below b

fc = frequency of median class

c = class width of median class

residuals

actual value - estimate value

more reliable if:

• there is a strong correlation

• estimates are within range of original data

• residuals are small

spearmans rank formula

6x sum of d squared/ n(n squared - 1)

probability distributions mean and variance formula

mean = sum of x * p(X=x)

variance = sum of x squared * p(X = x) - mean squared

conditions of a binomial distribution

• 2 possible outcomes

• fixed number of trials

• fixed probability of success

• trials are independent

binomial mean and variance

mean = np

var = np(1-p)

standardised normal distribution

Z ~ N (0,1)

formula: Z = x - mean / standard deviation

conditions for a poisson and exponential distribution

• events must occur one at a time

• each event must be independent

• the average rate must remain constant

poisson mean and variance

the average rate

exponential mean and variance

mean = 1/ average rate

var = 1/ average rate squared

exponential distribution formula

p(X < x) = 1 - e(-average rate * x)

p(X = x) = 0

p(X > x) = e(-average rate * x)

uniform distribution formulas

probability density, p = 1/ b - a

probability = area

mean = a + b / 2

var = (b - a) squared / 12

coding

adding/subtracting affects: mean & sd

multiplication/division affects: only mean

pmcc assumptions & how to calculate

assumptions:

• linear relationship between the bivariate data

• data is normally distributed

• no significant outliers in each data set

on calculator: calc> reg> x> a+bx

linear regression

• y = a + bx

• to find a & b: calc> reg> x> a + bx

• b represents the increase or decrease in y as x increases by 1

• a represents the value of y when x is 0

distribution of sample means

Xbar ~ N ( mean (standard deviation/ square root n) squared)

normal approximation to binomial distributions

conditions:

• n is large (n > 20) and p is 0.5

or

• np > 10 and n(1-p) > 10

X ~ N (np , np(1 - p) )

square root variance for sd