- The weights of a particular type of apple, \(A\) grams, and a particular type of orange, \(R\) grams, each follow independent normal distributions.
$$A \sim \mathrm {~N} \left( 160,12 ^ { 2 } \right) \quad R \sim \mathrm {~N} \left( 140,10 ^ { 2 } \right)$$
- Find the distribution of
- \(A + R\)
- the total weight of 2 randomly selected apples.
A box contains 4 apples and 1 orange only. Jesse selects 2 pieces of fruit at random from the box.
- Find the probability that the total weight of the 2 pieces of fruit exceeds 310 grams.
From a large number of apples and oranges, Celeste selects \(m\) apples and 1 orange at random. The random variable \(W\) is given by
$$W = \left( \sum _ { i = 1 } ^ { m } A _ { i } \right) - n \times R$$
where \(n\) is a positive integer.
Given that the middle \(95 \%\) of the distribution of \(W\) lies between 1100.08 and 1499.92 grams, (c) find the value of \(m\) and the value of \(n\)