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Maths Y13 Stats
Maths Spring Y13
| Question | Answer |
|---|---|
| Useful equation for union | P(AuB) = P(A) + P(B) - P(AnB) If mutually exclusive, = P(A) + P(B) because P(AnB) = 0 |
| Equation for given | Probability of A given B = P(A l B) = P(AnB)/(P(B) Limiting set to just P(B) then seeing what proportion of that is also P(A) If independent, P(A l B) = P(A) [just work it out] Super careful to do right way around if (B l A). Given under |
| Exhaustive | P(quehoawub) = 1 |
| What to always do for probability question | Draw venn diagram - really nice for working out the different sector values, and is especially vital for non independent |
| Sample spaces | Should draw for 'X of 2 numbers' questions For given: number of applicable values in the restricted space/restricted sample space |
| Does AnB' = (AnB)' | No |
| (AuBuC)' = | A'nB'nC' (also flips sign when expanding) |
| A few pointers for hypothesis testing | Don't assume significance is 5% - check Do both tails in calculator to choose most significant (most thorough is to check significance of > and < in the bcd menu when doing calc). Don't have to show why you chose - just write the smallest tail. |
| Rules if A and B are independent | P(AnB) = P(A) x P(B) P(A l B) = P(A) |
| Rules if A and B are mutually exclusive | P(AnB) = 0 P(AuB) = P(A) + P(B) |
| If you see 'given that' | IMMEDIATELY write out law for conditional probability (like even before finishing reading Q) Vital for preventing mistakes and working marks |
| If you see the words 'are independent' | IMMEDIATELY write out laws for independence (like even before finishing reading Q) Vital for preventing mistakes and working marks |
| nCr = | (n over r) = n!/r!(n-r)! |
| Hypothesis testing don't forget | X squiggle B |
| If random variable X is normally distributed, we write | X squiggle N(μ, σ^2) μ = mean σ = standard deviation |
| Normal distribution averages | Symmetrical so mean = mode = median |
| +- 1 standard deviation | Contains 68% of the data (in formula booklet i think) |
| +- 2 standard deviation | Contains 95% of the data |
| +- 3 standard deviation | Contains 99.7% of the data |
| +- 5 standard deviation | Contains all of the data (pretty much) |
| Ways to prove not normally distributed | Skews and discrete (positive skew means bump to the left) Discrete not normal but can approximate |
| What is the y axis on the normal distribution | Probability density (like freq density but the probability of it lying within the area. Total area under curve is therefore 1) |
| Straight line of the probability density | *Uniform* distribution |
| When using distribution menu on calc | ALWAYS view value with the OPTN F1 - it truncates otherwise |
| Using distribution menu for inverse normal | Can edit the p value to get the x value(s) |
| e.g. Find c such that P(26<y<c) = 0.2 | Draw graph. P(Y<c) - P(Y<26) = 0.36 Use calc for P(Y<26) |
| e.g. Find d such that P(25-d < y < 25+d) = 0.6 | Draw graph. The outer bounds are 0.2 each. then P(Y<25+d) = 0.8 easy on calc |
| Probability | Not % |
| Standard normal distribution | If X squiggle N(μ, σ^2) and Z squiggle N(0,1) Then P(X < x) is the same as P(Z < (x - μ)/σ). The transformation of x into (x - μ)/σ is called standardising |
| P(Z < (x - μ)/σ) [all of it] sometimes written as | ϕ((x - μ)/σ) Phi is a functionish thing meaning P(Z <. Allows for more coherent working when inversing. ONLY FOR STANDARD |
| e.g. standardise X squiggle N(10, 2^2) P(X < 6) | X squiggle N(0,1) 6-10/2 = -2 P(Z<-2) = P(X<6) |
| When to do standardising | Good for when mean and/or standard deviation are unknown Also good for comparisons - is just showing amount of standard deviations above/below the mean of zero. P(1 < Z < 2) means is between 1 and 2 standard deviations above |
| e.g. Length may be modelled by normal distribution with standard deviation = 3. 15% of components longer than 16cm. Find the mean. | X squiggle N(μ, 3^2) P(X > 16) = 0.15 P(X < 16) = 0.85 (because inverse does up to a point) Z squiggle N(0,1) P(Z < (16-μ)/3) = 0.85 16-μ/3 = ϕ^-1(0.85) = 1.036 [making sure to use (0,1)] etc |
| ϕ^-1(0.8) | Means area UP TO the result of this is 0.8. i.e. Only does P(X< ) Same style thing for normal inverse |
| What step to do after finding the mean with the standard normal | Do a plausibility check: for this question its about 1 standard deviation below 16cm, which makes sense because the ϕ^-1(0.85) is 1.036, which is close to one standard deviation above mean. |
| Normal distribution points of inflection | Points 1 standard deviation above and below mean |
| Normal approximations | Discrete values. Inclusive vs exclusive matters |
| Requirements for Normal approximations | Steps in possible values are small compared to standard deviation. Continuity corrections are made |
| Continuity corrects examples | (106 <= X <= 110) integers becomes (105.5 <= X <= 110.5) continuous 'less than 120' is P(X<119.5) [so it doesnt include the 120 itself] 'at least 110' is P(X >= 109.5) to catch the ones that round up to 110 so is 1 - P(X<109.5) |
| Binomial to normal approx | Bi(n,p) -> N(np, σ^2), where σ = sqrt(np(1-p)) |
| Distribution of sample means | For a random sample of size n taken from a random variable X, the sample mean X (with bar above) is approximately normally distributed with Xbar squiggle N(μ,(σ^2)/n) |
| Sample means | e.g. average of 3 numbers between 1 and 2 is a sample mean. Then tons of these sample means are made. We expect the sample means to vary about the population mean μ (mean doesnt change from the mean of the random variable) |
| Standard deviation of Xbar | sqrt(σ^2/n) = σ/sqrt(n) |
| Increasing sample size effect on sample means | Increasing sample size means increasing amount of random values drawn to make each sample. Sample means xbar (lower case for each one I think is the convention) will be closer to true mean μ, so standard deviation of Xbar reduces |
| Ways to prove normally distributed | Symmetry Almost all data within 3 standard deviations of mean (give values of upper and lower bounds) 95% of data within 2 standard deviations from mean (give upper and lower) Continuous |
| X and Xbar | Dont confuse them. X is each randomly chosen item. Xbar is the mean length of each sample. X might not be normally distributed but Xbar will be. |
| Normal hypothesis testing e.g. Rods believed to be 30cm, with standard deviation of 1. A sample of 20 rods gives mean of 29.5 Set-up of answer | Only talking about one sample. Hypotheses always in term of population parameter e.g. mean H0: μ = 30 H1: μ < 30 Let X be the length of the rod |
| Normal hypothesis testing e.g. Rods believed to be 30cm, with standard deviation of 1. A sample of 20 rods gives mean of 29.5 Answer | Assuming H0, X squiggle N(30,1^2) so Xbar squiggle N(30, 1^2/20) -> Xbar squiggle N(30, (sqrt0.05)^2) Use the μ and σ values for P(Xbar < 29.5) = 0.127 (table given in Q. The p value is the prob of the observed value or more extreme) Conclude |
| Why we did (sqrt0.05)^2 | To remind us of std dev not variance - ALWAYS do |
| Using critical region for normal hypothesis testing | Use inverse normal of the significance level to find the critical region then compare to the value. Critical regions good for if testing multiple samples. If looking for upper region of 10%, do inverse of 0.95 |
| How to state critical values | Critical region is Xbar < 232.7 or Xbar > 247.3 (have to be separated because no value does both) |
| Using critical region | Must prove with BOTH regions |
| 'Sample standard deviation' | Watch out for sample parameters rather than population |
| When can we use sample standard deviation as an estimate for the whole population | If the population standard deviation is not given and n is large enough (30+) Mention in your answer We still divide by sqrt(n) to find the standard deviation of the sample means |
| Function of binomial vs normal hypothesis test | Binomial is testing for a probability/proportion. Normal is testing for a population mean using a sample |
| r value | +ve r value means +ve gradient of lobf. aka 'product moment correlation coefficient' PMCC |
| r value hypothesis tests | H0 is always r = 0. They give you the critical values to do it with - essentially just choose the right tail |
| PMCC vs SRCC | PMCC is the r value, but the SRCC (R means Rank) is replacing the raw data with its rank in the data set |
| How to use SRCC | Replace raw data. Calculate PMCC using these ranks. Interpret the coefficient as rho = 0.86 for the original data and draw conclusions about association rho doesnt tell us about correlation - tells us if the data is strongly *associated* |
| How to adjust SRCC for repeat data points | Need to be adjusted so data isnt pushed out of sync |
| r = 1 | Perfectly straight line through origin (not necessarily y = x) |
| rho = 1 | Must be a *strictly* increasing function |
| r vs rho | r value for PMCC. r for the rank correlation (the correlation between the rank numbers, but doesnt tell us about the data set) rho for the SRCC of the original data |
| If you have the equation of the lobf | Dont estimate by drawing on graph, sub in instead |
| For something like an exponential graph, should r be used? | Probably not - would be too low. Ranks likely more reasonable. |
| If hypothesis says 'positive correlation' | Do SRCC and use rho for hypothesis ??? I dont know what this note means - check with someone smart if its correct |
Created by:
Pyrogearos2