What are some properties of a normal distribution? Include the area under the cu

Question

What are some properties of a normal distribution? Include the area under the curve, the Empirical rule, shape of the distribution, role that standard deviations play, and how to find the area under a normal curve. Monthly charges for cell phone service for cell plans in the U.S. are normally distributed with a population mean of exist62 and a population standard deviation of exist18? Address the following in the space below. a) Give those two values using their correct notation. b) Draw a normal with the parameters labeled. c) Shade the region that represents the proportion of plans that charge less than exist44. d) Suppose the area under the curve to the left of x = exist44 is 0.1587. Provide two interpretations of this result.

Explanation / Answer

Properties of Normal Distribution

1. It is symmetric about its mean – consequently, mean = median = mode.

2. If a random variable X ~ N(µ, 2), i.e., X has Normal Distribution with mean µ and variance 2, then, pdf of X, f(x) = {1/(2)}e^-[(1/2){(x - µ)/}2]

3. Z = (X - µ)/ ~ N(0, 1), N(0, 1) is called Standard Normal Distribution and Z is called Standard Normal Variate.

4. P(X or t) = P[{(X - µ)/ } or {(t - µ)/ }] = P[Z or {(t - µ)/ }]

5. X bar ~ N(µ,2/n), where X bar is average of a sample of size n from population of X.

6. Empirical rule, also known as 68 – 95 – 99.7 percent rule:

P{( µ - ) X ( µ + )} = 0.68;

P{( µ - 2) X ( µ + 2)} = 0.95;

P{( µ - 3) X ( µ + 3)} = 0.997

7. If X ~ N(µ1, 12), Y ~ N(µ2, 22) and X and Yare independent, then

(aX + bY) ~ N(µ, 2) µ = aµ1 + bµ2, and 2 = a212 + b222

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