Normal approximation to normal-derived binomial

5 The heights of books in a library, in cm, have a normal distribution with mean 21.7 and standard deviation 6.5. A book with a height of more than 29 cm is classified as 'large'.

1. Find the probability that, of 8 books chosen at random, fewer than 2 books are classified as large.
2. \(n\) books are chosen at random. The probability of there being at least 1 large book is more than 0.98 . Find the least possible value of \(n\).

5 Plastic drinking straws are manufactured to fit into drinks cartons which have a hole in the top. A straw fits into the hole if the diameter of the straw is less than 3 mm . The diameters of the straws have a normal distribution with mean 2.6 mm and standard deviation 0.25 mm .

1. A straw is chosen at random. Find the probability that it fits into the hole in a drinks carton.
2. 500 straws are chosen at random. Use a suitable approximation to find the probability that at least 480 straws fit into the holes in drinks cartons.
3. Justify the use of your approximation.

3 Tennis balls are dropped from a standard height, and the height of bounce, \(H \mathrm {~cm}\), is measured. \(H\) is a random variable with the distribution \(\mathrm { N } \left( 40 , \sigma ^ { 2 } \right)\). It is given that \(\mathrm { P } ( H < 32 ) = 0.2\).

1. Find the value of \(\sigma\).
2. 90 tennis balls are selected at random. Use an appropriate approximation to find the probability that more than 19 have \(H < 32\).

1. A machine puts liquid into bottles of perfume. The amount of liquid put into each bottle, \(D \mathrm { ml }\), follows a normal distribution with mean 25 ml

Given that \(15 \%\) of bottles contain less than 24.63 ml

1. find, to 2 decimal places, the value of \(k\) such that \(\mathrm { P } ( 24.63 < D < k ) = 0.45\) A random sample of 200 bottles is taken.
2. Using a normal approximation, find the probability that fewer than half of these bottles contain between 24.63 ml and \(k \mathrm { ml }\)