We also note the following about the normal distribution 1 The mean \u03bc is

# We also note the following about the normal

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We also note the following about the normal distribution. 1 The mean, μ , is independent of the variance, σ 2 . 2 A normal variate with mean, μ , and variance, σ 2 is denoted by X ND ( μ, σ 2 ) . By Arthur Mpazi Yambayamba The Normal Probability Distribution November 4, 2015 7 / 32
Areas Under a Normal Distribution Curve By Arthur Mpazi Yambayamba The Normal Probability Distribution November 4, 2015 8 / 32
The Standard Normal Distribution Since each normally distributed variable has its own mean and standard deviation, as stated earlier, the shape and location of these curves will vary. In practical applications, then, you would have to have a table of areas under the curve for each variable. To simplify this situation, statisticians use what is called the Standard Normal Distribution . The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. By Arthur Mpazi Yambayamba The Normal Probability Distribution November 4, 2015 9 / 32
The Standard Normal Distribution All normally distributed variables can be transformed into the standard normally distributed variable by using the formula for the standard score or z score given by: z = value - mean standard deviation or z = x - μ σ The z score or value gives the number of standard deviations that a particular value of X , x , is away from the mean. By Arthur Mpazi Yambayamba The Normal Probability Distribution November 4, 2015 10 / 32
The Standard Normal Distribution By Arthur Mpazi Yambayamba The Normal Probability Distribution November 4, 2015 11 / 32
Finding Areas Under the Standard Normal Distribution Curve The first step is to draw a sketch of the normal distribution curve and shade the area. Second step, use the procedure as illustrated in the following figures. (1) To the left of any z value, look up the z value in the table and use the area given. By Arthur Mpazi Yambayamba The Normal Probability Distribution November 4, 2015 12 / 32
Finding Areas Under the Standard Normal Distribution Curve (2) To the right of any z value, look up the z value in the table and subtract the area from 1. By Arthur Mpazi Yambayamba The Normal Probability Distribution November 4, 2015 13 / 32
Finding Areas Under the Standard Normal Distribution Curve (3) Between any two z values, look up both z values and subtract the corresponding areas. By Arthur Mpazi Yambayamba The Normal Probability Distribution November 4, 2015 14 / 32
Finding Areas Under the Standard Normal Distribution Curve Examples (1) Find the area to the left of z = 1 . 99 . (2) Find the area to the right of z = - 1 . 16 . (3) Find the area between z = +1 . 68 and z = - 1 . 37 . By Arthur Mpazi Yambayamba The Normal Probability Distribution November 4, 2015 15 / 32
Calculating Probabilities Using the Standard Normal Distribution Curve A normal distribution curve can be used as a probability distribution curve for normally distributed variables.

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• Fall '18
• F. TAILOKA
• Normal Distribution, normal probability distribution, Arthur Mpazi Yambayamba