Contact
CoCalc Logo Icon
StoreFeaturesDocsShareSupport News Sign UpSign In
| Download
Views: 218

The Empirical Rule - ELYSE LORENZO

What is it?

  • About 68% of the area lies within one standard deviation of the mean.

  • About 95% of the area lies within two standard deviations of the mean.

  • About 99.7% of the area lies within three standard deviations of the mean.

Graph

Walk Through (Easy Questions)

IQ scores follow a normal distribution with mean 100 and standard deviation 15.

  1. What is the probability that a person has an IQ score greater than 100?

  • The probability that a person has an IQ score greater than 100 is 50%.

  1. What is the probability that a person has an IQ score that falls between 85 and 115?

  • The probability that a person has an IQ score that falls between 85 and 115 is 68%.

  1. A special program for gifted and talented children automatically admits students with IQ scores above 130. About what percentage is automatically admitted to the program? Sketch the normal curve. Your sketch should include the mean, the number 130, and the area that represents the percentage admitted.

  • Based on the sketch that I made, 2.55% is automatically admitted to the program.

  1. What percentage of the population is beyond three standard deviations? What about just the right side?

  • The 0.3% of the population is beyond three standard deviations. Just the right side would be 0.15% of the population.

Group Work (Hard Questions)

Apply the empirical rule in the following problems, and sketch the curves.

  1. Explain how to determine the amount 13.5% seen in the graph.

  • The amount 13.5% that's seen in the graph is two standard deviations away from the mean.

  1. Women’s heights have a mean of 165 cm and a standard deviation of 6.5 cm. Typical heights fall within one standard deviation of the mean. a. What is the probability that a randomly selected woman has a typical height in this distribution?

  • The probability that a randomly selected woman has a typical height in this distribution would be 68%.

b. What are typical heights? (Give a lower and an upper bound to define an interval of typical heights.)

  • "Typical" heights would range from 158.5 cm to 171.5 cm.

  1. A tall woman with a height of 184.5 cm is nearly 6’ 1” tall. What is the probability that a randomly selected woman is taller than 184.5 cm?

  • The probability that a randomly selected woman is taller than 184.5 cm would be 0.15%.

  1. A pant manufacturer makes pants that fit women with heights between 145.5 cm and 184.5 cm. What is the probability that a randomly selected woman will be able to wear pants from this manufacturer?

  • The probability that a randomly selected woman will be able to wear pants from this manufacturer is 99.7%.

  1. A pant manufacturer makes sweat pants in sizes S, M, L. Small pants (S) are designed to fit women with heights less than 152 cm. What is the probability that a randomly selected woman will wear size S?

  • The probability that a randomly selected woman would wear a size S would be 2.55%.

  1. Make up a probability problem in the context of women’s heights that cannot be solved using the 68- 95-99.7 Empirical Rule. Explain why your problem cannot be solved using the Empirical Rule.

  • A dress manufacturer makes t-shirt dresses for women that have the height of 185 cm and up only. What is the probability that a randomly selected woman will be able to wear pants from this manufacturer?

  • My problem cannot be solved using the Empirical Rule because it has nothing to do with the Rule and the probability would be very small. It won't include those who are way less or way above "typical."

  1. People often superimpose a normal curve on discrete data by calculating the mean and standard deviation from the data, and using these as the mean and standard deviation for the normal. They then use the normal curve to estimate probabilities for the original data. Sometimes, teachers use this approach to assign letter grades to exams. It is called “grading on the curve,” and it works something like what is described in the quote below. Using this scheme, and assuming the scores themselves appear normal, about what percent of the students will receive each letter grade?

  • To receive an A, the percent of the students would be 16.05%. To receive a B, the percent of the students would be 34%. To receive a C, the percent of the students would be 34% also. To receive a D, the percent of the students would be 13.5%. To receive a F, the percent of the students would be 2.55%.

The instructor calculates the mean m and standard deviation s of the exam scores. Assuming the exam scores looked bell-shaped, the instructor assigns grades as follows. Students with scores above m + s receive an A. Those with scores between m and m + s receive a B. Those between m − s and m receive a C. Between m − 2s and m − s a D. Finally, those whose scores were below m − 2s receive an F.