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Section 6.1

Section 6.1. Introduction to the Normal Distribution. With extra good stuff added by D.R.S., University of Cordele. Properties of a Normal Distribution. Properties of a Normal Distribution 1. A normal distribution is bell-shaped and symmetric about its mean.

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Section 6.1

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  1. Section 6.1 Introduction to the Normal Distribution With extra good stuff added by D.R.S., University of Cordele.

  2. Properties of a Normal Distribution Properties of a Normal Distribution 1. A normal distribution is bell-shaped and symmetric about its mean. 2. A normal distribution is completely defined by its mean, m, and standard deviation, s. 3. The total area under a normal distribution curve equals 1. 4. The x-axis is a horizontal asymptote for a normal distribution curve.

  3. The Standard Normal Distribution Properties of the Standard Normal Distribution 1. The standard normal distribution is bell-shaped and symmetric about its mean. 2. The standard normal distribution is completely defined by its mean, m = 0, and standard deviation, s = 1. 3. The total area under the standard normal distribution curve equals 1. 4. The x-axis is a horizontal asymptote for the standard normal distribution curve. The two things that make it “The Standard…”

  4. Example 6.1: Calculating and Graphing z-Values Given a normal distribution with μ = 48 and s = 5, convert an x-value of 45 to a z-value and indicate where this z-value would be on the standard normal distribution. Solution Begin by finding the z-score for x = 45 as follows. TI-84 Needs Extra Parentheses!!!( 4 5 ─ 4 8 ) / 5

  5. Example 6.1: Calculating and Graphing z-Values (cont.) Now draw each of the distributions, marking a standard score of z = −0.60 on the standard normal distribution.

  6. Example 6.1: Calculating and Graphing z-Values (cont.) The distribution on the left is a normal distribution with a mean of 48 and a standard deviation of 5. The distribution on the right is a standard normal distribution with a standard score of z = −0.60 indicated.

  7. Recommend two parallel axes, z and x both RECOMMENDED: Sketch a normal distribution

  8. Recommend two parallel axes, z and x both RECOMMENDED: Sketch a normal distribution A z-axis, labelingz = 0 and z = +1, -1, +2, -2, +3, -3, etc., as needed. For this particular problem, z = -1 and z = +1 are good to visualize, and z = -0.60 is meaningful.

  9. Recommend two parallel axes, z and x both RECOMMENDED: Sketch a normal distribution A z-axis, labelingz = 0 and z = +1, -1, +2, -2, +3, -3, as needed. An x-axis, too, labeling the mean and other key values Observe the correspondence between z values and x values. Corresponding to those fourz-values are these four x-values.

  10. Recommend two parallel axes, z and x both RECOMMENDED: Sketch a normal distribution A z-axis, labelingz = 0 and z = +1, -1, +2, -2, +3, -3, as needed. An x-axis, too, labeling the mean and other key values. Observe the correspondence between z values and x values. It’s clear to the reader that for x=45, z=-0.60. It’s clear to the reader that the mean is x = 48.

  11. Excel: STANDARDIZE(x value, mean, standard deviation)

  12. Going the other way: converting z back to an x If you take this formula: And solve to get x by itself on one side:

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