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EDEXCEL MATH’S A LEVEL MIDTERM
REVISION
Linear Regression - ANSWERS-y = axⁿ logy = loga + nlogx Exponential Regression - ANSWERS-y = ab^x logy = loga + xlogb Normal Approximation - ANSWERS-μ = np σ =√(np(1-p)) Mean - ANSWERS-∑x ÷ n GF: ∑xf ÷ ∑f Variance - ANSWERS-(∑x²/n) - (∑x/n)² Standard Deviation - ANSWERS-√variance Histograms: Height - ANSWERS-Area = k x frequency Frequency Density - ANSWERS-frequency ÷ class width Population - ANSWERS-Whole set of items of interest. Census - ANSWERS-Observes/measures every member of a population Sample - ANSWERS-Selection of observations taken from a subset
of the population which is used to find out info about the population. Sampling Frame - ANSWERS-A list of individuals (named or numbered) from whom the sample is drawn Random Sampling - ANSWERS-Every member of the population has an equal chance of being selected Systematic Sampling - ANSWERS-Every nth person is chosen. Stratified Sampling - ANSWERS-Population is divided into mutually exclusive Strat and a random sample is taken from each. Quota Sampling - ANSWERS-Interviewer selects a sample that reflects the characteristics of the population Opportunity Sampling - ANSWERS-Choosing whoever is available Continuous Variable - ANSWERS-Can take any value in a given range Discrete Variable - ANSWERS-Takes specific values in a given range Conditions for Binomial - ANSWERS-Fixed no. of trials 2 possible outcomes Outcomes are independent Fixed probability of success Probability: Independent if... - ANSWERS-P(A∩B) = P(A) X P(B) P(A|B) = P(A) Probability: Mutually exclusive if... - ANSWERS-P(A∩B) = 0
n= number of trials standard deviation - ANSWERS-a computed measure of how much scores vary around the mean score general probability addition rule - ANSWERS-P(A∪B)=P(A)+P(B) −P(A∩B) P(A∩B)=P(A)+P(B)−P(A∪B) what is an event - ANSWERS-a set of possible outcomes - not necessarily equally likely sample space - ANSWERS-set of all possible outcomes , all equally likely A ∪ B - ANSWERS-means A or B or both A ∩ B - ANSWERS-means both A and B The rules for tree diagrams are - ANSWERS-Select which branches you need Multiply along each branch Add the results of each branch needed. Make sure that you include enough working to show which branches you are using (method). Be careful to allow for selection with and without replacement. if you are told what has happened for the first choice and need to find second you read off tree diagram
if you are told what has happened for the second choice use formula sample space diagram - ANSWERS- mutually exclusive events - ANSWERS-Two events that cannot occur at the same time what is P(A∩B) if two events are mutually exclusive - ANSWERS- P(A∩B) = 0 Sampling units - ANSWERS-Individual units of a population what does a census do - ANSWERS-observes and measures every member of a population advantages / disadvantages of census - ANSWERS-advantages : gives a completely accurate result disadvantages -time consuming and expensive -hard to process large quantity of data advantages/disadvantages of a sample - ANSWERS-advantages : less time consuming and expensive than census fewer people have to respond less data to process than in a census
disadvantages sampling frame is needed not suitable when population size/sample size is large as it would be potentially tie consuming, disruptive and expensive 3 different types of random sampling - ANSWERS-Simple random sampling Systematic sampling
- Stratified sampling systematic sampling - ANSWERS-The required elements are chosen at regular intervals from an ordered list e.g if sample size was 20 out of population of 100 you would take one out of every 5 as 100/20= advantages/disadvantages systematic sampling - ANSWERS- advantages
- easy and quick to use
- suitable for large samples/populations disadvantages sample frame needed
can introduce bias if sampling frame not random stratified sampling - ANSWERS-a variation of random sampling; the population is divided into mutually exclusive strata - males and females for example and a random sample is taken from each number sampled in a stratum/total sample size = number in stratum/number in population advantages/disadvantages stratified sampling - ANSWERS- advantages
- sample accurately reflects the population structure
- guarantees proportional representation of groups within a population disadvantages -population must be clearly classified into distinct strata -selection from strata suffers from same disadvantages as simple random sampling quota sampling - ANSWERS-interviewer/reasearcher selects a sample that reflects the characteristics of the whole population quota sampling advantages/disadvantages - ANSWERS- Advantages:
continuous variable - ANSWERS-a quantitative variable that has an infinite number of possible values that are not countable discrete variable - ANSWERS-variable that only have set values class boundaries - ANSWERS-Tell you the maximum and minimum values that belong in each class class width - ANSWERS-the difference between upper and lower class limits. - remember if continuous above 0.5 and below by 0. words to describe scatter diagrams - correlation - ANSWERS- Strong - close points to each other Weak Positive Negative Fairly Very Extremely - how steep None how to describe correlations in questions - ANSWERS-- always link to question context e.g house prices go up the closer they are to the station regression lines - ANSWERS-The line of best fit drawn through a scatterplot they have equation y=a + bx
b is gradient if positive then they are correlated positively if negative then negatively correlated interpolation - ANSWERS-estimating a value within the range of measured data
- USUALLY MORE RELIABLE You are usually asked to give an interpretation of the gradient Eg for every 1 degree rise there will be 22 more ice creams sold per hour For every 1 minute passed the temperature will cool by 1. degrees For every hour passed the bacteria will have increased by 15 - does this seem right? LOOK AT THE SCALE eg 15 (million) they were in millions extrapolation - ANSWERS-Estimating a value outside the range of measured data.
- usually unreliable how to identify outliers - ANSWERS-- if it is 1.5*IQR above/below the upper/lower quartiles it is an outlier
- if it is more than 2 standard deviations from the mean it is outlier coding - inverse function - ANSWERS-means it is the exact inverse function e.g if the x was formed from 2y+4 you would do x-4 /2 to
- Compare Skew
- Put comparison into context histograms what to know - ANSWERS-class width * frequency density = frequency might need to use ratios class width shown to you might need to see that if a certain area represents a certain frequency in a certain ratio , how you can apply this ratio to the others need to practice this with pp cumulative frequency tables to remember - ANSWERS-- keep a running total
- plot against upper boundaries
- make a smooth shaped s curve unimodal data - ANSWERS-data with one mode uniform data - ANSWERS-no mode / data is symmetrical bimodal data - ANSWERS-A histogram will have two peaks as the data has two modes.
- peaks do not have to be equal how to interpolate data e.g find the 17th value - ANSWERS-go across each frequency and when you find the area upon which the
17th value lies, and lets say the first category contained 10 and the next 12,
- you would do 17-10 =7 to get remainder of 7 , then do 7/12 * class width = x
- then you would add x to the starting value in the class width and that is answer traps you could fall for interpolation - ANSWERS-- nearest minute - make sure to get correct class width for this mean - ANSWERS-the arithmetic average of a distribution, obtained by adding the scores and then dividing by the number of scores
- pros uses every data item
- cons affected by extreme values median - ANSWERS-the middle score in a distribution; half the scores are above it and half are below it pros - unaffected by extreme values
- could still calculate if couple data items missing mode - ANSWERS-the most frequently occurring score(s) in a distribution
- pros unaffected by extreme values, easy to calculate
- cons only useful if there are relatively high frequencies involved
15-16th october 1987 was storm so high windspeeds rainfall etc Heathrow is warmest Leuchars coldest Hurn is by sea so more windy Are there cities that have very different weather to others - ANSWERS-beijing vs leuchars What is UK temp range - ANSWERS-3.8-28.7 degrees celcius which cities are the wettest? - ANSWERS-Camborne - 3.4 mm Jacksonville - 5.9 mm If you are rolling a dice , when writing a probability mass function after naming the probabilities of 1-6 each with 1/6 , what must you state? - ANSWERS-must state probability otherwise is 0 driest city in Uk - ANSWERS-Leeming difference temp in 2015 vs 1987 - ANSWERS-slightly higher temps in 2015 what cloud cover measured in - ANSWERS-0-8 oktas
- each 1/9 chance windiest months 2015 - ANSWERS-May in UK and Beijing -October for Jacksonville
- September for perth
Binomial Distribution questions how to answer : If Ben rolls dice six times and wants to see how many 5s he gets (fair dice), explain the distribution you would use? - ANSWERS-- binomial distrbution would be use
- all distributions are modelled with 'X'
- X~B(n,p) where n is number of trials p is p(success) for this question it would be X~B(6,1/6) Common Binomial Distribution question: Probability of desired outcome = P find the smallest number of weeks/months/rolls/attempts after which it will be - some percentage e.g 85 percent - certain he has obtained his desired outcome at least once. - ANSWERS-how to solve ; P and percentage as decimal would be either solved/given in question 1st Method -
√(Sxx/n) if data is coded in format y= [x-a]/b what is the mean and standard deviation of the coded data - ANSWERS-the mean of y [y bar] = [[mean of x] - a]/b
- basically it is effected by both whereas with standard deviation just affected by multiplier/divider [standard deviation of y] = [standard deviation of x] /b THIS IS ONLY WHEN y= [x-a]/b IT IS JUST AN EXAMPLE How to find upper quartile - ANSWERS-if it is discrete data find 3/4n. if this is a whole number, the upper quartile is half way between this data point and the one above.If it is not a whole number round up and pick this data point. how to find LQ - ANSWERS-if it is discrete data find 1/4n. if this is a whole number, the lower quartile is half way between this data point and the one above.If it is not a whole number round up and pick this data point. definiton of an outlier - ANSWERS-- more than 2 standard deviations from the mean
- either greater than the UQ by the IQR or lower than LQ by IQR what is cleaning the data - ANSWERS-The process of removing anomalies from a data set
how to calculate the height of each bar[aka the frequency density] what formula you use - ANSWERS-area of bar = k * frequency how to form a frequency polygon from a histogram - ANSWERS-- joining the middle of the top of each bar in a histogram forms a frequency polygon when comparing sets of data what do you comment on - ANSWERS-- a measure of location
- a measure of spread What is bivariate data? - ANSWERS-Data which has pairs of values for two variables what is correlation - ANSWERS-relationship between two variables what is a regression line - ANSWERS-line of best fit , written in form y=bx + a
- if b is positive data is positively correlated
- if b is negative data is negatively correlated why do we use regression lines - ANSWERS-- to make prediction for values of the dependant variable that are within the range of the given data purpose of a venn diagram - ANSWERS-- represents events graphically
- frequencies or probabilities can be placed in the regions of the venn diagram
if events are mutually exclusive what does P[A OR B] = - ANSWERS-P[A OR B] = P[A] + P[B]
