In this question you should state the parameters of any distributions that you use. Crispy Cream Donut Shop sells 2 types of donuts: Ring Donuts and Filled Donuts. The masses in grams of Ring Donuts and Filled Donuts have normal distributions $N(54,1.5^2)$ and $N(86,2^2)$ respectively. 12 donuts are packed into a paper box. The mass in grams of an empty paper box has a normal distribution $N(80,5^2)$.

1. Define the random variable representing the difference in mass and find its expected value.

Let $W = R_1 + R_2 + R_3 - 2F$, where $R_i \sim N(54, 1.5^2)$ and $F \sim N(86, 2^2)$.
$E(W) = 3E(R) - 2E(F)$
$E(W) = 3(54) - 2(86) = 162 - 172 = -10$

2. Calculate the variance of the linear combination.

$Var(W) = 3Var(R) + 2^2 Var(F)$
$Var(W) = 3(1.5^2) + 4(2^2)$
$Var(W) = 6.75 + 16 = 22.75$

3. State the parameters of the normal distribution for $W$ and compute the required probability.

Therefore, the distribution is $W \sim N(-10, 22.75)$.
We need to find $P(R_1 + R_2 + R_3 > 2F)$, which is equivalent to $P(W > 0)$.
Using a graphing calculator,
$P(W > 0) \approx 0.018015 \approx 0.0180 \text{ (to 3 s.f.)}$

1. Define the random variable for the total mass and calculate its expected value

Let $T$ be the total mass of the box and 12 donuts.
$T = R_1 + \dots + R_6 + F_1 + \dots + F_6 + B$
$E(T) = 6E(R) + 6E(F) + E(B)$
$E(T) = 6(54) + 6(86) + 80 = 920$

2. Calculate the variance of the total mass

$Var(T) = 6Var(R) + 6Var(F) + Var(B)$
$Var(T) = 6(1.5^2) + 6(2^2) + 5^2$
$Var(T) = 6(2.25) + 6(4) + 25 = 13.5 + 24 + 25 = 62.5$

3. State the distribution of $T$ and set up the probability equation

Since the individual masses are normally distributed and independent, their sum is also normally distributed. As required by the question preamble, we state the parameters of the distribution:
$T \sim N(920, 62.5)$
We are given:
$P(T > m) = 0.95$
$P(T \le m) = 0.05$

4. Solve for $m$ using the inverse normal distribution

Using the inverse normal function on a graphing calculator:
$m \approx 906.996$
$m = 907 \text{ (to 3 s.f.)}$

1. State the independence assumption required for the variance formulas used in parts (i) and (ii).

To calculate the variance of the combined masses (e.g. $R_1 + R_2 + R_3 - 2F$ or the total mass in a box), we add the individual variances. This requires the assumption that the mass of any given donut or box does not affect the mass of another.