Edexcel FS1 AS (Further Statistics 1 AS) 2022 June

1. Stuart is investigating a treatment for a disease that affects fruit trees. He has 400 fruit trees and applies the treatment to a random sample of these trees. The remainder of the trees have no treatment. He records the number of years, \(y\), that each fruit tree remains free from this disease.

The results are summarised in the table below. The data are to be used to determine whether or not there is an association between the application of the treatment and the number of years that a fruit tree remains free from this disease.

1. Calculate the expected frequencies for
   1. Applied and \(y < 1\)
   2. Not applied and \(1 \leqslant y < 2\) The value of \(\sum \frac { ( O - E ) ^ { 2 } } { E }\) for the other four classes is 2.642 to 3 decimal places.
2. Test, at the \(5 \%\) level of significance, whether or not there is an association between the application of the treatment and the number of years a fruit tree remains free from this disease. You should state your hypotheses, test statistic, critical value and conclusion clearly.

1. Xena catches fish at random, at a constant rate of 0.6 per hour.
   1. Find the probability that Xena catches exactly 4 fish in a 5 -hour period.
   The probability of Xena catching no fish in a period of \(t\) hours is less than 0.16
2. Find the minimum value of \(t\), giving your answer to one decimal place. Independently of Xena, Zion catches fish at random with a mean rate of 0.8 per hour.
   Xena and Zion try using new bait to catch fish. The number of fish caught in total by Xena and Zion after using the new bait, in a randomly selected 4-hour period, is 12
3. Use a suitable test to determine, at the \(5 \%\) level of significance, whether or not there is evidence that the rate at which fish are caught has increased after using the new bait. State your hypotheses clearly and the \(p\)-value used in your test.

1. In a game, a coin is spun 5 times and the number of heads obtained is recorded. Tao suggests playing the game 20 times and carrying out a chi-squared test to investigate whether the coin might be biased.
   1. Explain why playing the game only 20 times may cause problems when carrying out the test.
   Chris decides to play the game 500 times. The results are as follows

   Number of heads	0	1	2	3	4	5
   Observed frequency	2	27	93	181	146	51

   Chris decides to test whether or not the data can be modelled by a binomial distribution, with the probability of a head on each spin being 0.6 She calculates the expected frequencies, to 2 decimal places, as follows

   Number of heads	0	1	2	3	4	5
   Expected frequency	5.12	38.40	115.20	172.80	129.60	38.88

2. State the number of degrees of freedom in Chris' test, giving a reason for your answer.
3. Carry out the test at the \(5 \%\) level of significance. You should state your hypotheses, test statistic, critical value and conclusion clearly.
4. Showing your working, find an alternative model which would better fit Chris’ data.

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

1. Find in terms of \(q\)
   1. \(\mathrm { E } ( X )\)
   2. \(\mathrm { E } \left( X ^ { 2 } \right)\) Given that \(\operatorname { Var } ( X ) = 3.66\)
2. show that \(q = 0.3\) In a game, the score is given by the discrete random variable \(X\)
   Given that games are independent,
3. calculate the probability that after the 4th game has been played, the total score is exactly 20 A round consists of 4 games plus 2 bonus games. The bonus games are only played if after the 4th game has been played the total score is exactly 20 A prize of \(\pounds 10\) is awarded if 6 games are played in a round and the total score for the round is at least 27 Bobby plays 3 rounds.
4. Find the probability that Bobby wins at least \(\pounds 10\)