Chapter Six Probability Distributions

Transcription

1 6.1 Probability Distributions Discrete Random Variable Chapter Six Probability Distributions x P(x) Practice. Construct a probability distribution for the number of dogs each person has in the classroom. Mean or Expected Value Standard Deviation Binomial Distribution Four Characteristics 1.) 2.) 3.) 4.)

2 Practice. Determine if the following experiments represents a binomial experiment. If not, explain why. 1.) A random sample of 15 college seniors is obtained, and the individuals selected are asked to state their ages. 2.) An experimental drug is administered to 100 randomly selected individuals, with the number of individuals responding favorably recorded. 3.) A state lottery randomly selects 6 balls numbered 1 to 40. You choose six numbers and purchase a lottery ticket. The random variable represents the number of matches on your ticket to the numbers drawn in the lottery. 4.) From past records, a clothing store finds that 26% of people who enter the store will make a purchase. During a one-hour period, 18 people enter the store. The random variable represents the number of people who do not make a purchase. Formula P(x) ( n C x )(p x )(q n x ) n = p = q = x = Practice. Find the indicated probabilities. 5.) A surgical technique is performed on seven patients. You are told there is a 70% chance of success. Find the probability that the surgery is successful for a.) exactly five patients. b.) at least five patients. c.) less than five patients.

3 6.) Ten percent of adults say oatmeal raisin is their favorite cookie. You randomly select 12 adults and ask each to name his or her favorite cookie. Find the probability that the number who say oatmeal raisin is their favorite cookie is a.) exactly three. b.) at least three. c.) less than three. Mean Standard Deviation

4 The Normal Distribution (review) Chapter Seven The Normal Distribution The Standard Normal Distribution Mean = Standard Deviation = z-score - Area Under the Curve Area to the Left Practice. Find the following areas under the standard normal curve. 1.) To the left of z = ) To the left of z = ) To the left of z = -1.04

5 Area to the Right Practice. Find the following areas under the standard normal curve. 4.) To the right of z = ) To the right of z = ) To the right of z = Area Between Practice. Find the following areas under the standard normal curve. 7.) Between z = 0 and z = ) Between z = and z = ) Between z = and z = 0.04

6 Practice. Find the following probabilities. 10.) P(z 1.15) 11.) P(z 2.56) 12.) P( 2.71 z 1.5) 13.) P(z 3.88) 14.) P(z 4.71) The Normal Distribution (Again) Mean = Standard Deviation = x

7 Convert from a Normal Distribution to a Standard Normal Distribution z x Practice. Find the following probabilities. 1.) The weights of adult male beagles are normally distributed, with a mean of 25 pounds and a standard deviation of 3 pounds. A beagle is randomly selected. a.) Find the probability that the beagle s weight is less than 23 pounds. b.) Find the probability that the beagle s weight is between 23 and 25 pounds. c.) Find the probability that the beagle s weight is more than 27 pounds.

8 2.) The monthly utility bills in a city are normally distributed, with a mean of $100 and a standard deviation of $12. A utility bill is randomly selected. a.) Find the probability that the utility bill is less than $70. b.) Find the probability that the utility bill is between $90 and $120. c.) Find the probability that the utility bill is more than $ Backwards curve. Practice. Find the z-score(s) that corresponds to the given area under the 1.) The area to the left is

9 2.) The area to the right is ) The area to the right is ) The area to the left is ) P 42 6.) P 90 7.) P 5

10 Converting from a Standard Normal Distribution to a Normal Distribution x z σ μ 7.) The weights of the contents of a cereal box are normally distributed with a mean weight of 20 ounces and a standard deviation of 0.07 ounce. Boxes in the lower 5% do not meet the minimum weight requirements and must be repackaged. What is the minimum weight requirement for a cereal box? 8.) In a survey of men in the U.S. (ages 20-29), the mean height was 69.2 inches with a standard deviation of 2.9 inches. What height represents the 80 th percentile? 9.) On a dry surface, the braking distance (in meters) of a Pontiac Grand AM SE can be approximated by a normal distribution with a mean of 45.1 m and a standard deviation of 0.5 m. What is the shortest braking distance of a Pontiac Grand AM SE that can be in the top 15% of braking distances?

11 7.4 - The Central Limit Theorem - Example. Replacement times for TV sets are normally distributed with a mean of 8.2 years and a standard deviation of 2.1 years. a.) Find the probability that a randomly selected TV will have a replacement time less than 7 years. b.) If 28 TVs are randomly selected, find the probability that their sample mean is less than 7 years. The Central Limit Theorem Used when finding the probability of the mean of a sample. x μ x z where μ μ and σ x x σ x σ n or x μ z σ n

12 Practice. Find the following probabilities. 1.) Cans of Coca-Cola are filled so that the actual amount have a mean of 12.0 oz and a standard deviation of 0.31 oz. Find the probability that sample of 36 cans will have a mean amount of at least oz. 2.) Standardized scores are normally distributed with a mean of 550 and a standard deviation of 110. If 25 test scores are randomly chosen, find the probability the mean score is between 385 and ) The reading speed of second grade students is approximately normal, with a mean of 90 words per minute (wpm) and a standard deviation of 10 wpm. a.) What is the probability a randomly selected student will read more than 95 words per minute?

13 b.) What is the probability that a random sample of 14 second grade students results in a mean reading rate of less than 95 words per minute? c.) What is the probability that a random sample of 34 second grade students results in a mean reading rate more than 95 words per minute?