Edexcel S2 (Statistics 2) 2016 June

1. During a typical day, a school website receives visits randomly at a rate of 9 per hour.

The probability that the school website receives fewer than \(v\) visits in a randomly selected one hour period is less than 0.75

1. Find the largest possible value of \(v\)
2. Find the probability that in a randomly selected one hour period, the school website receives at least 4 but at most 11 visits.
3. Find the probability that in a randomly selected 10 minute period, the school website receives more than 1 visit.
4. Using a suitable approximation, find the probability that in a randomly selected 8 hour period the school website receives more than 80 visits.
   

2. The random variable \(X \sim \mathrm {~B} ( 10 , p )\)

1. 1. Write down an expression for \(\mathrm { P } ( X = 3 )\) in terms of \(p\)
   2. Find the value of \(p\) such that \(\mathrm { P } ( X = 3 )\) is 16 times the value of \(\mathrm { P } ( X = 7 )\) The random variable \(Y \sim \operatorname { Po } ( \lambda )\)
2. Find the value of \(\lambda\) such that \(\mathrm { P } ( Y = 3 )\) is 5 times the value of \(\mathrm { P } ( Y = 5 )\) The random variable \(W \sim \mathrm {~B} ( n , 0.4 )\)
3. Find the value of \(n\) and the value of \(\alpha\) such that \(W\) can be approximated by the normal distribution, \(\mathrm { N } ( 32 , \alpha )\)
   

3. A single observation \(x\) is to be taken from \(X \sim \mathrm {~B} ( 12 , p )\) This observation is used to test \(\mathrm { H } _ { 0 } : p = 0.45\) against \(\mathrm { H } _ { 1 } : p > 0.45\)

1. Using a \(5 \%\) level of significance, find the critical region for this test.
2. State the actual significance level of this test. The value of the observation is found to be 9
3. State the conclusion that can be made based on this observation.
4. State whether or not this conclusion would change if the same test was carried out at the
   1. 10\% level of significance,
   2. \(1 \%\) level of significance.

1. The waiting times, in minutes, between flight take-offs at an airport are modelled by the continuous random variable \(X\) with probability density function

$$f ( x ) = \begin{cases} \frac { 1 } { 5 } & 2 \leqslant x \leqslant 7
0 & \text { otherwise } \end{cases}$$

1. Write down the name of this distribution. A randomly selected flight takes off at 9am
2. Find the probability that the next flight takes off before 9.05 am
3. Find the probability that at least 1 of the next 5 flights has a waiting time of more than 6 minutes.
4. Find the cumulative distribution function of \(X\), for all \(x\)
5. Sketch the cumulative distribution function of \(X\) for \(2 \leqslant x \leqslant 7\) On foggy days, an extra 2 minutes is added to each waiting time.
6. Find the mean and variance of the waiting times between flight take-offs on foggy days.

5. A bag contains a large number of coins. It contains only \(1 \mathrm { p } , 5 \mathrm { p }\) and 10 p coins. The fraction of 1 p coins in the bag is \(q\), the fraction of 5 p coins in the bag is \(r\) and the fraction of 10p coins in the bag is \(s\). Two coins are selected at random from the bag and the coin with the highest value is recorded. Let \(M\) represent the value of the highest coin. The sampling distribution of \(M\) is given below

1. List all the possible samples of two coins which may be selected.
2. Find the value of \(q\), the value of \(r\) and the value of \(s\)

6. A continuous random variable \(X\) has probability density function $$f ( x ) = \begin{cases} a x - b x ^ { 2 } & 0 \leqslant x \leqslant 2
0 & \text { otherwise } \end{cases}$$ Given that the mode is 1

1. show that \(a = 2 b\)
2. Find the value of \(a\) and the value of \(b\)
3. Calculate F(1.5)
4. State whether the upper quartile of \(X\) is greater than 1.5, equal to 1.5, or less than 1.5 Give a reason for your answer.
   

7. Last year \(4 \%\) of cars tested in a large chain of garages failed an emissions test. A random sample of \(n\) of these cars is taken. The number of cars that fail the test is represented by \(X\) Given that the standard deviation of \(X\) is 1.44

1. 1. find the value of \(n\)
   2. find \(\mathrm { E } ( X )\) A random sample of 20 of the cars tested is taken.
2. Find the probability that all of these cars passed the emissions test. Given that at least 1 of these cars failed the emissions test,
3. find the probability that exactly 3 of these cars failed the emissions test. A car mechanic claims that more than \(4 \%\) of the cars tested at the garage chain this year are failing the emissions test. A random sample of 125 of these cars is taken and 10 of these cars fail the emissions test.
4. Using a suitable approximation, test whether or not there is evidence to support the mechanic's claim. Use a \(5 \%\) level of significance and state your hypotheses clearly.