6. A company has 20 boats to hire out. Payment is always taken in advance and all 20 boats are hired out each day. A manager at the company notices that \(10 \%\) of groups do not turn up to take the boats, despite having already paid to hire them. The manager wishes to investigate whether the numbers of boats that do not get taken each day can be modelled by the binomial distribution \(B ( 20,0 \cdot 1 )\). The numbers of boats that were not taken for 110 randomly selected days are given below.

1. State suitable hypotheses to carry out a goodness of fit test.
2. Here is part of the table for a \(\chi ^ { 2 }\) goodness of fit test on the data.

   Number of boats not taken	0	1	2	3	4	5 or more
   Observed	10	35	29	25	8	3
   Expected	\(f\)	29.72	\(g\)	20.91	9.88	\(4 \cdot 75\)

   1. Calculate the values of \(f\) and \(g\).
   2. By completing the test, give the conclusion the manager should reach. The cost of hiring a boat is \(\pounds 15\). Since demand is high and the proportion of groups that do not turn up is also relatively high, the manager decides to take payment for 22 boats each day. She would give \(\pounds 20\) (a full refund and some compensation) to any group that has paid and turned up, but cannot take a boat out due to the overselling. Assume that the proportion of groups not turning up stays the same.
3. 1. Suggest a binomial model that the manager could use for the number of groups arriving expecting to hire a boat.
   2. Hence calculate the expected daily net income for the company following the manager's decision.
4. Is the manager justified in her decision? Give a reason for your answer.