Edexcel S1 (Statistics 1) Specimen

1. (a) Explain what you understand by a statistical model.
   (2)
   (b) Write down a random variable which could be modelled by
   1. a discrete uniform distribution,
   2. a normal distribution.
   3. A group of students believes that the time taken to travel to college, \(T\) minutes, can be assumed to be normally distributed. Within the college \(5 \%\) of students take at least 55 minutes to travel to college and \(0.1 \%\) take less than 10 minutes.
   Find the mean and standard deviation of \(T\).

3. The discrete random variable \(X\) has probability function
\(\mathrm { P } ( X = x ) = \begin{cases} k x , & x = 1,2,3,4,5 ,
0 , & \text { otherwise } . \end{cases}\)

1. Show that \(k = \frac { 1 } { 15 }\). Find the value of
2. \(\mathrm { E } ( 2 X + 3 )\),
3. \(\operatorname { Var } ( 2 X - 4 )\).
   (6)

4. A drilling machine can run at various speeds, but in general the higher the speed the sooner the drill needs to be replaced. Over several months, 15 pairs of observations relating to speed, \(s\) revolutions per minute, and life of drill, \(h\) hours, are collected. For convenience the data are coded so that \(x = s - 20\) and \(y = h - 100\) and the following summations obtained.
\(\Sigma x = 143 ; \Sigma y = 391 ; \Sigma x ^ { 2 } = 2413 ; \Sigma y ^ { 2 } = 22441 ; \Sigma x y = 484\).

1. Find the equation of the regression line of \(h\) on \(s\).
2. Interpret the slope of your regression line. Estimate the life of a drill revolving at 30 revolutions per minute.
   (2)

5. (a) Explain briefly the advantages and disadvantages of using the quartiles to summarise a set of data.
(b) Describe the main features and uses of a box plot. The distances, in kilometres, travelled to school by the teachers in two schools, \(A\) and \(B\), in the same town were recorded. The data for School \(A\) are summarised in Diagram 1. \section*{Diagram 1} For School \(B\), the least distance travelled was 3 km and the longest distance travelled was 55 km . The three quartiles were 17, 24 and 31 respectively. An outlier is an observation that falls either \(1.5 \times\) (interquartile range) above the upper quartile or \(1.5 \times\) (interquartile range) below the lower quartile.
(c) Draw a box plot for School B.
(d) Compare and contrast the two box plots.
(4)

6. For any married couple who are members of a tennis club, the probability that the husband has a degree is \(\frac { 3 } { 5 }\) and the probability that the wife has a degree is \(\frac { 1 } { 2 }\). The probability that the husband has a degree, given that the wife has a degree, is \(\frac { 11 } { 12 }\). A married couple is chosen at random.

1. Show that the probability that both of them have degrees is \(\frac { 11 } { 24 }\).
2. Draw a Venn diagram to represent these data. Find the probability that
3. only one of them has a degree,
4. neither of them has a degree. Two married couples are chosen at random.
5. Find the probability that only one of the two husbands and only one of the two wives have degrees.