OCR FS1 AS (Further Statistics 1 AS) 2021 June

A book reviewer estimates that the probability that he receives a delivery of books to review on any one weekday is \(0.1\). The first weekday in September on which he receives a delivery of books to review is the \(X\)th weekday of September.

1. State an assumption needed for \(X\) to be well modelled by a geometric distribution. [1]
2. Find \(P(X = 11)\). [2]
3. Find \(P(X \leq 8)\). [2]
4. Find \(\text{Var}(X)\). [2]
5. Give a reason why a geometric distribution might not be an appropriate model for the first weekday in a calendar year on which the reviewer receives a delivery of books to review. [1]

The probability distribution for the discrete random variable \(W\) is given in the table.

\(w\)	1	2	3	4
\(P(W = w)\)	0.25	0.36	\(x\)	\(x^2\)

1. Show that \(\text{Var}(W) = 0.8571\). [7]
2. Find \(\text{Var}(3W + 6)\). [1]

In this question you must show detailed reasoning. The random variable \(T\) has a binomial distribution. It is known that \(E(T) = 5.625\) and the standard deviation of \(T\) is \(1.875\). Find the values of the parameters of the distribution. [5]

The table shows the results of a random sample drawn from a population which is thought to have the distribution \(U(20)\).

Range	\(1 \leq x \leq 8\)	\(9 \leq x \leq 12\)	\(13 \leq x \leq 20\)
Observed frequency	12	\(y\)	\(28 - y\)

Find the range of values of \(y\) for which the data are not consistent with the distribution at the \(5\%\) significance level. [9]