A chemical bottling plant produces two types of cleaning fluid concentrates, Alpha and Beta. The volume, XXX centilitres, of liquid in a bottle of Type Alpha follows a normal distribution such that X∼N(10,12)X \sim N(10, 1^2)X∼N(10,12). The volume, YYY centilitres, of liquid in a bottle of Type Beta follows a normal distribution such that Y∼N(15,22)Y \sim N(15, 2^2)Y∼N(15,22). The random variables XXX and YYY are independent.
Find the probability that a randomly selected bottle of Type Alpha contains more liquid than a randomly selected bottle of Type Beta.
A technician selects 5 bottles of Type Alpha and 1 bottle of Type Beta.
Find the probability that the total volume of the 5 bottles of Type Alpha is greater than 3.4 times the volume of the bottle of Type Beta.
The technician defines a linear combination S=k1X+k2YS = k_1 X + k_2 YS=k1X+k2Y, where k1k_1k1 and k2k_2k2 are constants. The technician requires SSS to have a mean of 400 and a minimum variance.
Determine the value of k1k_1k1 and the value of k2k_2k2 that satisfy these requirements.
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.