358 questions · 35 question types identified
Questions requiring calculation of two or more different probabilities (combinations of single tail and interval probabilities) in separate parts.
Questions requiring both finding percentages for given boundaries AND finding unknown boundaries from given percentages within the same problem.
Calculate probabilities involving multiple independent observations from a normal distribution (e.g., probability all exceed a value, or exactly k exceed it).
Two probability statements are directly given (e.g., P(X > a) = p and P(X > b) = q), requiring standardization and solving simultaneous equations for μ and σ.
Assess whether a normal distribution is appropriate given context (e.g., variable cannot be negative) or compare model predictions with observed data.
Questions asking for a single one-sided probability, either P(X < a) or P(X > a), requiring one standardisation and one table lookup.
Calculate expected frequency where one parameter (usually σ or μ) must first be found from given percentage information before computing the expected count.
Given a normal distribution with unknown parameter(s), find the parameter(s) using a given probability or percentile, then use a probability from this distribution as p in a binomial calculation to find probabilities of events in repeated trials.
Given a fully specified normal distribution (all parameters known), calculate a probability from the normal distribution to use as p in a binomial calculation for repeated trials, without needing to find any unknown parameters first.
Given a normal distribution with known mean and a probability statement involving a specific value, find the unknown standard deviation σ.
The constraint is a linear relationship between mean and standard deviation (μ = kσ or σ = kμ), requiring standardization and using Z-tables to find the parameter.
Questions that require comparing probabilities or outcomes between two normal distributions to determine which is more/less likely or to find relative positions (e.g., which car is more likely to reach a destination, comparing percentiles).
Probabilities are expressed as percentages or proportions of a population (e.g., '20% weigh more than', '1 in 5', '75% are greater than') requiring conversion to probability statements before solving.
Find P(A|B) where both events involve a normally distributed variable, using the definition of conditional probability.
Use symmetry of the normal distribution about its mean to deduce probabilities without calculation (e.g., if P(X > μ+a) is given, find P(X < μ-a)).
Given information about proportions or frequencies of observations falling in certain ranges (e.g., '42 out of 500 are less than 4 cm'), estimate μ and σ by setting up equations using the inverse normal distribution.
Given a normal distribution with known standard deviation and a probability statement involving a specific value, find the unknown mean μ.
Given a normal distribution with known mean and standard deviation, and a sample size n, calculate the expected number of observations satisfying a single condition (e.g., greater than a value, within a range) using P(condition) × n.
Given raw summary statistics like Σx, Σx², or Σ(x-a), Σ(x-a)² for a sample, calculate unbiased estimates of μ and σ² using standard formulas (sample mean and sample variance).
Questions that ask for separate probability calculations for two or more independent normal distributions without combining or comparing them directly (e.g., find P(X < a) and P(Y < b) separately).
Use normal distribution properties to identify outliers, find quartiles for box plots, or interpret box plot features in context of normal model.
Given a frequency table with class intervals, estimate μ and σ using mid-interval values and grouped data formulas, then model with a normal distribution.
| \(m < 25\) | \(25 \leq m < 35\) | \(35 \leq m < 45\) | \(45 \leq m < 55\) | \(55 \leq m < 65\) | \(65 \leq m < 75\) | \(m \geq 75\) | ||
| 0 | 3 | 29 | 36 | 30 | 2 | 0 |
Questions involving probability of combined events from multiple normal distributions, such as both/all satisfying conditions simultaneously, or selecting from multiple populations (e.g., probability both a child and adult meet criteria, exactly one of several meeting a condition).
The constraint is a quadratic relationship between mean and variance (μ = kσ² or similar), requiring substitution into the standardized form and solving a quadratic equation.
Use a normal distribution to approximate binomial probabilities when n is large, typically with continuity correction.
Given a normal distribution with known mean and standard deviation, find the unknown value k from a probability statement like P(X < k) = p or P(X > k) = p.
Questions that require finding unknown boundary values (e.g., 'greatest possible weight of small') given percentages or proportions, using inverse normal distribution.
Find probabilities for rounded values or integer marks when the underlying variable is continuous normal (requires considering intervals).
Work with a linear function of a normal variable (e.g., cost = price × quantity) to find the distribution and probabilities of the transformed variable.
Questions asking for the probability that X lies between two values, requiring two standardisations and subtraction of cumulative probabilities.
One probability statement involves a symmetric interval around the mean (e.g., P(μ - 2k < X < μ + 2k) = 0.6) or explicitly states P(X < μ) = 0.5, simplifying one equation.
Questions that only require finding percentages/proportions for given boundary values using direct normal distribution calculations, without needing to find unknown boundaries.
Given a probability or percentage, find the corresponding value (percentile, quartile, or quantile) in the distribution.
Given a specific value or percentile in the distribution, find the corresponding probability.
Use given quantile information (such as quartiles or interquartile range) to determine the mean or standard deviation of the distribution.