3 Stopoff owns a chain of hotels. Guests are presented with the bills for their stays when they check out.
- Assume that the number of bills that contain errors may be modelled by a binomial distribution with parameters \(n\) and \(p\), where \(p = 0.30\).
Determine the probability that, in a random sample of 40 bills:
- at most 10 bills contain errors;
- at least 15 bills contain errors;
- exactly 12 bills contain errors.
- Calculate the mean and the variance for each of the distributions \(\mathrm { B } ( 16,0.20 )\) and \(B ( 16,0.125 )\).
- Stan, who is a travelling salesperson, always uses Stopoff hotels. He holds one of its diamond customer cards and so should qualify for special customer care. However, he regularly finds errors in his bills when he checks out.
Each month, during a 12-month period, Stan stayed in Stopoff hotels on exactly 16 occasions. He recorded, each month, the number of occasions on which his bill contained errors. His recorded values were as follows.
$$\begin{array} { l l l l l l l l l l l l }
2 & 1 & 4 & 3 & 1 & 3 & 0 & 3 & 1 & 0 & 5 & 1
\end{array}$$
- Calculate the mean and the variance of these 12 values.
- Hence state with reasons which, if either, of the distributions \(\mathrm { B } ( 16,0.20 )\) and \(B ( 16,0.125 )\) is likely to provide a satisfactory model for these 12 values.