Mathematics 9709 · AS & A Level · The normal distribution
The normal distribution — practice question
For one type of leaf, lengths measured in cm follow the distribution $N(5.2,\,1.5^2)$. A second leaf type is also represented by a normal distribution. A scientist records the lengths of a random sample of 500 leaves of this type and discovers that 46 are shorter than $3\text{ cm}$ and 95 are longer than $8\text{ cm}$.
(a)[2]
Find the probability that a leaf chosen at random from this type has a length below $6\text{ cm}$.
(b)[5]
Find estimates for the mean and standard deviation of the lengths of leaves belonging to this type.
(c)[4]
In a random sample of 2000 leaves of this second type, how many would the scientist expect to find with lengths more than $1$ standard deviation away from the mean?
Worked solution & mark scheme
This 11-mark question has a full step-by-step worked solution and mark scheme. One marking point: “Appropriate standardisation, for example $P\!\left(Z<\dfrac{6-5.2}{1.5}\right)$” …