Question

To represent normal variable (distribution) with mean() and standard deviation (0) we use shorthand notation X-N . a). For example, suppose the vitamin C content of a particular variety of orange is distributed normally with mean 720 IU and standard deviation 46 IU. If we designate the vitamin C content of a randomly selected orange, then our shorthand notation is X-N[720 IU, 46 IU). To represent probability between two values a and b we use notation Pla< X <b), to represent probability greater than a we use notation P a), and to represent probability less than a, we use notation PpXca) Use the example and notation to answer the following questions: 2pt 1) What is the probability that a randomly selected orange will have less than 660 IU? 2 pt 2) Using X as the random variable, what probability notation would you use to represent the probability in part 12 2pt 2) What is the BOth percentile of the of the distribution of vitamin C content of the oranges? 1pt 3) What proportion of oranges exceed the vitamin C content you found in part (2) above? 3pt 4) What range of vitamin C content values represent the middle 80% of the distribution? State your answer as a probability statement using the probability notation
Please answer all parts, questions 1-5. Thank you!

Answer 1

- 1.300 0.20 & Solution :- Let x. Vitamin C contest in randomly seleded orange - X N (M = 720, 02 = 462) The prob that randomly selected orange will have less than 660 1U = P(x < 660] =P[Xavy < 660-43 =P [z_< 660 - 720] 46 = [z < -1.30] = P(Z > 1-30) (As Normal dist is symmetric) "P(X < 660) = 0.0968 2) Probability notation is P(x <660) Let 80th porcentle of the dists of vitamin C be Pro such that P(x > P80] = 0.20 or PL x <P 8) = 0.80 Consider, P[X> P8] = 0.20 P [X-4 > Pso-47= 0.20 / 0 Pro ...PLZ > Peo - 720) - 0.20 – Pso - 720 = 0.84 46 . Pro = 720+ 0.84 (46) Pgo = 758.64 - The proportion of oranges exceed vitamin C contest than 758.64 is 20% To find the range of vitamin C contest that represest middle 80% of dist consider, Pla<x<b] =0.80 Plany base 0.80 (As normal dists b=778.88 i. 1-2 P zob-l = 0.80 is symmetne e at o P[zyball )= o. lo - a= M- 1.280 = 720-1428046) = 220 +1.28 (46) a = 661.12 a ob ( az b-4 = 1.28 =) b = Mt 1.285