Edexcel S2 (Statistics 2) 2024 June

1 A garage sells tyres. The number of customers arriving at the garage to buy tyres in a 10-minute period is modelled by a Poisson distribution with mean 2

1. Find the probability that
   1. fewer than 4 customers arrive to buy tyres in the next 10 minutes,
   2. more than 5 customers arrive to buy tyres in the next 10 minutes. The manager randomly selects 20 non-overlapping, 30-minute periods.
2. Find the probability that there are between 4 and 7 (inclusive) customers arriving to buy tyres in exactly 15 of these 30-minute periods. The manager believes that placing an advert in the local paper will lead to a significant increase in the number of customers arriving at the garage.
   A week after the advert is placed, the manager randomly selects a 25 -minute period and finds that 10 customers arrive at the garage to buy tyres.
3. Test, at the \(5 \%\) level of significance, whether or not there is evidence to support the manager's belief.
   State your hypotheses clearly.
4. Explain why the Poisson distribution is unlikely to be valid for the number of tyres sold during a 10-minute period.

2 The continuous random variable \(H\) has cumulative distribution function given by $$\mathrm { F } ( h ) = \left\{ \begin{array} { l r } 0 & h \leqslant 0
\frac { h ^ { 2 } } { 48 } & 0 < h \leqslant 4
\frac { h } { 6 } - \frac { 1 } { 3 } & 4 < h \leqslant 5
\frac { 3 } { 10 } h - \frac { h ^ { 2 } } { 75 } - \frac { 2 } { 3 } & 5 < h \leqslant d
1 & h > d \end{array} \right.$$ where \(d\) is a constant.

1. Show that \(2 d ^ { 2 } - 45 d + 250 = 0\)
2. Find \(\mathrm { P } ( H < 1.5 \mid 1 < H < 4.5 )\)
3. Find the probability density function \(\mathrm { f } ( h )\) You may leave the limits of \(h\) in terms of \(d\) where necessary.

3 Jian owns a large group of shops. She decides to visit a random sample of the shops to check if the stocktaking system is being used incorrectly.

1. Suggest a suitable sampling frame for Jian to use.
2. Identify the sampling units.
3. Give one advantage and one disadvantage of taking a sample rather than a census. Jian believes that the stocktaking system is being used incorrectly in \(40 \%\) of the shops.
   To investigate her belief, a random sample of 30 of the shops is taken.
4. Using a 5\% level of significance, find the critical region for a two-tailed test of Jian’s belief.
   You should state the probability in each tail, which should each be as close as possible to 2.5\% The total number of shops, in the sample of 30, where the stocktaking system is being used incorrectly is 20
5. Using the critical region from part (d), state what this suggests about Jian’s belief. Give a reason for your answer. Jian introduces a new, simpler, stocktaking system to all the shops.
   She takes a random sample of 150 shops and finds that in 47 of these shops the new stocktaking system is being used incorrectly.
6. Using a suitable approximation, test, at the \(5 \%\) level of significance, whether or not there is evidence that the proportion of shops where the stocktaking system is being used incorrectly is now less than 0.4 You should state your hypotheses and show your working clearly.

4 A bag contains 50 counters, each with one of the numbers 4,7 or 10 written on it in the ratio \(2 : 3 : 5\) respectively. A random sample of 2 counters is taken from the bag. The numbers on the 2 counters are recorded as \(D _ { 1 }\) and \(D _ { 2 }\) The random variable \(M\) represents the mean of \(D _ { 1 }\) and \(D _ { 2 }\)

1. Show that \(\mathrm { P } ( M = 4 ) = \frac { 9 } { 245 }\)
2. Find the sampling distribution of \(M\) A random sample of \(n\) sets of 2 counters is taken. The random variable \(T\) represents the number of these \(n\) sets of 2 counters that have a mean of 4 Given that each set of 2 counters is replaced after it is drawn,
3. calculate the minimum value of \(n\) such that \(\mathrm { P } ( T = 0 ) < 0.15\)

5 A receptionist receives incoming telephone calls and should connect them to the appropriate department. The probability of them being connected to the wrong department on the first attempt is 0.05 A random sample of 8 calls is taken.

1. Find the probability that at least 2 of these calls are connected to the wrong department on the first attempt. The receptionist receives 1000 calls each day.
2. Use a Poisson approximation to find the probability that exactly 45 callers are connected to the wrong department on the first attempt in a day. The total time, \(T\) seconds, taken for a call to be answered by a department has a continuous uniform distribution over the interval [10,50]
3. Find \(\mathrm { P } ( T > 16 )\) The number of calls the receptionist receives in a one-minute interval is modelled by a Poisson distribution with mean 6 The receptionist receives a call from Jia and tries to connect it to the right department.
4. Find the probability that in the next 40 seconds Jia's call is answered by the right department on the first attempt and the receptionist has received no other calls.

6 In this question solutions relying entirely on calculator technology are not acceptable.
The continuous random variable \(X\) has the following probability density function $$f ( x ) = \begin{cases} a + b x & - 1 \leqslant x \leqslant 3
0 & \text { otherwise } \end{cases}$$ where \(a\) and \(b\) are constants.

1. Show that \(4 a + 4 b = 1\) Given that \(\mathrm { E } \left( X ^ { 2 } \right) = \frac { 17 } { 5 }\)
2. 1. find an equation in terms of \(a\) only
   2. hence show that \(b = 0.1\)
3. Sketch the probability density function \(\mathrm { f } ( x )\) of \(X\)
4. Find the value of \(k\) for which \(\mathrm { P } ( X \geqslant k ) = 0.8\)