AQA A Level Mathematics Statistics Unit 1 Comprehensive Questions with Answers

What are the parameters for a binomial distribution? - ANS X~B(n,p) (n = number, p = probability) What are the parameters for a normal distribution? - ANS X~N(µ, σ²) (µ = population mean, σ² = variance) What is the binomial equation? - ANS What is the mean of a binomial distribution? - ANS µ = np What's the variance of a binomial distribution? - ANS σ² = np(1-p) What values are given in the binomial tables? - ANS P(X x) - if you want x then you need to do 1 - P(X x) Sigma notation for sample mean? - ANS -x- = ∑x ÷ n Sigma notation for sample mean of grouped data? - ANS -x- = ∑ƒx ÷ ∑ƒ Formula for frequency density? (= height on a histrogram) - ANS Class frequency ÷ class width (More complex) formula for sample standard deviation? - ANS Simpler formula for sample standard deviation? - ANS s = √(1/n-1) (∑x² - (∑x)²/n) (More complex) formula for population standard deviation? - ANS Simpler formula for population standard deviation? - ANS σ = √(1/n)(∑x² - µ²) About what proportion of values lie within 1 standard deviation of the mean? - ANS About ²/₃ About what proportion of values lie within 2 standard deviations of the mean? - ANS About 95% About what proportion of values lie within 3 standard deviations of the mean? - ANS Almost all values Formula for sample standard deviation of grouped data? - ANS 1/n-1 × (∑fx² - (∑ƒx)²/n) Formula for population standard deviation of grouped data? - ANS 1/n × (∑fx² - (∑ƒx)²/n) What happens to the mean when adding or subtracting a constant, c from every value? - ANS The mean changes by c What happens to the standard deviation when adding or subtracting a constant, c from every value? - ANS No change to the standard deviation What happens to the mean when multiplying every value by a constant, k? - ANS Multiplies the mean by k What happens to the standard deviation when multiplying every value by a constant, k? - ANS Multiplies the standard deviation by k What happens to the variance when multiplying every value by a constant, k? - ANS Multiplies every value by k² Advantages of the mode? - ANS Certain to have a value that was actually observed Advantages of the median? - ANS Can be calculated where the median or mean can't - think pigeons. Advantages of the mean? - ANS Equally influenced by every value in the data. Statistical modelling (distributions) usually requires the mean Disadvantages of the mode? - ANS May not be unique Often doesn't give useful info - e.g. mean no. eggs broken in any supermarket is probably 0 Disadvantages of the mean? - ANS Can be greatly affected by a mistaken or unusual observation - e.g. a can 'being' 4000g instead of 400g, or a boss' salary being included with the workers' Disadvantages of the median? - ANS Can be difficult to calculate when there's a lot of data Advantages of the range? - ANS Easy to calculate Advantages of IQR? - ANS Doesn't involve squares or roots Not affected by extreme values Advantages of standard deviation? - ANS Equally influenced by each value Important in statistical modelling (sometimes as the variance) Disadvantages of the range? - ANS Too dependant on extreme values Disadvantages of IQR and standard deviation? - ANS Difficuly to calculate without a spreadsheet and/or calculator when there are many values Addition rule for non- mutually exclusive events? - ANS P(A ∪ B) = P(A) + P(B) - P(A ∩ B) Addition rule for mutually exclusive events? - ANS P(A ∪ B) = P(A) + P(B) What are exhaustive events? - ANS When it is certain that at least one of them will occur Multiplication rule for independent events? - ANS P(A ∩ B) = P(A) × P(B) What are independent events? - ANS When knowledge that one event does or doesn't occur doesn't affect the probability that the other occurs Conditions for a binomial random variable? - ANS A fixed number, n, of trials Each trial is either a success or a failure The events are independent (probability of success, p, is the same for each trial) What is represented by the area under a normal distribution curve? - ANS Probability (total area = 1) What is the (theoretical) range of a normal distribution? - ANS -∞ to +∞ What is represented by µ in N(µ, σ²)? - ANS µ is the mean, median and mode What is the mean, variance, and standard deviation of the standard normal distribution? - ANS µ = 0, σ² = 1, ∴ σ = 1 Formula for Z? - ANS Z = (X - µ)/σ What proportion of the normal distribution is in the interval µ - σ, µ + σ? - ANS About 2/3 What proportion of the normal distribution is in the interval µ - 2σ, µ + 2σ? - ANS About 95% What proportion of the normal distribution is in the interval µ - 3σ, µ + 3σ? - ANS Almost all (about 99.7%) What is ⁻X⁻? - ANS An unbiased estimator of the population mean, µ What is S²? - ANS An unbiased estimator of the population variance, σ² What is the variance of the distribution of ⁻X⁻? - ANS σ²/n What are the parameters for the sampling distribution of a normal distribution? - ANS ⁻X⁻ ~ N(µ, σ²/n) What is the Central Limit Theorem? - ANS As n increases (e.g. n ≥ 30), the distribution of the sample mean ⁻X⁻ increasingly resembles a normal distribution REGARDLESS OF THE DISTRIBUTION OF X What is the formula for Z, in terms of ⁻X⁻? - ANS Z = (⁻X⁻ - µ) / (σ/√n) What is the product moment correlation coefficient (r)? - ANS A measure of the linear association between two random variables, X and Y What is the range of the PMCC (r)? - ANS -1 for perfect negative correlation to 1for perfect positive correlation r = 0 means no /linear/ correlation (could still be a non-linear relationship) What is a spurious correlation? - ANS A 'nonsense' correlation (e.g. between female life expectancy and North Sea stocks of haddock) What is the response (dependent) variable? - ANS Y - it responds to a change in X What is the explanatory (independent) variable? - ANS X What does a represent in the equation y = a + bx? - ANS The intercept with the y axis - the (theoretical) value of y when x = 0 What does b represent in the equation y = a + bx? - ANS The slope or gradient (the 'regression coefficient') What point on the graph does the least squares regression line always pass through? - ANS (⁻x⁻, ⁻y⁻) What is the name given to the distance of a point from the least squares regression line? - ANS A residual (observed y-value - value on regression line) What is prediction? - ANS Prediction is estimating a value of Y given the value of x provided that: - The regression line is a good fit - The observation is from the same population as the current data - The value of x should be in, or close to, the range of measured values What is extrapolation? - ANS Predicting outside of the measured values of x used in the calculation of the regression line - can give very misleading values What is an outlier? - ANS A data point with a relatively large residual What is an influential data point? - ANS One with a much smaller or greater x-value than the other x-values What effect does linear scaling have on the PMCC (r)? - ANS No effect