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Using the Empirical Rule

This comprehensive guide covers the fundamentals of normal distributions and the Empirical Rule, illustrating their significance in real-world data analysis. Normal distributions, characterized by their bell-shaped curves, are defined by their mean (μ) and standard deviation (σ), affecting their horizontal positioning and spread. The Empirical Rule states that approximately 68%, 95%, and 99.7% of observations fall within one, two, and three standard deviations of the mean, respectively. This understanding aids in interpreting data from various fields like biology, psychology, and sports.

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Using the Empirical Rule

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  1. Using the Empirical Rule

  2. Normal Distributions • These are special density curves. • They have the same overall shape • Symmetric • Single-Peaked • Bell-Shaped • They are completely described by giving its mean () and its standard deviation (). • We abbreviate it N(,)

  3. Normal Curves…. • Changing the mean without changing the standard deviation simply moves the curve horizontally. • The Standard deviation controls the spread of a Normal Curve.

  4. Standard Deviation • It’s the natural measure of spread for Normal distributions. • It can be located by eye on a Normal curve. • It’s the point at which the curve changes from concave down to concave up.

  5. Why is the Normal Curve Important? • They are good descriptions for some real data such as • Test scores like SAT, IQ • Repeated careful measurements of the same quantity • Characteristics of biological populations (height) • They are good approximations to the results of many kinds of chance outcomes • They are used in many statistical inference procedures.

  6. Empirical Rule • What percent do you think…… • www.whfreeman.com/tps4e

  7. Empirical Rule (68-95-99.7 Rule) • In the Normal distribution with mean () and standard deviation (): • Within 1 of  ≈ 68% of the observations • Within 2 of  ≈ 95% of the observations • Within 3 of  ≈ 99.7% of the observations

  8. The distribution of batting average (proportion of hits) for the 432 Major League Baseball players with at least 100 plate appearances in the 2009 season is normally distributed defined N(0.261, 0.034). • Sketch a Normal density curve for this distribution of batting averages. Label the points that are 1, 2, and 3 standard deviations from the mean. • What percent of the batting averages are above 0.329? • What percent are between 0.227 and .295?

  9. Scores on the Wechsler adult Intelligence Scale (a standard IQ test) for the 20 to 34 age group are approximately Normally distributed. N(110, 25). • What percent are between 85 and 135? • What percent are below 185? • What percent are below 60?

  10. Standard Normal Distribution • It is the Normal distribution with mean 0 and standard deviation 1. • If a variable x has any Normal distribution N(, ), then the standardized variable has the standard Normal distribution.

  11. A standard Normal table give the area under the curve to the left of z. Find the area to the left of z = 0.21

  12. Using the chart • Find P(z<1.23) • Find P(z > 2.01)

  13. More examples • Find P(z< -0.13) • Find P(z > -1.72)

  14. More examples • Find P(-1.56 < z < 1.01) • Find P(-2.23 < z < -0.27)

  15. Find the z-score that correlates with the 20th percentile.

  16. Normal Distribution Calculations

  17. When Tiger Woods hits his driver, the distance the ball travels follows a Normal Distribution with mean 304 yards and standard deviation 8 yards. What percent of Tiger’s drives travel at least 290 yards? Let x = the distance Tiger’s ball travels and it has a distribution N(304, 8). We want P(x  290). About 96% of Tigers drives travel at least 290 yards.

  18. Scores for a test have a mean of 100 and standard deviation of 15. Find the probability that a score is below 112.

  19. Every month, American households generate an average of 28 pounds of newspaper for garbage or recycling. Assume =2 pounds. If a household is selected at random, find the probability that it generates between 27 and 31 pounds per month.

  20. 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.

  21. 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 reading that would qualify a person to be in the study.

  22. Homework • Page 131 (47-59)

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