The Normal Distribution Review

Size: px

Start display at page:

Download "The Normal Distribution Review"

Ella Stokes
6 years ago
Views:

1 The Normal Distribution Review A. Properties of the Normal Distribution Many continuous variables have distributions that are bell-shaped and are called approximately normally distributed variables. The theoretical curve, called the normal distribution curve, can be used to study many variables that are not normally distributed but are approximately normal. B. Mathematical Equation for the Normal Distribution you don't want to work with it, really One equation for the standard bell curve (this one is for mean = 0 and variance = 1) is: Y = (e^(-x 2 /2))/(sqrt(2π)) E = Π = 3.14 C. Properties of the Normal Distribution The shape and position of the normal distribution curve depend on two parameters, the mean and the standard deviation. Each normally distributed variable has its own normal distribution curve, which depends on the values of the variable s mean and standard deviation. The normal distribution curve is. The,, and are equal and located at the center of the distribution. The curve is symmetrical about the mean. The curve is continuous. The curve never touches the x-axis. The total area under the normal distribution curve is equal to 1. The area under the normal curve that lies within one standard deviation of the mean is approximately. two standard deviations of the mean is approximately.

2 three standard deviations of the mean is approximately. D. The Standard Normal Distribution The standard normal distribution is a normal distribution with a mean of and a standard deviation of. All normal distributed variables can be transformed into the standard normally distributed variable by using the formula for the standard score: That formula is: Find the area under the standard normal curve between z = 0 and z = 2.34 Use your table at the end of the text to find the area. Find the area under the standard normal curve between z = 0 and z = -1.5 Use the symmetric property of the normal distribution and your table at the end of the text to find the area. Find the area to the right of z = 1.11 Find the area to the left of z = -1.93

3 Find the area between z = 2 and z = 2.47 Find the area between z = 1.68 and z = Find the area to the left of z = 1.99 Find the area to the right of z = E. Applications of the Normal Distribution - Example

4 Each month, an American household generates an average of 28 pounds of newspaper for garbage or recycling. Assume the standard deviation is 2 pounds. Assume the amount generated is normally distributed. If a household is selected at random, find the probability of its generating: More than 30.2 pounds per month. First find the z-value for Between 27 and 31 pounds per month. The American Automobile Association reports that the average time it takes to respond to an emergency call is 25 minutes. Assume the variable is approximately normally distributed and the standard deviation is 4.5 minutes. If 80 calls are randomly selected, approximately how many will be responded to in less than 15 minutes?

5 An exclusive college desires to accept only the top 10% of all graduating seniors based on the results of a national placement test. This test has a mean of 500 and a standard deviation of 100. Find the cutoff score for the exam. Assume the variable is normally distributed. Work backward to solve this problem. Example: For a medical study, a researcher wishes to select people in the middle 60% of the population based on blood pressure. If the mean systolic blood pressure is 120 and the standard deviation is 8, find the upper and lower readings that would qualify people to participate in the study. Notation z-score used throughout statistics in a variety of ways. Need convenient notation to indicate the area under the standard normal distribution. z(α) is the token, or algebraic name, for the z-score (point on the z axis) such that there is α of the area (probability) to the of z(α).

6 Illustration of Notation a. draw a picture of z(.10) b. draw a picture of z(.80) c. Find the numerical value of z(.10) d. Find the numerical value of z(.80) e. Find the numerical value of z(.99)

The Normal Distribution

Chapter 7 The Normal Distribution Section 7-1 Continuous Random Variables and Density Functions Recall the definitions Let S be the sample space of a probability experiment. A random variable X is a function