Edexcel FS1 (Further Statistics 1) 2023 June

1. The discrete random variable \(X\) has probability distribution

where \(a\) and \(b\) are probabilities.

1. Find \(\mathrm { E } ( X )\) Given that \(\operatorname { Var } ( X ) = 3.9\)
2. find the value of \(a\) and the value of \(b\) The independent random variables \(X _ { 1 }\) and \(X _ { 2 }\) each have the same distribution as \(X\)
3. Find \(\mathrm { P } \left( X _ { 1 } + X _ { 2 } > 3 \right)\)

1. Telephone calls arrive at a call centre randomly, at an average rate of 1.7 per minute. After the call centre was closed for a week, in a random sample of 10 minutes there were 25 calls to the call centre.
   1. Carry out a suitable test to determine whether or not there is evidence that the rate of calls arriving at the call centre has changed.
      Use a \(5 \%\) level of significance and state your hypotheses clearly.
   Only 1.2\% of the calls to the call centre last longer than 8 minutes.
   One day Tiang has 70 calls.
2. Find the probability that out of these 70 calls Tiang has more than 2 calls lasting longer than 8 minutes. The call centre records show that \(95 \%\) of days have at least one call lasting longer than 30 minutes.
   On Wednesday 900 calls arrived at the call centre and none of them lasted longer than 30 minutes.
3. Use a Poisson approximation to estimate the proportion of calls arriving at the call centre that last longer than 30 minutes.

1. In a class experiment, each day for 170 days, a child is chosen at random and spins a large cardboard coin 5 times and the number of heads is recorded.
   The results are summarised in the following table.

Marcus believes that a \(\mathrm { B } ( 5,0.5 )\) distribution can be used to model these data and he calculates expected frequencies, to 2 decimal places, as follows

1. Find the value of \(r\) and the value of \(s\)
2. Carry out a suitable test, at the \(5 \%\) level of significance, to determine whether or not the \(\mathrm { B } ( 5,0.5 )\) distribution is a good model for these data.
   You should state clearly your hypotheses, the test statistic and the critical value used. Nima believes that a better model for these data would be \(\mathrm { B } ( 5 , p )\)
3. Find a suitable estimate for \(p\) To test her model, Nima uses this value of \(p\), to calculate expected frequencies as follows

   Number of heads	0	1	2	3	4	5
   Expected frequency	2.07	14.65	41.44	58.63	41.47	11.74

   The test statistic for Nima’s test is 1.62 (to 3 significant figures)
4. State,
   1. giving your reasons, the degrees of freedom
   2. the critical value
      that Nima should use for a test at the 5\% significance level.
5. With reference to Marcus' and Nima's test results, comment on
   1. the probability of the coin landing on heads,
   2. the independence of the spins of the coin. Give reasons for your answers.

1. There are 32 students in a class.

Each student rolls a fair die repeatedly, stopping when their total number of sixes is 4 Each student records the total number of times they rolled the die. Estimate the probability that the mean number of rolls for the class is less than 27.2

1. A machine fills cartons with juice.

The amount of juice in a carton is normally distributed with mean \(\mu \mathrm { ml }\) and standard deviation 8 ml . A manager wants to test whether or not the amount of juice in the cartons, \(X \mathrm { ml }\), is less than 330 ml . The manager takes a random sample of 25 cartons of juice and calculates the mean amount of juice \(\bar { x } \mathrm { ml }\).

1. Using a \(5 \%\) level of significance, find the critical region of \(\bar { X }\) for this test. State your hypotheses clearly. The Director is concerned about the machine filling the cartons with more than 330 ml of juice as well as less than 330 ml of juice. The Director takes a sample of 55 cartons, records the mean amount of juice \(\bar { y } \mathrm { ml }\) and uses a test with a critical region of $$\{ \bar { Y } < 328 \} \cup \{ \bar { Y } > 332 \}$$
2. Find P (Type I error) for the Director's test. When \(\mu = 325 \mathrm { ml }\)
3. find P (Type II error) for the test in part (a)

1. The discrete random variable \(X\) has probability generating function

$$\mathrm { G } _ { X } ( t ) = \frac { t ^ { 2 } } { ( 3 - 2 t ) ^ { 2 } }$$

1. Specify the distribution of \(X\) A fair die is rolled repeatedly.
2. Describe an outcome that could be modelled by the random variable \(X\)
3. Use calculus and \(\mathrm { G } _ { X } ( t )\) to find
   1. \(\mathrm { E } ( X )\)
   2. \(\operatorname { Var } ( X )\) The discrete random variable \(Y\) has probability generating function $$\mathrm { G } _ { Y } ( t ) = \frac { t ^ { 10 } } { \left( 3 - 2 t ^ { 3 } \right) ^ { 2 } }$$
4. Find the exact value of \(\mathrm { P } ( Y = 19 )\)

1. Each time a spinner is spun, the probability that it lands on red is 0.2
   1. Find the probability that the spinner lands on red
      1. for the 1st time on the 4th spin
      2. for the 3rd time on the 8th spin
      3. exactly 4 times during 10 spins
   Each time the spinner is spun, the probability that it lands on yellow is 0.4
   In a game with this spinner, a player must choose one of two events
   \(R\) is the event that the spinner lands on red for the \(\mathbf { 1 s t }\) time in at most 4 spins
   \(Y\) is the event that the spinner lands on yellow for the 3rd time in at most 7 spins
2. Showing your calculations clearly, determine which of these events has the greater probability.