Question

The mass of a Grade A Gala apple is Normally distributed with mean 205 grams and standard deviation 5 grams. M1A1 packages of apples are intended to have no more than 2000 grams, but can exceed that mass 4% of the time. What is the maximum number of these Gala apples that should be placed into one M1A1 package, so that the mass limit is exceeded less than 4% of the time. Justify your answer by giving the probability of your value of n and also for n+1.

Answer 1

Let X be the mass of Grade A Gala Apple,

$\fn_phv \mu = 205$

$\fn_phv \sigma = 5$

Thus, X ~ $\fn_phv N(205, 5^{2})$

Let Y be the sum of a sample of n Grade A Gala Apple then,

Y =$\fn_phv N(n*205, n*5^{2})$  ( As per Reproductive property of the normal distribution)

So, P(Y >= 2000) < 0.04

Z = (Y-n*$\fn_phv \mu$)/($\fn_phv \sqrt{n}*\sigma$)

For p = 0.04 on std. normal curve

Z = 1.75

To satisfy the above condition,

Z <= 1.75

(Y-n*$\fn_phv \mu$)/($\fn_phv \sqrt{n}*\sigma$) <= 1.75

Soving the above equation,

n < 9.62

That is the maximum number of these Gala apples that should be placed into one M1A1 package is 9.

For n = 9,

z = (2000-205*9)/$\fn_phv \sqrt{9}$ * 5

z = 10.33

Probability of P(Z>=10.33) < .00001

while for n = 10

z = (2000-205*10)/$\fn_phv \sqrt{10}$ *5

z = -3.16

Probability of P(Z>=-3.16) = 0.9992

Hence, the maximum number of these Gala apples that should be placed into one M1A1 package so that the mass limit is exceeded less than 4% of the time is 9.