The continuous random variable TTT represents the deviation, in micrometres, of a high-precision digital caliper reading. The distribution of TTT is a continuous uniform distribution over the interval [0,2.5][0, 2.5][0,2.5].
Determine P(T<1.0)P(T < 1.0)P(T<1.0).
State the value of E(T)E(T)E(T).
Calculate Var(T)Var(T)Var(T).
A random sample of 20 readings taken by this caliper is recorded.
Find the probability that fewer than 9 of these readings show a deviation of more than 1.0 μm\mu \text{m}μm.
For a different model of caliper, the deviation X μmX \, \mu \text{m}Xμm is represented by the cumulative distribution function F(x)F(x)F(x) defined by:
F(x)={0x<00.8x−0.16x20≤x≤2.51x>2.5 F(x) = \begin{cases} 0 & x < 0 \\ 0.8x - 0.16x^2 & 0 \le x \le 2.5 \\ 1 & x > 2.5 \end{cases} F(x)=⎩⎨⎧00.8x−0.16x21x<00≤x≤2.5x>2.5Using this model, find the value of P(X>1.0)P(X > 1.0)P(X>1.0).
A large batch of 200 readings is collected from this different caliper.
Using a suitable approximation, find the probability that at least 80 of these readings show a deviation of more than 1.0 μm\mu \text{m}μm.
616 exam-style questions on Edexcel A Level Maths The Normal Distribution, covering 3.1 The Normal Distribution, 3.2 Finding Probabilities for Normal Distributions, 3.3 The Inverse Normal Distribution Function, 3.4 The Standard Normal Distribution, 3.5 Finding the mean and standard deviation, 3.6 Approximating a Binomial Distribution, and 3.7 Hypothesis Testing with the Normal Distribution. Each one has a worked solution and a mark scheme showing where the marks go.