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Chapter 6: The Normal Distribution Section 6.1: Normal Distributions Definitions: 1. Recall, that a continuous variable can assume all values between any two given values of the variables. Many continuous variables have distributions that are bell-shaped, and these are called approximately distributed variables. 2. If a random variable has a probability distribution whose graph is continuous, bell-shaped, and symmetric, it is called a normal distribution. The graph is called a normal distribution curve. See below.

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Summary of the Properties of the Theoretical Normal Distribution 1. A normal distribution curve is bell-shaped. 2. The mean, median, and mode are equal and are located at the center of the distribution. 3. A normal distribution curve is unimodal. 4. The curve is symmetric about the mean. 5. The curve is continuous. 6. The curve never touches the x-axis. 7. The total area under a normal distribution curve is equal to 1.00 or 100%. 8. The area under the part of a normal curve that lies within 1 standard deviation of the mean is approximately 0.68, or 68%; within 2 standard deviations, about 0.95, or 95%; and within 3 standard deviations, about 0.997, or 99.7%.

Areas Under a Normal Distribution Curve

Normal and Skewed Distribution

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3. The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1.

Standard Normal Distribution

All normally distributed variables can be transformed into the standard normally distribution variable by using the formula for the standard score:

z = value − meanstandard deviation

or z = X − µσ

4. The z-value or z-score is actually the number of standard deviations that a particular X value is away from the mean. The two step method for finding areas under the standard normal distribution curve. Step 1: Draw the normal distribution curve and shad the area Step 2: Find the appropriate figure in the Procedure Table Note: There are three basic types of problems, and all three are summarized in the Procedure Table. Note that this table is presented as an aid in understanding how to use the standard normal distribution table and in visualizing the problems. After learning the procedures, you should not find it necessary to refer to the Procedure Table for every problem.

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Procedure Table: Finding the Area Under the Standard Normal Distribution Curve 1. To the left of any z-value: Look up the z-value in the table and use the area given.

2. To the right of any z-value: Look up the z-value and subtract the area from 1.

3. Between any two z-value: Look up both z-value and subtract the corresponding areas.

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Exercise 1. Find the area under the standard normal distribution curve. a) To the left of z = 1.73 . b) To the right of z = −1.24 . c) Between z = 1.62 and z = −1.35 Exercise 2. Find the area under the standard normal distribution curve. a) To the left of z = 2.09 b) To the right of z = 1.14 c) Between z = 1.62 and z = 1.35 d) Between z = 0 and z = 1.77 e) To the right of z = 2.01

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Moreover, the area under the standard distribution curve can also be though as a probability or as the proportion of the population with a given characteristic. Exercise 3. Find the probability of each. (Assume this is a standard normal distribution.) a) P(0 < z < 2.35) b) P(z <1.73) c) P(z >1.98) Exercise 4. Find the z-value that corresponds to the given area. a)

b)

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Exercise 5. Find the z-value so that a) 54.78% of the area under the distribution curve lies to the left of it. b) 82.12% of the area under the distribution curve lies to the right of it. Exercise 6. Many times in statistics it is necessary to see if a set of data values is approximately normally distributed. There are special techniques that can be used. One technique is to draw a histogram for the data and see if it is approximately bell-shaped. (Note: It does not have to be exactly symmetric to be bell-shaped.)

a) Construct a frequency distribution for the data.

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b) Construct a histogram for the data. c) Describe the shape of the histogram. d) Based on your answer to part (c), do you feel that the distribution is approximately normal?

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e) Find the mean and standard deviation for the data.

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f) What percent of the data values fall within 1 standard deviation of the mean? g) What percent of the data values fall within 2 standard deviations of the mean? h) What percent of the data values fall within 3 standard deviations of the mean? i) How do you answers to parts (f), (g), and (h) compare to 68, 95, and 100%, respectively? j) Does your answer help support the conclusion you reached in part (d)? Explain.

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Z-table

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Z-table