SPS SPS FM Statistics (SPS FM Statistics) 2021 June

Employees at a company were asked how long their average commute to work was. The table below gives information about their answers.

Time taken (\(t\) minutes)	Number of people
\(0 < t \leq 10\)	\(x\)
\(10 < t \leq 20\)	30
\(20 < t \leq 30\)	35
\(30 < t \leq 50\)	28
\(50 < t \leq 90\)	12

The company estimates that the mean time for employees commuting to work is 28 minutes. Work out the value of \(x\), showing your working clearly. [4]

Events \(A\) and \(B\) are such that \(P(A \cup B) = 0.95\), \(P(A \cap B) = 0.6\) and \(P(A|B) = 0.75\).

1. Find \(P(B)\). [3]
2. Find \(P(A)\). [3]
3. Show that the events \(A'\) and \(B\) are independent. [2]

The letters of the word CHAFFINCH are written on cards.

1. In how many ways can the letters be rearranged with no restrictions. [1]
2. In how many difference ways can the letters be rearranged if the vowels are to have at least one consonant between them. [3]

The weights of sacks of potatoes are normally distributed. It is known that one in five sacks weigh more than 6kg and three in five sacks weigh more than 5.5kg.

1. Find the mean and standard deviation of the weights of potato sacks. [5]
2. The sacks are put into crates, with twelve sacks going into each crate. What is the probability that a given crate contains two or more sacks that weigh more than 6kg? You must explain your reasoning clearly in this question. [4]

Eleven students in a class sit a Mathematics exam and their average score is 67% with a standard deviation of 12%. One student from the class is absent and sits the paper later, achieving a score of 85%.

1. Find the mean score for the whole class and the standard deviation for the whole class. [5]
2. Comment, with justification, on whether the score for the paper sat later should be considered as an outlier. [2]

Only two airlines fly daily into an airport. AMP Air has 70 flights per day and Volt Air has 65 flights per day. Passengers flying with AMP Air have an 18% probability of losing their luggage and passengers flying with Volt Air have a 23% probability of losing their luggage. You overhear a passenger in the airport complaining about her luggage being lost. Find the exact probability that she travelled with Volt Air, giving your answer as a rational number. [6]

A continuous random variable \(X\) has probability density function \(f\) given by $$f(x) = \begin{cases} \frac{x^2}{a} + b, & 0 \leq x \leq 4 \\ 0 & \text{otherwise} \end{cases}$$ where \(a\) and \(b\) are positive constants. It is given that \(P(X \geq 2) = 0.75\).

1. Show that \(a = 32\) and \(b = \frac{1}{12}\). [5]
2. Find \(E(X)\). [3]
3. Find \(P(X > E(X)|X > 2)\) [4]