Edexcel FS1 AS (Further Statistics 1 AS) 2019 June

1. A leisure club offers a choice of one of three activities to its 150 members on a Tuesday evening. The manager believes that there may be an association between the choice of activity and the age of the member and collected the following data.

1. Write down suitable hypotheses for a test of the manager's belief. The manager calculated expected frequencies to use in the test.
2. Calculate the expected frequency of members aged 60 or over who choose snooker, used by the manager.
3. Explain why there are 6 degrees of freedom used in this test. The test statistic used to test the manager's belief is 19.583
4. Using a 5\% level of significance, complete the test of the manager's belief.

1. A spinner used for a game is designed to give scores with the following probabilities

The spinner is spun 80 times and the results are as follows Test, at the \(10 \%\) level of significance, whether or not the spinner is giving scores as it is designed to do. Show your working and state your hypotheses clearly.

1. Andreia's secretary makes random errors in his work at an average rate of 1.7 errors every 100 words.
   1. Find the probability that the secretary makes fewer than 2 errors in the next 100 -word piece of work.
   Andreia asks the secretary to produce a 250 -word article for a magazine.
2. Find the probability that there are exactly 5 errors in this article. Andreia offers the secretary a choice of one of two bonus schemes, based on a random sample of 40 pieces of work each consisting of 100 words. In scheme \(\mathbf { A }\) the secretary will receive the bonus if more than 10 of the 40 pieces of work contain no errors. In scheme \(\mathbf { B }\) the bonus is awarded if the total number of errors in all 40 pieces of work is fewer than 56
3. Showing your calculations clearly, explain which bonus scheme you would advise the secretary to choose. Following the bonus scheme, Andreia randomly selects a single 500 -word piece of work from the secretary to test if there is any evidence that the secretary's rate of errors has decreased.
4. Stating your hypotheses clearly and using a \(5 \%\) level of significance, find the critical region for this test.

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

where \(q\) and \(r\) are probabilities.

1. Write down, in terms of \(q , \mathrm { P } ( X \leqslant 0 )\)
2. Show that \(\mathrm { E } \left( X ^ { 2 } \right) = \frac { 7 } { 15 } + 13 q + 16 r\) Given that \(\mathrm { E } \left( X ^ { 3 } \right) = \mathrm { E } \left( X ^ { 2 } \right) + \mathrm { E } ( 6 X )\)
3. find the value of \(q\) and the value of \(r\)
4. Hence find \(\mathrm { P } \left( X ^ { 3 } > X ^ { 2 } + 6 X \right)\)