Edexcel S1 (Statistics 1)

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.
   2. Find the percentage of the households that do not have a PC.
   3. Find the probability that a household chosen at random has a CD player or a PC but not both.
   4. The random variable \(X\) has the normal distribution \(\mathrm { N } \left( 2,1 \cdot 7 ^ { 2 } \right)\).
   5. State the standard deviation of \(X\).
   6. Find \(\mathrm { P } ( X < 0 )\).
   7. Find \(\mathrm { P } ( 0 \cdot 6 < X < 3 \cdot 4 )\).
   8. The discrete random variable \(X\) has probability function
   $$\mathrm { P } ( X = x ) = \left\{ \begin{array} { c l } c x ^ { 2 } & x = - 3 , - 2 , - 1,1,2,3
   0 & \text { otherwise. } \end{array} \right.$$
2. Show that \(c = \frac { 1 } { 28 }\).
3. Calculate
   1. \(\mathrm { E } ( X )\),
   2. \(\mathrm { E } \left( X ^ { 2 } \right)\).
4. Calculate
   1. \(\operatorname { Var } ( X )\),
   2. \(\operatorname { Var } ( 10 - 2 X )\).
      (3 marks)

4. The heights, \(h \mathrm {~m}\), of eight children were measured, giving the following values of \(h\) : $$1 \cdot 20,1 \cdot 12,1 \cdot 43,0 \cdot 98,1 \cdot 31,1 \cdot 26,1 \cdot 02,1 \cdot 41$$

1. Find the mean height of the children.
2. Calculate the variance of the heights. The children were also weighed. It was found that their masses, \(w \mathrm {~kg}\), were such that $$\sum w = 324 , \quad \sum w ^ { 2 } = 13532 , \quad \sum w h = 403 .$$
3. Calculate the product-moment correlation coefficient between \(w\) and \(h\).
4. Comment briefly on the value you have obtained.

5. 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.
2. If there are 200 residents, find how many are over 80 years old. \section*{STATISTICS 1 (A) TEST PAPER 5 Page 2}

1. 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,
   2. failed the test, given that it should have passed.
   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.
2. Find the value of \(x\).

7. 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.
2. Find the median and the quartiles for each company.
3. On graph paper, draw box plots for the two companies. Show your scale.
4. Use your plots to compare the two sets of data briefly.
5. Describe the skewness of each company's distribution of times.