Question 4 - A-Level Maths

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
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
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.
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}

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