Edexcel S1 (Statistics 1)

70% of the households in a town have a CD player and 45% have both a CD player and a personal computer (PC). 18% have neither a CD player nor a PC.

1. Illustrate this information using a Venn diagram. [3 marks]
2. Find the percentage of the households that do not have a PC. [2 marks]
3. Find the probability that a household chosen at random has a CD player or a PC but not both. [2 marks]

The random variable \(X\) has the normal distribution \(N(2, 1.7^2)\).

1. State the standard deviation of \(X\). [1 mark]
2. Find \(P(X < 0)\). [2 marks]
3. Find \(P(0.6 < X < 3.4)\). [4 marks]

The discrete random variable \(X\) has probability function $$P(X = x) = \begin{cases} cx^2 & x = -3, -2, -1, 1, 2, 3 \\ 0 & \text{otherwise.} \end{cases}$$

1. Show that \(c = \frac{1}{28}\). [3 marks]
2. Calculate
   1. \(E(X)\),
   2. \(E(X^2)\).
   [3 marks]
3. Calculate
   1. \(\text{Var}(X)\),
   2. \(\text{Var}(10 - 2X)\).
   [3 marks]

The heights, \(h\) m, of eight children were measured, giving the following values of \(h\): 1.20, 1.12, 1.43, 0.98, 1.31, 1.26, 1.02, 1.41.

1. Find the mean height of the children. [2 marks]
2. Calculate the variance of the heights. [3 marks]

The children were also weighed. It was found that their masses, \(w\) kg, were such that $$\sum w = 324, \quad \sum w^2 = 13532, \quad \sum wh = 403.$$

1. Calculate the product-moment correlation coefficient between \(w\) and \(h\). [4 marks]
2. Comment briefly on the value you have obtained. [1 mark]

The ages of the residents of a retirement community are assumed to be normally distributed. 15% of the residents are under 60 years old and 5% are over 90 years old.

1. Using this information, find the mean and the standard deviation of the ages. [7 marks]
2. If there are 200 residents, find how many are over 80 years old. [3 marks]

Of the cars that are taken to a certain garage for an M.O.T. test, 87% pass. However, 2% of these have faults for which they should have been failed. 5% of the cars which fail are in fact roadworthy and should have passed. Using a tree diagram, or otherwise, calculate the probabilities that a car chosen at random

1. should have passed the test, regardless of whether it actually did or not, [4 marks]
2. failed the test, given that it should have passed. [3 marks]

The garage is told to improve its procedures. When it is inspected again a year later, it is found that the pass rate is still 87% overall and 2% of the cars passed have faults as before, but now 0.3% of the cars which should have passed are failed and \(x\)% of the cars which are failed should have passed.

1. Find the value of \(x\). [8 marks]

The back-to-back stem and leaf diagram shows the journey times, to the nearest minute, of the commuter services into a big city provided by the trains of two operating companies. Key: \(4|3|6\) means 34 minutes for Company \(A\) and 36 minutes for Company \(B\).

1. Write down the numbers needed to complete the diagram. [1 mark]
2. Find the median and the quartiles for each company. [6 marks]
3. On graph paper, draw box plots for the two companies. Show your scale. [6 marks]
4. Use your plots to compare the two sets of data briefly. [2 marks]
5. Describe the skewness of each company's distribution of times. [2 marks]