Rounded or discrete from continuous

5 The random variable \(X\) is such that \(X \sim \mathrm {~N} ( 82,126 )\).

1. A value of \(X\) is chosen at random and rounded to the nearest whole number. Find the probability that this whole number is 84 .
2. Five independent observations of \(X\) are taken. Find the probability that at most one of them is greater than 87.
3. Find the value of \(k\) such that \(\mathrm { P } ( 87 < X < k ) = 0.3\).

3 In an English language test for 12-year-old children, the raw scores, \(X\), are Normally distributed with mean 45.3 and standard deviation 11.5.

1. Find
   (A) \(\mathrm { P } ( X < 50 )\),
   (B) \(\mathrm { P } ( 45.3 < X < 50 )\).
2. Find the least raw score which would be obtained by the highest scoring \(10 \%\) of children.
3. The raw score is then scaled so that the scaled score is Normally distributed with mean 100 and standard deviation 15. This scaled score is then rounded to the nearest integer. Find the probability that a randomly selected child gets a rounded score of exactly 111 .
4. In a Mathematics test for 12-year-old children, the raw scores, \(Y\), are Normally distributed with mean \(\mu\) and standard deviation \(\sigma\). Given that \(\mathrm { P } ( Y < 15 ) = 0.3\) and \(\mathrm { P } ( Y < 22 ) = 0.8\), find the values of \(\mu\) and \(\sigma\).

3 The scores, \(X\), in Paper 1 of an English examination have an underlying Normal distribution with mean 76 and standard deviation 12. The scores are reported as integer marks. So, for example, a score for which \(75.5 \leqslant X < 76.5\) is reported as 76 marks.

1. Find the probability that a candidate's reported mark is 76 .
2. Find the probability that a candidate's reported mark is at least 80 .
3. Three candidates are chosen at random. Find the probability that exactly one of these three candidates' reported marks is at least 80 . The proportion of candidates who receive an A* grade (the highest grade) must not exceed \(10 \%\) but should be as close as possible to \(10 \%\).
4. Find the lowest reported mark that should be awarded an A* grade. The scores in Paper 2 of the examination have an underlying Normal distribution with mean \(\mu\) and standard deviation 12.
5. Given that \(20 \%\) of candidates receive a reported mark of 50 or less, find the value of \(\mu\).

3 The wing lengths of native English male blackbirds, measured in mm , are Normally distributed with mean 130.5 and variance 11.84.

1. Find the probability that a randomly selected native English male blackbird has a wing length greater than 135 mm .
2. Given that \(1 \%\) of native English male blackbirds have wing length more than \(k \mathrm {~mm}\), find the value of \(k\).
3. Find the probability that a randomly selected native English male blackbird has a wing length which is 131 mm correct to the nearest millimetre. It is suspected that Scandinavian male blackbirds have, on average, longer wings than native English male blackbirds. A random sample of 20 Scandinavian male blackbirds has mean wing length 132.4 mm . You may assume that wing lengths in this population are Normally distributed with variance \(11.84 \mathrm {~mm} ^ { 2 }\).
4. Carry out an appropriate hypothesis test, at the \(5 \%\) significance level.
5. Discuss briefly one advantage and one disadvantage of using a \(10 \%\) significance level rather than a \(5 \%\) significance level in hypothesis testing in general.

3 Each day, Margot completes the crossword in her local morning newspaper. Her completion times, \(X\) minutes, can be modelled by a normal random variable with a mean of 65 and a standard deviation of 20 .

1. Determine:
   1. \(\mathrm { P } ( X < 90 )\);
   2. \(\mathrm { P } ( X > 60 )\).
2. Given that Margot's completion times are independent from day to day, determine the probability that, during a particular period of 6 days:
   1. she completes one of the six crosswords in exactly 60 minutes;
   2. she completes each crossword in less than 60 minutes;
   3. her mean completion time is less than 60 minutes.
      
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