Edexcel S2 (Statistics 2) 2019 January

1. A bus company sells tickets for a journey from London to Oxford every Saturday. Past records show that \(5 \%\) of people who buy a ticket do not turn up for the journey.

The bus has seats for 48 people.
Each week the bus company sells tickets to exactly 50 people for the journey.
The random variable \(X\) represents the number of these people who do not turn up for the journey.

1. State one assumption required to model \(X\) as a binomial distribution. For this week's journey find,
2. 1. the probability that all 50 people turn up for the journey,
   2. \(\mathrm { P } ( X = 1 )\) The bus company receives \(\pounds 20\) for each ticket sold and all 50 tickets are sold. It must pay out \(\pounds 60\) to each person who buys a ticket and turns up for the journey but does not have a seat.
3. Find the bus company's expected total earnings per journey.

1. During morning hours, employees arrive randomly at an office drinks dispenser at a rate of 2 every 10 minutes.

The number of employees arriving at the drinks dispenser is assumed to follow a Poisson distribution.

1. Find the probability that fewer than 5 employees arrive at the drinks dispenser during a 10-minute period one morning. During a 30 -minute period one morning, the probability that \(n\) employees arrive at the drinks dispenser is the same as the probability that \(n + 1\) employees arrive at the drinks dispenser.
2. Find the value of \(n\) During a 45-minute period one morning, the probability that between \(c\) and 12, inclusive, employees arrive at the drinks dispenser is 0.8546
3. Find the value of \(C\)
4. Find the probability that exactly 2 employees arrive at the drinks dispenser in exactly 4 of the 6 non-overlapping 10-minute intervals between 10 am and 11am one morning.

3. Figure 1 shows an accurate graph of the cumulative distribution function, \(\mathrm { F } ( x )\), for the continuous random variable \(X\) \begin{figure}[h] \end{figure}

1. Find \(\mathrm { P } ( 3 < X < 7 )\) The probability density function of \(X\) is given by $$\mathrm { f } ( x ) = \begin{cases} a & 2 \leqslant x < 4
   b & 4 \leqslant x < 6
   c & 6 \leqslant x \leqslant 8
   0 & \text { otherwise } \end{cases}$$ where \(a\), \(b\) and \(c\) are constants.
2. Find the value of \(a\), the value of \(b\) and the value of \(c\)
3. Find \(\mathrm { E } ( X )\)

1. At a shop, past figures show that \(35 \%\) of customers pay by credit card. Following the shop’s decision to no longer charge a fee for using a credit card, a random sample of 20 customers is taken and 11 are found to have paid by credit card.

Hadi believes that the proportion of customers paying by credit card is now greater than 35\%

1. Test Hadi's belief at the \(5 \%\) level of significance. State your hypotheses clearly. For a random sample of 20 customers,
2. show that 11 lies less than 2 standard deviations above the mean number of customers paying by credit card.
   You may assume that \(35 \%\) is the true proportion of customers who pay by credit card.

1. The continuous random variable \(X\) is uniformly distributed over the interval \([ a , b ]\) where \(0 < a < b\)

Given that \(\mathrm { P } ( X < b - 2 a ) = \frac { 1 } { 3 }\)

1. 1. show that \(\mathrm { E } ( X ) = \frac { 5 a } { 2 }\)
   2. find \(\mathrm { P } ( X > b - 4 a )\) The continuous random variable \(Y\) is uniformly distributed over the interval [3, c] where \(c > 3\) Given that \(\operatorname { Var } ( Y ) = 3 c - 9\), find
2. 1. the value of \(c\)
   2. \(\mathrm { P } ( 2 Y - 7 < 20 - Y )\)
   3. \(\mathrm { E } \left( Y ^ { 2 } \right)\)

1. (i) (a) State the conditions under which the Poisson distribution may be used as an approximation to the binomial distribution.

A factory produces tyres for bicycles and \(0.25 \%\) of the tyres produced are defective. A company orders 3000 tyres from the factory.
(b) Find, using a Poisson approximation, the probability that there are more than 7 defective tyres in the company’s order.
(ii) At the company \(40 \%\) of employees are known to cycle to work. A random sample of 150 employees is taken. The random variable \(C\) represents the number of employees in the sample who cycle to work.
(a) Describe a suitable sampling frame that can be used to take this sample.
(b) Explain what you understand by the sampling distribution of \(C\) Louis uses a normal approximation to calculate the probability that at most \(\alpha\) employees in the sample cycle to work. He forgets to use a continuity correction and obtains the incorrect probability 0.0668 Find, showing all stages of your working,
(c) the value of \(\alpha\)
(d) the correct probability.

1. The continuous random variable \(X\) has probability density function

$$f ( x ) = \begin{cases} c ( x + 3 ) & - 3 \leqslant x < 0
\frac { 5 } { 36 } ( 3 - x ) & 0 \leqslant x \leqslant 3
0 & \text { otherwise } \end{cases}$$ where \(c\) is a positive constant.

1. Show that \(c = \frac { 1 } { 12 }\)
2. 1. Sketch the probability density function.
   2. Explain why the mode of \(X = 0\)
3. Find the cumulative distribution function of \(X\), for all values of \(x\)
4. Find, to 3 significant figures, the value of \(d\) such that \(\mathrm { P } ( X > d \mid X > 0 ) = \frac { 2 } { 5 }\)

   Leave blank

   Q7

   
   \hline &
   \hline \end{tabular}