Edexcel FS1 AS (Further Statistics 1 AS) Specimen

1. A university foreign language department carried out a survey of prospective students to find out which of three languages they were most interested in studying.

A random sample of 150 prospective students gave the following results. A test is carried out at the \(1 \%\) level of significance to determine whether or not there is an association between gender and choice of language.

1. State the null hypothesis for this test.
2. Show that the expected frequency for females choosing Spanish is 30.6
3. Calculate the test statistic for this test, stating the expected frequencies you have used.
4. State whether or not the null hypothesis is rejected. Justify your answer.
5. Explain whether or not the null hypothesis would be rejected if the test was carried out at the \(10 \%\) level of significance. \section*{Q uestion 1 continued} \section*{Q uestion 1 continued} \section*{Q uestion 1 continued}

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

The random variable \(Y = 2 - 5 X\)
Given that \(\mathrm { E } ( \mathrm { Y } ) = - 4\) and \(\mathrm { P } ( \mathrm { Y } \geqslant - 3 ) = 0.45\)

1. find the probability distribution of X . Given also that \(\mathrm { E } \left( \mathrm { Y } ^ { 2 } \right) = 75\)
2. find the exact value of \(\operatorname { Var } ( \mathrm { X } )\)
3. Find \(\mathrm { P } ( \mathrm { Y } > \mathrm { X } )\) \section*{Q uestion 2 continued}

1. Two car hire companies hire cars independently of each other.

Car Hire A hires cars at a rate of 2.6 cars per hour.
Car Hire B hires cars at a rate of 1.2 cars per hour.

1. In a 1 hour period, find the probability that each company hires exactly 2 cars.
2. In a 1 hour period, find the probability that the total number of cars hired by the two companies is 3
3. In a 2 hour period, find the probability that the total number of cars hired by the two companies is less than 9 On average, 1 in 250 new cars produced at a factory has a defect.
   In a random sample of 600 new cars produced at the factory,
4. 1. find the mean of the number of cars with a defect,
   2. find the variance of the number of cars with a defect.
5. 1. Use a Poisson approximation to find the probability that no more than 4 of the cars in the sample have a defect.
   2. Give a reason to support the use of a Poisson approximation. \section*{Q uestion 3 continued}

1. The discrete random variable \(X\) follows a Poisson distribution with mean 1.4
   1. Write down the value of
      1. \(\mathrm { P } ( \mathrm { X } = 1 )\)
      2. \(\mathrm { P } ( \mathrm { X } \leqslant 4 )\)
   The manager of a bank recorded the number of mortgages approved each week over a 40 week period.

   Number of mortgages approved	0	1	2	3	4	5	6
   Frequency	10	16	7	4	2	0	1

2. Show that the mean number of mortgages approved over the 40 week period is 1.4 The bank manager believes that the Poisson distribution may be a good model for the number of mortgages approved each week. She uses a Poisson distribution with a mean of 1.4 to calculate expected frequencies as follows.

   Number of mortgages approved	0	1	2	3	4	5 or more
   Expected frequency	9.86	r	9.67	4.51	1.58	s

3. Find the value of r and the value of s giving your answers to 2 decimal places. The bank manager will test, at the \(5 \%\) level of significance, whether or not the data can be modelled by a Poisson distribution.
4. Calculate the test statistic and state the conclusion for this test. State clearly the degrees of freedom and the hypotheses used in the test. \section*{Q uestion 4 continued} \section*{Q uestion 4 continued}