Edexcel FS1 (Further Statistics 1) 2024 June

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

1. Find \(\operatorname { Var } ( X )\)
2. Find \(\operatorname { Var } \left( X ^ { 2 } \right)\)

1. The number of errors made by a secretary is modelled by a Poisson distribution with a mean of 2.4 per 100 words.

A 100-word piece of work completed by the secretary is selected at random.

1. Find the probability that
   1. there are exactly 3 errors,
   2. there are fewer than 2 errors. After a long holiday, a randomly selected piece of work containing 250 words completed by the secretary is examined to see if the rate of errors has changed.
2. Stating your hypotheses clearly, and using a \(5 \%\) level of significance, find the critical region for a suitable test.
3. Find P (Type I error) for the test in part (b)

1. Tisam took a survey of students' favourite colours. The results are summarised in the table below.

Tisam carries out a suitable test to see if there is any association between favourite colour and year group.

1. Write down the hypotheses for a suitable test. For her table, Tisam only needs to check one cell to show that none of the expected frequencies are less than 5
2. 1. Identify this cell, giving your reason.
   2. Calculate the expected frequency for this cell. The test statistic for Tisam's test is 38.449
3. Using a \(1 \%\) level of significance, complete the test. You should state your critical value and conclusion clearly.

1. Every morning Geethaka repeatedly rolls a fair, six-sided die until he rolls a 3 and then he stops. The random variable \(X\) represents the number of times he rolls the die each morning.
   1. Suggest a suitable model for the random variable \(X\)
   2. Show that \(\mathrm { P } ( X \leqslant 3 ) = \frac { 91 } { 216 }\)
   After 64 mornings Geethaka will calculate the mean number of times he rolled the die.
2. Estimate the probability that the mean number of rolls is between 5.6 and 7.2 Nira wants to check Geethaka's die to decide whether or not the probability of rolling a 3 with his die is less than \(\frac { 1 } { 6 }\) Nira rolls the die repeatedly until she rolls a 3
   She obtains \(x = 16\)
3. By carrying out a suitable test, determine what Nira's conclusion should be. You should state your hypotheses clearly and use a \(5 \%\) level of significance.

1. Some of the components produced by a factory are defective. The management requires that no more than \(3 \%\) of the components produced are defective.
   Niluki monitors the production process and takes a random sample of \(n\) components.
   1. Write down the hypotheses Niluki should use in a test to assess whether or not the proportion of defective components is greater than 0.03
   Niluki defines the random variable \(D _ { n }\) to represent the number of defective components in a sample of size \(n\). She considers two tests \(\mathbf { A }\) and \(\mathbf { B }\) In test \(\mathbf { A }\), Niluki uses \(n = 100\) and if \(D _ { 100 } \geqslant 5\) she rejects \(H _ { 0 }\)
2. Find the size of test \(\mathbf { A }\) In test B, Niluki uses \(n = 80\) and
   • if \(D _ { 80 } \geqslant 5\) she rejects \(\mathrm { H } _ { 0 }\)
   • if \(D _ { 80 } \leqslant 3\) she does not reject \(\mathrm { H } _ { 0 }\)
   • if \(D _ { 80 } = 4\) she takes a second random sample of size 80 and if \(D _ { 80 } \geqslant 1\) in this second sample then she rejects \(\mathrm { H } _ { 0 }\) otherwise she does not reject \(\mathrm { H } _ { 0 }\)
   • Find the size of test \(\mathbf { B }\)
   Given that the actual proportion of defective components is 0.06
3. 1. find the power of test \(\mathbf { A }\)
   2. find the expected number of components sampled using test \(\mathbf { B }\) Given also that, when the actual proportion of defective components is 0.06 , the power of test \(\mathbf { B }\) is 0.713
4. suggest, giving your reasons, which test Niluki should use.

1. The random variable \(X\) has probability generating function \(\mathrm { G } _ { X } ( t )\) where

$$\mathrm { G } _ { X } ( t ) = \frac { 1 } { \sqrt { 4 - 3 t } }$$

1. Use calculus to find \(\operatorname { Var } ( X )\) Show your working clearly.
2. Find the exact value of \(\mathrm { P } ( X \leqslant 2 )\) The independent random variables \(X _ { 1 }\) and \(X _ { 2 }\) each have the same distribution as \(X\) The random variable \(Y = X _ { 1 } + X _ { 2 } + 1\)
3. By finding the probability generating function of \(Y\), state the name of the distribution of \(Y\)
4. Hence, or otherwise, find \(\mathrm { P } \left( X _ { 1 } + X _ { 2 } > 5 \right)\)

1. The probability of winning a prize when playing a single game of Pento is \(\frac { 1 } { 5 }\)

When more than one game is played the games are independent.
Sam plays 20 games.

1. Find the probability that Sam wins 4 or more prizes. Tessa plays a series of games.
2. Find the probability that Tessa wins her 4th prize on her 20th game. Rama invites Sam and Tessa to play some new games of Pento. They must pay Rama \(\pounds 1\) for each game they play but Rama will pay them \(\pounds 2\) for the first time they win a prize, \(\pounds 4\) for the second time and \(\pounds ( 2 w )\) when they win their \(w\) th prize ( \(w > 2\) ) Sam decides to play \(n\) games of Pento with Rama.
3. Show that Sam's expected profit is \(\pounds \frac { 1 } { 25 } \left( n ^ { 2 } - 16 n \right)\) Given that Sam chose \(n = 15\)
4. find the probability that Sam does not make a loss. Tessa agrees to play Pento with Rama. She will play games until she wins \(r\) prizes and then she will stop.
5. Find, in terms of \(r\), Tessa's expected profit.