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

Question

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 .

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Isabella 3 weeks 2021-12-29T00:16:04+00:00 1 Answer 0 views 0

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    2021-12-29T00:17:54+00:00

    Answer:

    a) P(x > 120) = 0.74537

    b) P(100 < x < 110) = 0.46777

    c) Value of a = 110.7

    Step-by-step explanation:

    Let x = amount of miraculin produced (measured in micro grams per gram of fresh weight)

    We are given Mean, \mu = 105.3  and  Standard Deviation, \sigma = 8.0

    Also, since x is normally distributed so;

                        Z = \frac{x - \mu}{\sigma} follows N(0,1)

    a) P(x > 100) = P( \frac{x - \mu}{\sigma} > \frac{100 - 105.3}{8.0} ) = P(Z > -0.6625) = P(Z < 0.66) = 0.74537

    b) 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( \frac{x - \mu}{\sigma} < \frac{110 - 105.3}{8.0} ) = P(Z < 0.59) = 0.72240

    Hence, P(100 < x < 110) = 0.72240 – 0.25463 = 0.46777

    c) Given expression is P(x < a ) = 0.25

                           ⇒ P( \frac{x - \mu}{\sigma} < \frac{a - 105.3}{8.0} ) = 0.25

                           ⇒ P(Z < \frac{a - 105.3}{8.0} ) = 0.25

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

                         \frac{a - 105.3}{8.0} = 0.6745

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

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