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Maths Y12 Stats
Maths Autumn Y12
| Question | Answer |
|---|---|
| Assumption for probability question with interpolation | Assuming uniformly distributed |
| What to label outer box of Venn diagram with | S |
| Mutually exclusive | Can't happen at same time. P(AnB) = 0 P(AuB) = P(A) + P(B) [non mutually exclusive ones would have a -P(AnB) at the end] |
| How to phrase probability | P(x=5) = 0.7 |
| Independent events | When one event has no effect on another happening P(AnB) = P(A) x P(B) Inverse also true - if equation not satisfied, not independent |
| Discrete uniform distributions | When all the probabilities are the same |
| How to give probability distribution in table form | Top row is x, 1, 2, 3, etc (values of x) Bottom row is P(X = x) X = score of dice rolled e.g. |
| Probability mass function | P(X = x) = { 1/8 if x = 0, 3 Make sure to include 0 otherwise |
| A biased 4 sided die rolled. Number on bottom face random variable x. Given P(X = x) = k/x, find k | First draw table. The values of P(X = x) will be k/1, k/2, etc k/1 + k/2 + k/3 + k/4 = 1. Easy |
| 'Write down the sample space' | {0, 1, 2, 3, 4} |
| 2/3 chance of winning 5 games. What's the prob of winning exactly 3? | 5C3 [amount of ways to do it] x (2/3)^3 x (2/3)^2 [chance of doing one] |
| Requirements to use binomial distribution | Fixed number of trials 2 possible outcomes Prob of success is the same each time Trials independent |
| What to write for every question using binomial distribution for probability | X ~ B(n, p) The first part means 'if x is binomially distributed' n = number of trials p = probability of success |
| If X ~ B(n, p), then | P(X = r) = nCr x p^r x (1-p)^(n-r) Probability of getting r successful trials is nCr = number of ways of choosing r successes from n trials p^r = prob of success to the power r (1-p) = prob of failure In formula booklet |
| P(X <= 1) first line of working | P(X = 0) + P(X = 1) = P(X <= 1) DONT FORGET x = 0 This line of working isn't needed for like 2 markers, where the only working is X ~ B(n, p) and P(X <= 1) = answer |
| For questions about what value of r for a certain chance of winning | Just trial and error and show that it works: P(X > r) < 0.05 P(X > 7) = 0.038 |
| Always for probabilities | Check is decimal not percent. Write P(x = eefhqohf) = and 5 < 6 |
| Population parameter | A statistical measure relating to a population e.g. a mean The lower case p |
| Hypothesis | A statement made about a population parameter |
| Null hypothesis (H subscript 0) | The 'default' position which we usually initially assume to be true. For rolling a dice and getting a 6, then H0 : p = 1/6 |
| Alternative hypothesis (h subscript 1) | Tells us about the parameter/situation if our null hypothesis turns out to be incorrect For rolling a dice and getting a 6, then H1: p =/= 1/6 Not necessarily true just exists as disgreement |
| Test statistic | The result of the experiment we are using which we use to test H0 If we tossed a coin 4 times and got 3 heads then that value/proportion is the test statistic The capital X |
| Test the claim made at 5% significance | One-tailed hypothesis test: Test the likelihood of the test statistic against H0. If the chance of it happening by coincidence with H0 rules is less than 5%, then 'we can reject H0 and can accept H1' Then ALWAYS CONTEXTUALISE 'yes new drug is better' |
| loga(n) = x | Means a^x = n |
| Multiplication law | loga(xy) = loga(x) + loga(y) |
| Division law | loga(x/y) = loga(x) - loga(y) |
| Power law | loga(x^k) = k * loga(x) |
| How to solve equations with logs | When equation has weird powers consider it. Take logs. Bring powers down with power law. Multiply out brackets. Rearrange xs together. factorise to isolate x term. |
| When to take ln | When you need to 'cancel out' an e base number of an indices. loge(e) = 1 |
| For answers with e | Leave in terms of e unless asked for decimal |
| e^x can never | Be negative as it is a positive base number so no power will make it negative |
| Differentiating e^kx | The power doesn't reduce afterwards so answer is ke^kx Don't bring down the variable (x) |
| P = 160e^-0.006t P = density of pesticide t = days after application Interpret the meaning of 160 in this model | Sub in t = 0 When t = 0, we get 160 as the answer. t = 0 implies that no time has passed, so 160 must be the original amount of pesticide sprayed in the area |
| P = 160e^-0.006t P = density of pesticide t = days after application Find dP/dt | Remember power doesn't change |
| P = 160e^-0.006t P = density of pesticide t = days after application Interpret the significance of the sign of your answer to part c | The sign is -ve so gradient function downsloping so level of decay of the pesticide is decreasing |
| What other points you HAVE to say for a hypothesis test Q | 'X is the number of votes who support Mr Evans' when X ~ B State H0 and H1 even for two-tailed. When stating these, also say 'P is the probability that a voter selected at random supports mr Evans' When X ~ B, beforehand say 'Under H0,' |
| What is a population | A set of items that are of interest |
| What is a census | Measures every member of a population |
| What is a sample | A selection of the population used to estimate information about the population as a whole |
| What is a sampling frame | The source material or device from which a sample is drawn. It is a list of all those within a population who can be sampled |
| Census pros | Completely accurate |
| Census cons | Time consuming Processing a lot of data takes a long time Can't be used if sampling process would render items unusable |
| Sample pros | Less time consuming Less data to process Fewer responses needed |
| Sample cons | Sample might not be properly representative of the population |
| Types of random sample | Must represent population Simple random sampling Systematic sampling Stratified sampling |
| Simple random sampling | In a simple random sample, every element in the set has an equal chance of being selected. Involves assigning number to all members then generate random numbers |
| Systematic sampling | Members chosen at regular intervals from an ordered list |
| Stratified sampling | Population divided into groups. Quantities chosen randomly from each group should mean it represents whole population |
| When saying to increase sample size | Increase sample size *by testing more harnesses* |
| e.g. of sampling frame for council asking residents about opinions Identify the sampling units | A list of residents Each individual resident |
| For sample data | Consider removing anomalies |
| Pros of simple random sampling | Free of bias Easy + cheap to implement Every unit has equal selection chance |
| Cons of simple random sampling | Not suitable for large populations or sample size A sampling frame is needed |
| Pros of systematic sampling | Simple and quick to use Suitable for large samples and populations |
| Cons of systematic sampling | Sampling frame needed Possible bias as units do not have equal chance of selection |
| Pros of stratified sampling | Sample accurately reflects population Guarantees proportional representation of groups |
| Cons of stratified sampling | Classification into groups is time consuming Selection within group has same issues as simple random sampling |
| For calculating stratified | Find total. Multiply by % sample to find how many total will be chosen. Then do (amount wanted)/(total) and multiply by population of each group |
| When describing pros and cons of particular sampling method | Always refer to context of question |
| A 5 digit membership number where members ending 000 are selected is not systematic. Why? How to reduce bias? | First person is not selected at random and the required elements of the sample are not being chosen at regular intervals Take simple random sample using list of members as sampling frame |
| Non random sampling types | Quota sampling Opportunity sampling |
| Quota sampling | An interviewer/researcher selects a sample that reflects the characteristics of the group (e.g. interviewer may meet people to assess characteristics then choose sample from that) Once a quota has been filled, no more people in that group interviewed |
| Opportunity sampling | Sample taken from people available at the time and who fit the criteria needed (e.g. people leaving a supermarket) |
| Pros of quota sampling | Allows small sample to represent population No sampling frame required Quick and inexpensive Allows comparison between groups |
| Cons of quota sampling | Potential for bias Can be more time to divide population into groups after More in-depth studies need an increasing number of different groups Some people may not be willing to take part |
| Pros of opportunity sampling | Easy to carry out Inexpensive |
| Cons of opportunity sampling | Unlikely to be proportional sample Researcher's ability can affect answer People might not want to be interviewed/asked |
| Asking first 50 people you see on Monday morning outside Tesco. Suggest 2 improvements | Variety of places, variety of days/times |
| When finding class boundaries for continuous data (e.g. class 34 - 36) | Careful - they've been rounded to nearest mm (unless you were given it as inequalities, so : class boundaries 33.5, 36.5 midpoint 0.5(33.5 + 36.5) = 35 Class width 3 |
| What is discrete data | Can only be certain values e.g. integers (unless an average of discrete data). "Bar charts to graph discrete data because the separate bars emphasize the distinct nature of each value" |
| What is continuous data | Any value. "histograms and scatterplots because continuous spectrum" |
| Numerical data name and spelling | Quantitative |
| Non-numerical data name and spelling | Qualitative |
| What are class boundaries | The max and min values that belong in a group |
| Describe how random numbers are used to select a sample | Select ____ random numbers from computer generation. Ignore repeats. Match to people. |
| How to express a modal class example | 300 - 500 |
| Midpoint of discrete points equation | (n + 1)/2 (e.g. 1, 2, 3, 4, 5) |
| Midpoint of continuous data equation | n/2 (e.g. halfway between 36 and 37) |
| How to find mean of a histogram | Either do midpoint of each bar x density or write out table with class intervals next to frequency |
| When weird class boundaries are given for standard deviation | Remember you're trying to simplify down to the 5 points you will sum |
| How to do standard deviation using calculator | Menu 2 (stat) Calc SET 1 var freq: list 2 1 variable |
| What stuff do you need to say for a standard deviation question | Sigma f x^2 = Sigma fx = Sigma f = Then plug in |
| Outliers | +- 2 standard deviations from mean MORE THAN Q3 + 1.5(IQR) LESS THAN Q1 - 1.5(IQR) |
| Continuous data how to find UQ and LQ | Divide n by 4 Can then x3 |
| Discrete data how to find UQ and LQ | For even n: Divide n by 4 (or 3n by 4 for UQ). If whole, quartile is between this value and the one above. If not whole, round up and take that data point. For odd n, use (n+1) instead of n |
| How to work out skews | If median equidistant from quartiles, symmetrical. If median closer to LQ then positively skewed. If closer to UQ then negatively skewed. |
| What is the 10th percentile | 10th lowest percent |
| How to find median and quartiles from histogram | Find position (continuous so is simple). LQ = Lower end of LQ class + (Frequency into class/Frequency density of LQ class) |
| How is standard deviation affected by the coding of data and what is data coding? | Data coding is changing the form of data to make it simpler to work with. Mean is just subbed in as normal to formula to find other side of coding. Standard deviation not affected by + or - (everything just shifted - spread the same) |
| What is the modal group of a histogram | The group with the highest frequency density |
| After excluding outliers from data, what things change? | Range changes but median and quartiles stay the same as if no outliers |
| When finding standard deviation, what must you take care to do with the average of x? | Keep it super accurate because will have huge impacts on result |
| why does grouped data use Xm instead of Xi? | It doesn't really need to be, but it shows it's using class midpoints instead of actual stuff |
| If you see lower case sigma(n - 1) or lower case sigma | the n-1 one is the same as s. The lower case sigma often is used for population standard deviation (different equation) |
| what does Xi mean for standard deviation? | Just a standin for whatever's on the x axis e.g. £ |
| What to call bar graph with clusters of bars | Compound bar graph |
| What to add in Q about usefulness of certain graphical methods | So patterns over time can be identified more easily |
| When asked to identify value of change in a graph | Say increase/decrease |
| Downsides of adding much higher data entries to a graph | Would have to expand y axis of graph too. Would make small changes harder to detect |
| Pie chart purpose and why not suitable for time based population count | Can be used to show how total quantity is divided into categories Total for time based count not meaningful due to some people being double counted. |
| Half of population male. 1/4 of population over 65. Can you work out how much of population is men over 65? | No evidence they are independent. (can't assume that half of over 65s are male) |
| When describing correlations | Mention strength Must say what the correlation is between e.g. negative correlation between population density and distance from capital |
| Regression lines (aka 'least squares regression line') | Lobf type. Straight line that minimizes sum of squares of distances of each point from the line. Can only be used to estimate the dependent variable for a known independent value - not vice versa - would have to flip graph and redraw line. |
| What form does a linear regression line take? | y = a + bx Generally emphasis is on interpreting line in context, not working out a and b |
| R values for correlations | -1 =< r =< 1 r = 1 perfect positive correlation r = 0 no correlation r = -1 perfect negative correlation |
| e.g. for why polling your friends might not yield good conclusions about age at which education/training ended vs hourly pay | Not varied ages and low sample size. Also due to similar ages, if they are low ages, then people who just left education will earn less than someone in workforce for ages at first but not forever |
| 'Give an interpretation of the value of the gradient of this regression line' | IN CONTEXT (must say), as the daily mean windspeed increases by 1 knot, the daily maximum gust increases by 1.82 knots |
| Justify the use of a linear regression line in this case | Strong positive correlation so lobf will be accurate |
| What does the y intercept represent on a linear regression line equation? | Daily maximum gust is 7.23 when daily mean windspeed is 0 |
| When calculation least squares regression line | Weirdly the distance is measured straight up/down |
| What type of data set can you use regression lines on? | Bivariate data sets |
| Extrapolation vs interpolation | Making prediction outside range of the given data gives much less reliable estimate (y intercept still works - that is there to show the positioning of the line, though it can't be used to estimate at independent = 0 if range not there) |
| 5 pick 3 | Permutation. how many ways to choose where order matters. n!/(n-r)! r objects chosen from set of n objects |
| 5 choose 3 | Choosing 3 letters where order doesn't matter n!/((n-r)!r!) [2 ns then 2 rs] |
| Binomial expansion formula | (a+b)^n = a^n + (nC1)(a^(n-1))b etc |
| Combinations definition | a selection of a number of objects where order of selection isn't important |
| Permutations definition | ORDERED arrangement of a number of object |
| Find first 4 terms of binomial expansion in ascending powers of x | Includes x^0 |
| What is a correlation that looks like a quadratic called? | Quadratic correlation |
| Defining positive correlation | LINEAR. |
| When comparing data | Always reference SKEW, iqr, and median |
| 'Expected result' | The mean of X. Literally just np |
| Stuff to remember for hypothesis tests | Define p. Define hypotheses. Define X. Assuming H0, X ~ (). 0.05 > 0.04 so it is significant, sufficient evidence at the 5% significance level to reject H0 and accept H1, contextualise. |
| What to add if the hypothesis test is two-tailed | If two-tailed, say the H1 is =/= because the test is to investigate if the proportion is different rather than higher/lower |
| What is 'the p value' | The chance of the event occurring following H0 |
| What is the critical region | Range of whole numbers which are significant. First value inside region called critical value. two-tailed has 2 critical regions. Can see if test statistic falls in critical region. Expressed as inequalities |
| What is the actual significance level | Probability of data being in the critical region if H0 is true. For continuous, same as significance level. For discreet, might differ |
Created by:
Pyrogearos2