Edexcel S2 (Statistics 2) 2017 January

1. The continuous random variable \(W\) has the normal distribution \(\mathrm { N } \left( 32,4 { } ^ { 2 } \right)\)
   1. Write down the value of \(\mathrm { P } ( W = 36 )\)
   The discrete random variable \(X\) has the binomial distribution \(\mathrm { B } ( 20,0.45 )\)
2. Find \(\mathrm { P } ( X = 8 )\)
3. Find the probability that \(X\) lies within one standard deviation of its mean.

2. The continuous random variable \(X\) is uniformly distributed over the interval \([ \alpha , \beta ]\) where \(\beta > \alpha\) Given that \(\mathrm { E } ( X ) = 8\)

1. write down an equation involving \(\alpha\) and \(\beta\) Given also that \(\mathrm { P } ( X \leqslant 13 ) = 0.7\)
2. find the value of \(\alpha\) and the value of \(\beta\)
3. find \(\operatorname { Var } ( X )\)
4. find \(\mathrm { P } ( 5 \leqslant X \leqslant 35 )\)

3. (a) State the condition under which the normal distribution may be used as an approximation to the Poisson distribution. The number of reported first aid incidents per week at an airport terminal has a Poisson distribution with mean 3.5
(b) Find the modal number of reported first aid incidents in a randomly selected week. Justify your answer. The random variable \(X\) represents the number of reported first aid incidents at this airport terminal in the next 2 weeks.
(c) Find \(\mathrm { P } ( X > 5 )\)
(d) Given that there were exactly 6 reported first aid incidents in a 2 week period, find the probability that exactly 4 were reported in the first week.
(e) Using a suitable approximation, find the probability that in the next 40 weeks there will be at least 120 reported first aid incidents.

1. The time, in thousands of hours, that a certain electrical component will last is modelled by the random variable \(X\), with probability density function

$$f ( x ) = \begin{cases} \frac { 3 } { 64 } x ^ { 2 } ( 4 - x ) & 0 \leqslant x \leqslant 4
0 & \text { otherwise } \end{cases}$$ Using this model, find, by algebraic integration,

1. the mean number of hours that a component will last,
2. the standard deviation of \(X\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{ce1f9aa7-cf16-4293-98b1-157eed35b761-06_478_974_1069_479} \captionsetup{labelformat=empty} \caption{Figure 1}
   \end{figure} Figure 1 shows a sketch of the probability density function of the random variable \(X\).
3. Give a reason why the random variable \(X\) might be unsuitable as a model for the time, in thousands of hours, that these electrical components will last.
4. Sketch a probability density function of a more realistic model.

1. In the manufacture of cloth in a factory, defects occur randomly in the production process at a rate of 2 per \(5 \mathrm {~m} ^ { 2 }\)

The quality control manager randomly selects 12 pieces of cloth each of area \(15 \mathrm {~m} ^ { 2 }\).

1. Find the probability that exactly half of these 12 pieces of cloth will contain at most 7 defects. The factory introduces a new procedure to manufacture the cloth. After the introduction of this new procedure, the manager takes a random sample of \(25 \mathrm {~m} ^ { 2 }\) of cloth from the next batch produced to test if there has been any change in the rate of defects.
2. 1. Write down suitable hypotheses for this test.
   2. Describe a suitable test statistic that the manager should use.
   3. Explain what is meant by the critical region for this test.
3. Using a 5\% level of significance, find the critical region for this test. You should choose the largest critical region for which the probability in each tail is less than 2.5\%
4. Find the actual significance level for this test.

1. A seed producer claims that \(96 \%\) of its bean seeds germinate.

To test the producer's claim, a random sample of 75 bean seeds was planted and 66 of these seeds germinated. Use a suitable approximation to test, at the \(1 \%\) level of significance, whether or not the producer is overstating the probability of its bean seeds germinating. State your hypotheses clearly.

7. The continuous random variable \(X\) has probability density function \(\mathrm { f } ( x )\) given by $$f ( x ) = \begin{cases} \frac { 1 } { 20 } x ^ { 3 } & 0 \leqslant x \leqslant 2
\frac { 1 } { 10 } ( 6 - x ) & 2 < x \leqslant 6
0 & \text { otherwise } \end{cases}$$

1. Sketch the graph of \(\mathrm { f } ( x )\) for all values of \(x\).
2. Write down the mode of \(X\).
3. Show that \(\mathrm { P } ( X > 2 ) = 0.8\)
4. Define fully the cumulative distribution function \(\mathrm { F } ( x )\). Given that \(\mathrm { P } ( X < a \mid X > 2 ) = \frac { 5 } { 8 }\)
5. find the value of \(\mathrm { F } ( a )\).
6. Hence, or otherwise, find the value of \(a\). Give your answer to 3 significant figures.