2. The weight of candy apples is normally distributed with a mean of 100 grams a

Question

2. The weight of candy apples is normally distributed with a mean of 100 grams and a standard deviation of 10 grams. The weight of caramel apples is normally distributed with a mean of 120 grams and a standard deviation of 10 grams, 50% of all the apples are caramel and 50% are candy. a. What is the probability the weight of a candy apple is between 107 and 108 grams? b. What is the probability the weight of a caramel apple is between 107 and 108 grams? c. If you weigh an apple chosen at random and find it weighs 107 grams, what is the probability it's a candy apple? You get a new sample of apples and are told that 90% of them are caramel. You pick an apple from this sample at random and find it weighs 107 grams. What is the probability that it's a candy apple? d.

Explanation / Answer

Q.2 Let say weight of a random candy apple is X and weight of a random caramel apple is Y grams.

(a) Pr(107 <= X< = 108) = NORM (107 <= X <= 108 ; 100 ; 10) = Pr(X < 108 ; 100; 10) - Pr(X < 107; 100; 10)

Z2 = (108 - 100)/10 = 0.8 and Z1 = (107 - 100)/ 10 = 0.7

Pr(107 <= X< = 108) = (0.8) - (0.7)

where is the standard normal cumulative distribution.

Pr(107 <= X< = 108) = (0.8) - (0.7) = 0.7881 - 0.7580 = 0.0301

(b)

Pr(107 <= Y< = 108) = NORM (107 <= Y <= 108 ; 100 ; 10) = Pr(Y < 108 ; 100; 10) - Pr(Y < 107; 100; 10)

Z2 = (108 - 120)/10 = -1.2 and Z1 = (107 - 120)/ 10 = -1.3

Pr(107 <= X< = 108) = (-1.2) - (-1.3)

where is the standard normal cumulative distribution.

Pr(107 <= X< = 108) = (-1.2) - (-1.3) = 0.1151- 0.0968 = 0.0183

(c) Pr(candy apple) = Pr(caramel apple) = 0.5

Now the apple weighs = 107 grams.

So, now we have to find that probability it is a candy apple.

Pr(Candy apple l weighs 107 grams) = Pr(candy apple and weighs 107 grams)/ Pr(weighs 107 grams)

= 0.5 * Pr(X > 107; 100; 10)/ [0.5 * Pr(X > 107; 100 ; 10) + 0.5* Pr(X <107; 120; 10)]

= (1 - (0.7)/ [(1 -(0.7)) + (-1.3)]

= 0.2420/ [ 0.2420 + 0.0968]

= 0.7143

(d) if Pr(Candy ) = 0.1 and Pr(Caramel) = 0.9

then Pr(Candy apple l weighs 107 grams) = Pr(candy apple and weighs 107 grams)/ Pr(weighs 107 grams)

= 0.1 * Pr(X > 107; 100; 10)/ [0.9 * Pr(X > 107; 100 ; 10) + 0.1* Pr(X <107; 120; 10)]

= (1 - (0.7))/ [  (1 - (0.7)) + 9 * (1.3) ]

= 0.242/ [0.242 + 9 * 0.0968]  

= 0.242/ 1.1132

= 0.2174

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