7 Mr Alott and Miss Fewer work in a postal sorting office.
- The number of letters per batch, \(R\), sorted incorrectly by Mr Alott when sorting batches of 50 letters may be modelled by the distribution \(\mathrm { B } ( 50,0.15 )\).
Determine:
- \(\mathrm { P } ( R < 10 )\);
- \(\mathrm { P } ( 5 \leqslant R \leqslant 10 )\).
- It is assumed that the probability that Miss Fewer sorts a letter incorrectly is 0.06 , and that her sorting of a letter incorrectly is independent from letter to letter.
- Calculate the probability that, when sorting a batch of \(\mathbf { 2 2 }\) letters, Miss Fewer sorts exactly 2 letters incorrectly.
- Calculate the probability that, when sorting a batch of \(\mathbf { 3 5 }\) letters, Miss Fewer sorts at least 1 letter incorrectly.
- Calculate the mean and the variance for the number of letters sorted correctly by Miss Fewer when she sorts a batch of \(\mathbf { 1 2 0 }\) letters.
- Miss Fewer sorts a random sample of 20 batches, each containing 120 letters. The number of letters sorted correctly per batch has a mean of 112.8 and a variance of 56.86 .
Comment on the assumptions that the probability that Miss Fewer sorts a letter incorrectly is 0.06 , and that her sorting of a letter incorrectly is independent from letter to letter.
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