Netherlands HAVO 5 (eindexamenjaar) Mathematics A
Chapters: 2
1. Relationships (part 2)
Linear relationships · Exponential relationships
- Linear Functions – A linear function adds the same amount to y every time x goes up by 1. Its rule is y = mx + c (also written f(x) = mx + b). m is the slope: rise ÷ run, the change in y for each 1 step in x. c is the y-intercept: the value of y when x = 0. Its graph is always a straight line.
- Exponential Functions – An exponential function has the form y = a·bˣ, where a ≠ 0 is the starting value and b > 0, b ≠ 1 is the growth factor. If b > 1 it grows; if 0 < b < 1 it decays. Each step of 1 in x multiplies y by b (a constant ratio), unlike a linear function which adds a constant. The graph of y = bˣ passes through (0, 1), has domain all real numbers, range y > 0, and the x-axis as a horizontal asymptote. The general form y = a·b^(k(x − d)) + c stretches, reflects and shifts it.
2. Statistics (part 2)
Data and distributions · Making statistical statements
- Normal Distribution – A normal distribution is a continuous, symmetric, bell-shaped distribution fixed by its mean μ (centre) and standard deviation σ (spread): X ~ N(μ, σ²). Mean = median = mode. About 68% of values lie within 1σ of μ, 95% within 2σ and 99.7% within 3σ. Any normal value is turned into a standard score z = (x − μ)/σ, which follows N(0, 1); probabilities are areas under the curve, read from a table or calculator as Φ(z).
- Statistical Inference – Statistical inference means using a sample to say something about a whole population. A number that describes the population (like the true proportion p or the mean μ) is a parameter. A number worked out from a sample (like p̂ or x̄) is a statistic, and we use it as a point estimate. Different random samples give different answers: this is sampling variability. If we took many samples, their statistics would form the sampling distribution, centred on the true value, with spread called the standard error: SE = √(p(1−p)/n) for a proportion and σ/√n for a mean. Bigger samples give smaller spread. A 95% confidence interval is estimate ± 1.96 × SE; about 95 of every 100 such intervals catch the true value. Simulation helps us check whether a claimed model fits the data. Good inference needs random sampling, and an association in data does not prove cause.