Edexcel S2 (Statistics 2) 2022 January

1 A local pottery makes cups. The number of faulty cups made by the pottery in a week follows a Poisson distribution with a mean of 6 In a randomly chosen week, the probability that there will be at least \(x\) faulty cups made is 0.1528

1. Find the value of \(x\)
2. Use a normal approximation to find the probability that in 6 randomly chosen weeks the total number of faulty cups made is fewer than 32 A week is called a "poor week" if at least \(x\) faulty cups are made, where \(x\) is the value found in part (a).
3. Find the probability that in 50 randomly chosen weeks, more than 1 is a "poor week".

2 The continuous random variable \(X\) has cumulative distribution function given by $$\mathrm { F } ( x ) = \left\{ \begin{array} { l r } 0 & x < - k
\frac { x + k } { 4 k } & - k \leqslant x \leqslant 3 k
1 & x > 3 k \end{array} \right.$$ where \(k\) is a positive constant.

1. Specify fully, in terms of \(k\), the probability density function of \(X\)
2. Write down, in terms of \(k\), the value of \(\mathrm { E } ( X )\)
3. Show that \(\operatorname { Var } ( X ) = \frac { 4 } { 3 } k ^ { 2 }\)
4. Find, in terms of \(k\), the value of \(\mathrm { E } \left( 3 X ^ { 2 } \right)\)
   

3 A photocopier in a school is known to break down at random at a mean rate of 8 times per week.

1. Give a reason why a Poisson distribution could be used to model the number of breakdowns. The headteacher of the school replaces the photocopier with a refurbished one and wants to find out if the rate of breakdowns has increased or decreased.
2. Write down suitable null and alternative hypotheses that the headteacher should use. The refurbished photocopier was monitored for the first week after it was installed.
3. Using a \(5 \%\) level of significance, find the critical region to test whether the rate of breakdowns has now changed.
4. Find the actual significance level of a test based on the critical region from part (c). During the first week after it was installed there were 4 breakdowns.
5. Comment on this finding in the light of the critical region found in part (c).

4 The continuous random variable \(X\) has a probability density function given by $$\mathrm { f } ( x ) = \begin{cases} \frac { 1 } { 2 } k ( x - 1 ) & 1 \leqslant x \leqslant 3
k & 3 < x \leqslant 6
\frac { 1 } { 4 } k ( 10 - x ) & 6 < x \leqslant 10
0 & \text { otherwise } \end{cases}$$ where \(k\) is a positive constant.

1. Sketch \(\mathrm { f } ( x )\) for all values of \(x\)
2. Show that \(k = \frac { 1 } { 6 }\)
3. Specify fully the cumulative distribution function \(\mathrm { F } ( x )\) of \(X\) Given that \(\mathrm { E } ( X ) = \frac { 61 } { 12 }\)
4. find \(\mathrm { P } ( X > \mathrm { E } ( X ) )\)
5. Describe the skewness of the distribution, giving a reason for your answer.
   

5 Applicants for a pilot training programme with a passenger airline are screened for colour blindness. Past records show that the proportion of applicants identified as colour blind is 0.045

1. Write down a suitable model for the distribution of the number of applicants identified as colour blind from a total of \(n\) applicants.
2. State one assumption necessary for this distribution to be a suitable model of this situation.
3. Using a suitable approximation, find the probability that exactly 5 out of 120 applicants are identified as colour blind.
4. Explain why the approximation that you used in part (c) is appropriate. Jaymini claims that 75\% of all applicants for this training programme go on to become pilots. From a random sample of 96 applicants for this training programme 67 go on to become pilots.
5. Using a suitable approximation, test Jaymini's claim at the \(5 \%\) level of significance. State your hypotheses clearly.

6

1. Explain what you understand by the sampling distribution of a statistic. At Sam's cafe a standard breakfast consists of 6 breakfast items. Customers can then choose to upgrade to a medium breakfast by adding 1 extra breakfast item or they can upgrade to a large breakfast by adding 2 extra breakfast items. Standard, medium and large breakfasts are sold in the ratio \(6 : 3 : 2\) respectively. A random sample of 2 customers is taken from customers who have bought a breakfast from Sam's cafe on a particular day.
2. Find the sampling distribution for the total number, \(T\), of breakfast items bought by these 2 customers. Show your working clearly.
3. Find \(\mathrm { E } ( T )\)

7 The sides of a square are each of length \(L \mathrm {~cm}\) and its area is \(A \mathrm {~cm} ^ { 2 }\) Given that \(A\) is uniformly distributed on the interval [10,30]

1. find \(\mathrm { P } ( L \geqslant 4.5 )\)
2. find \(\operatorname { Var } ( L )\)
   
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