Areas under normal distribution curves can be used calculate probabilities of many random variables in the natural and social sciences. The graph of the Standard Normal Curve, with a mean of 0 and a standard deviation of 1 is given below, and its function is given by 1 f(x) = √2π 1.5 1 The integral of L e da would be used to calculate the probability of an event that occurs √2π 1.5 between -1.5 to 1.5 standard deviations from the mean. 1 Unfortunately, e 2 dx has no proper closed form antiderivative, so we must use numerical √2TT 1.5 methods to approximate the values. (-1.5, 0.13) 1.5 -115 (-1,0.24) (-0.5, 0.35) Using Simpson's Rule, -0.5 04 -0-2 [ 15 e- 3 dr ~ [ е 0 (0, 0.4) 015 Using the Midpoint Rule with three rectangles of equal width, Using the Trapezoidal Rule with six trapezoids of equal width, 1.5 ·[e- (0.5, 0.35) 1.5 [tbe е (1, 0.24) da ≈ 115 (1.5, 0.13)

Weights of Lamb after eight weeks after born are normally distributed with mean µ = 14 kg and variance σ 2 = 4. a)Find the probability p that a single lamb has weight less than 12 kg. b) Find the probability that exactly 3 of the next 5 lambs examined will have weights less than 12 kg.

Let X be a normal random variable with mean 85 and a variance of 25 (i.e., X ∼ N (85, 25)). step1:  Let M = aX + b for some constants a, b not equal to 0. Write an expression for the pdf of M . step 2:  Suppose that X represents an approximate distribution of the final scores in a certain math course (ignore the fact that this approximation can technically have scores that are greater than 100 or less than 0). If the teacher were to curve the scores, it means he would determine a function to apply to the scores to achieve a desired distribution (assume an affine function in this case, as in the previous part). Suppose he wants the scores to be normally distributed with mean 80 and a standard deviation of 4. What should he choose for a and b? What students would see their score lowered, and what students would see their score increased?

Find P(11<X<15) If X is normally distributed with a mean of 14 and a standard deviation of 2?

A person's blood glucose level and diabetes are closely related. Let x be a random variable measured in milligrams of glucose per deciliter (1/10 of a liter) of blood. Suppose that after a 12-hour fast, the random variable x will have a distribution that is approximately normal with mean μ = 88 and standard deviation σ = 21. Note: After 50 years of age, both the mean and standard deviation tend to increase. For an adult (under 50) after a 12-hour fast, find the following probabilities. (Round your answers to four decimal places.) (a) x is more than 60(b) x is less than 110(c) x is between 60 and 110(d) x is greater than 140 (borderline diabetes starts at 140)