Suppose you take a trip to Stars Hollow Apple Orchard. Stars Hollow Apple Orchar

Question

Suppose you take a trip to Stars Hollow Apple Orchard. Stars Hollow Apple Orchard is a magical place where the weight of the apples exactly follows a normal distribution! They grow two types of apples, Fuji and Gala. The weight of the Fuji apples is normally distributed with a mean of 91 grams and a standard deviation of 3 grams. Let X be the weight of a randomly selected Fuji apple. The weight of the Gala apples is normally distributed with a mean of 89 grams and a standard deviation of 4 grams. Let Y be the weight of a randomly selected Gala apple. (a) Suppose you pick one Fuji and one Gala apple at random. (Assume independence.) What is the probability that the Gala apple weighs more than the Fuji apple? That is, find P(Y > X). b) Suppose you pick sixteen Fuji apples at random. (Assume independence.) What is the probability their total weight is less than 1.465 kilograms? c) Suppose you pick four Gala apples at random. (Assume independence.) What is the probability their average weight is greater than 92 grams? (d) Suppose you pick four Fuji and ten Gala apples at random. (Assume independence.) What is the probability their total weight is greater than 1.247 kilograms? (e) Suppose you pick six Fuji and four Gala apples at random. (Assume independence.) What is the probability that at most one of these apples weighs less than 85 grams?

Explanation / Answer

For fuji apple :

Mean = 91 grams and standard deviation = 3 grams

Let say weight of any random Fuji apples is = X gms

For Gala apples :

Mean = 89 grams and standard deviation = 4 grams  

Let say weight of any random Gala apples is = Ygms

(a) Here P(Y > X) = P(Y -X >0)

Here E(Y-X) = 89 - 91 = -2 gms

Var(Y -X) = Var(Y) + Var(X) = 32 + 42 = 25

Std. dev(Y -X) = sqrt(25) = 5

so Here P(Y -X >0) ~ Norm (Y - X > 0 ; -2; 5)

Z = (0 + 2)/5 = 0.4

P(Y -X >0) ~ Norm (Y - X > 0 ; -2; 5) = 1 - Z (0.4) = 1 - 0.6554 = 0.3446

(b) Here Lets say Z is equal to total weight of 16 Fuji apples.

Z = 16X

E(Z) = 16 * 0.091 = 1.456 kg

Var(Z) = 16 * 0.0032 = 0.144 kg

STD(Z) = 0.012 kg

Pr(Z < 1.465 kg) = Pr(Z < 1.465 kg ; 1.456 kg ; 0.012 kg)

here Z- value = (1.465 - 1.456)/ 0.012 = 0.75

Pr(Z < 1.465 kg) = Pr(Z < 1.465 kg ; 1.456 kg ; 0.012 kg) = Pr(Z < 0.75) = 0.7734

(c) Here Lets say we pick 4 Gala apples.

estimated average weight E(x?) = 89 grams

Standard error of the sample mean s = 4/ sqrt(4) = 2 gms

so now we have to calculate

Pr(x? > 92 grams) ~ Pr(x? > 92 gms; 89 gms ; 2 gms)

here Z- value = (92 - 89)/ 2 = 1.5

Pr(x? > 92 grams) = 1 - Pr(Z > 1.5) = 1 - 0.9332 =  0.0668

(d) Four fuji and ten gala apples at random.

Let say total weight is W grams.

W = 4X + 10Y

E(W) = E(4X + 10Y) = 4 * 91 + 10 * 89 = 1254 gms

Var(W) = 4 * Var(X) + 10 * Var(Y) = 4 * 32 + 10 * 42 = 196

STD(W) = sqrt(196) = 14 gm

so, Pr(W > 1.247 Kg) ~ Norm (W > 1.247 g, ; 1.254 gm; 0.014 gm)

Z = (1.247 - 1.254)/ 0.014 = -0.5

Pr(W > 1.247 Kg) = 1 - Pr(Z > -0.5) = 1 - 0.3085 = 0.6915

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