Edexcel S2 (Statistics 2) 2023 October

1. Sam is a telephone sales representative.

For each call to a customer

• Sam either makes a sale or does not make a sale
• sales are made independently

Past records show that, for each call to a customer, the probability that Sam makes a sale is 0.2

1. Find the probability that Sam makes
   1. exactly 2 sales in 14 calls,
   2. more than 3 sales in 25 calls. Sam makes \(n\) calls each day.
2. Find the minimum value of \(n\)
   1. so that the expected number of sales each day is at least 6
   2. so that the probability of at least 1 sale in a randomly selected day exceeds 0.95

1. The continuous random variable \(X\) has probability density function \(\mathrm { f } ( x )\) given by

$$\mathrm { f } ( x ) = \begin{cases} a x ^ { 3 } & 0 \leqslant x \leqslant 4
b x + c & 4 < x \leqslant d
0 & \text { otherwise } \end{cases}$$ where \(a\), \(b\), \(c\) and \(d\) are constants such that

• \(b x + c = a x ^ { 3 }\) at \(x = 4\)
• \(b x + c\) is a straight line segment with end coordinates ( \(4,64 a\) ) and ( \(d , 0\) )
  1. State the mode of \(X\)

Given that the mode of \(X\) is equal to the median of \(X\)

use algebraic integration to show that \(a = \frac { 1 } { 128 }\)

Find the value of \(d\)

Hence find the value of \(b\) and the value of \(c\)

1. Every morning Navtej travels from home to work. Navtej leaves home at a random time between 08:00 and 08:15

• It always takes Navtej 3 minutes to walk to the bus stop
• Buses run every 15 minutes and Navtej catches the first bus that arrives
• Once Navtej has caught the bus it always takes a further 29 minutes for Navtej to reach work

The total time, \(T\) minutes, for Navtej's journey from home to work is modelled by a continuous uniform distribution over the interval \([ \alpha , \beta ]\)

1. 1. Show that \(\alpha = 32\)
   2. Show that \(\beta = 47\)
2. State fully the probability density function for this distribution.
3. Find the value of
   1. \(\mathrm { E } ( T )\)
   2. \(\operatorname { Var } ( T )\)
4. Find the probability that the time for Navtej's journey is within 5 minutes of 35 minutes.

1. A manufacturer makes t -shirts in 3 sizes, small, medium and large.

20\% of the t -shirts made by the manufacturer are small and sell for \(\pounds 10 30 \%\) of the t -shirts made by the manufacturer are medium and sell for \(\pounds 12\) The rest of the t -shirts made by the manufacturer are large and sell for \(\pounds 15\)

1. Find the mean value of the t -shirts made by the manufacturer. A random sample of 3 t -shirts made by the manufacturer is taken.
2. List all the possible combinations of the individual selling prices of these 3 t-shirts.
3. Find the sampling distribution of the median selling price of these 3 t-shirts.

1. A supermarket receives complaints at a mean rate of 6 per week.
   1. State one assumption necessary, in order for a Poisson distribution to be used to model the number of complaints received by the supermarket.
   2. Find the probability that, in a given week, there are
      1. fewer than 3 complaints received by the supermarket,
      2. at least 6 complaints received by the supermarket.
   In a randomly selected week, the supermarket received 12 complaints.
2. Test, at the \(5 \%\) level of significance, whether or not there is evidence that the mean number of complaints is greater than 6 per week.
   State your hypotheses clearly. Following changes made by the supermarket, it received 26 complaints over a 6-week period.
3. Use a suitable approximation to test whether or not there is evidence that, following the changes, the mean number of complaints received is less than 6 per week. You should state your hypotheses clearly and use a 5\% significance level.

1. The continuous random variable \(Y\) has cumulative distribution function given by

$$\mathrm { F } ( y ) = \left\{ \begin{array} { l r } 0 & y < 0
\frac { 1 } { 21 } y ^ { 2 } & 0 \leqslant y \leqslant k
\frac { 2 } { 15 } \left( 6 y - \frac { y ^ { 2 } } { 2 } \right) - \frac { 7 } { 5 } & k < y \leqslant 6
1 & y > 6 \end{array} \right.$$

1. Find \(\mathrm { P } \left( \left. Y < \frac { 1 } { 4 } k \right\rvert \, Y < k \right)\)
2. Find the value of \(k\)
3. Use algebraic calculus to find \(\mathrm { E } ( Y )\)

20\% of the t -shirts made by the manufacturer are small and sell for \(\pounds 10 30 \%\) of the t -shirts made by the manufacturer are medium and sell for \(\pounds 12\) The rest of the t -shirts made by the manufacturer are large and sell for \(\pounds 15\)

1. Find the mean value of the t -shirts made by the manufacturer. A random sample of 3 t -shirts made by the manufacturer is taken.
2. List all the possible combinations of the individual selling prices of these 3 t-shirts.
3. Find the sampling distribution of the median selling price of these 3 t-shirts.
   1. A supermarket receives complaints at a mean rate of 6 per week.
   2. State one assumption necessary, in order for a Poisson distribution to be used to model the number of complaints received by the supermarket.
   3. Find the probability that, in a given week, there are
      1. fewer than 3 complaints received by the supermarket,
      2. at least 6 complaints received by the supermarket.
   In a randomly selected week, the supermarket received 12 complaints.
4. Test, at the \(5 \%\) level of significance, whether or not there is evidence that the mean number of complaints is greater than 6 per week.
   State your hypotheses clearly. Following changes made by the supermarket, it received 26 complaints over a 6-week period.
5. Use a suitable approximation to test whether or not there is evidence that, following the changes, the mean number of complaints received is less than 6 per week. You should state your hypotheses clearly and use a 5\% significance level.
   1. The continuous random variable \(Y\) has cumulative distribution function given by
   $$\mathrm { F } ( y ) = \left\{ \begin{array} { l r } 0 & y < 0
   \frac { 1 } { 21 } y ^ { 2 } & 0 \leqslant y \leqslant k
   \frac { 2 } { 15 } \left( 6 y - \frac { y ^ { 2 } } { 2 } \right) - \frac { 7 } { 5 } & k < y \leqslant 6
   1 & y > 6 \end{array} \right.$$
6. Find \(\mathrm { P } \left( \left. Y < \frac { 1 } { 4 } k \right\rvert \, Y < k \right)\)
7. Find the value of \(k\)
8. Use algebraic calculus to find \(\mathrm { E } ( Y )\)
   1. The discrete random variable \(X\) is given by
   $$X \sim \mathrm {~B} ( n , p )$$ The value of \(n\) and the value of \(p\) are such that \(X\) can be approximated by a normal random variable \(Y\) where $$Y \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)$$ Given that when using a normal approximation $$\mathrm { P } ( X < 86 ) = 0.2266 \text { and } \mathrm { P } ( X > 97 ) = 0.1056$$
9. show that \(\sigma = 6\)
10. Hence find the value of \(n\) and the value of \(p\)