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Chapter 7 The Normal Probability Distribution

Chapter 7 The Normal Probability Distribution. 7.2 The Standard Normal Distribution. Properties of the Normal Density Curve

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Chapter 7 The Normal Probability Distribution

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  1. Chapter 7The Normal Probability Distribution 7.2 The Standard Normal Distribution

  2. Properties of the Normal Density Curve 7. The Empirical Rule: About 68% of the area under the graph is between -1 and 1; about 95% of the area under the graph is between -2 and 2; about 99.7% of the area under the graph is between -3 and 3.

  3. The table gives the area under the standard normal curve for values to the left of a specified Z-score, zo, as shown in the figure.

  4. EXAMPLE Finding the Area Under the Standard Normal Curve Find the area under the standard normal curve to the left of Z = -0.38.

  5. Area under the normal curve to the right of zo = 1 – Area to the left of zo

  6. EXAMPLE Finding the Area Under the Standard Normal Curve Find the area under the standard normal curve to the right of Z = 1.25.

  7. EXAMPLE Finding the Area Under the Standard Normal Curve Find the area under the standard normal curve between Z = -1.02 and Z = 2.94.

  8. EXAMPLE Finding a Z-score from a Specified Area to the Left Find the Z-score such that the area to the left of the Z-score is 0.68.

  9. EXAMPLE Finding a Z-score from a Specified Area to the Right Find the Z-score such that the area to the right of the Z-score is 0.3021.

  10. EXAMPLE Finding a Z-score Find the Z-scores that separate the middle 92% of the area under the normal curve from the 8% in the tails.

  11. EXAMPLE Finding the Value of z Find the value of z0.25

  12. Notation for the Probability of a Standard Normal Random Variable • P(a < Z < b) represents the probability a standard normal random variable is between a and b • P(Z > a) represents the probability a standard normal random variable is greater than a. • P(Z < a) represents the probability a standard normal random variable is less than a.

  13. EXAMPLE Finding Probabilities of Standard Normal Random Variables Find each of the following probabilities: (a) P(Z < -0.23) (b) P(Z > 1.93) (c) P(0.65 < Z < 2.10)

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