## Miraculin – a protein naturally produced in a rare tropical fruit – can convert a sour taste into a sweet taste. Consequently, miraculin has

Miraculin – a protein naturally produced in a rare tropical fruit – can convert a sour taste into a sweet taste. Consequently, miraculin has the potential to be an alternative low-calorie sweetener. In Plant Science (May 2010), a group of Japanese environmental scientists investigated the ability of a hybrid tomato plant to produce miraculin. For a particular generation of the tomato plant, the amount x of miraculin produced (measured in micrograms per gram of fresh weight) had a mean of 105.3 and a standard deviation of 8.0. Assume that x is normally distributed. a. Find P(x > 120). b. Find P(100 < x < 110). c. Find the value a for which P(x < a) = .25 .

## Answers ( )

Answer:a) P(x > 120) = 0.74537b) P(100 < x < 110) = 0.46777c) Value of a = 110.7Step-by-step explanation:Let x = amount of miraculin produced (measured in micro grams per gram of fresh weight)

We are given

Mean,= 105.3 and Standard Deviation,= 8.0Also, since x is normally distributed so;

Z = follows N(0,1)

a)

P(x > 100)= P( > ) = P(Z > -0.6625) = P(Z < 0.66) = 0.74537b)

P(100 < x < 110) = P(x < 110) – P(x <= 100)P(x <= 100) = 1 – P(x > 100) = 1 – 0.74537 = 0.25463

P(x < 110) = P( < ) = P(Z < 0.59) = 0.72240

Hence,

P(100 < x < 110) = 0.72240 – 0.25463 = 0.46777c) Given expression is P(x < a ) = 0.25

⇒ P( < ) = 0.25

⇒ P(Z < ) = 0.25

By seeing the Z % table we find that the value of z which have an are of 25% is 0.6745i.e.= 0.6745

So, value of a = 0.6745*8 + 105.3 = 110.7 .