The random variable X has probability distribution P(X = x) = kx for x = 1, 2, 3, 4.
(a) Find k.
(b) Find E(X).
(c) Find P(X ≥ 3).
(d) Two independent values of X are taken. Find the probability that their sum is 5.
From Paper 2: Pure Mathematics and Statistics
910 marks
A machine produces bolts, of which 8% are defective. After an adjustment, a random sample of 50 bolts contains 1 defective bolt.
(a) Test, at the 5% significance level, whether the adjustment has reduced the proportion of defective bolts.
(b) Find the critical region for this test.
(c) Explain why the significance level of the test is not exactly 5%.
From Paper 2: Pure Mathematics and Statistics
1012 marks
The heights, H cm, of a species of plant are modelled by H ~ N(25, 42).
(a) Find P(H < 20).
(b) Find P(|H - 25| < 3).
(c) Six plants are chosen at random. Find the probability that exactly two of them are shorter than 20 cm.
(d) Find the height h such that 80% of plants are taller than h.
(e) A new fertiliser is used. A random sample of 20 plants has mean height 26.8 cm. Assuming the standard deviation is unchanged, test at the 5% level whether the mean height has increased.
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This paper is written by Plasmo in the style of the OCR A-Level Maths exam. It is not a past paper and is not produced or endorsed by the exam board.