Modeling Random Events: The Normal and Binomial Models Chapter 6 Modeling Random Events: The Normal and Binomial Models Copyright © 2014 Pearson Education, Inc. All rights reserved

Learning Objectives Be able to distinguish between discrete and continuous-valued variables. Know when a Normal model is appropriate and be able to apply the model to find probabilities. Know when the binomial model is appropriate and be able to apply the model to find probabilities.

Probability Distributions Are Models of Random Experiments 6.1 Probability Distributions Are Models of Random Experiments Copyright © 2014 Pearson Education, Inc. All rights reserved

Probability Models and Distributions A Probability Model is a description of how a statistician thinks data are produced. Uniform Linear Normal Other A Probability Distribution or Probability Distribution Function (pdf) is a table graph or formula that gives all the outcomes of an experiment and their probabilities.

Discrete vs. Continuous A random variable is called Discrete if the outcomes are values that can be listed or counted. Number of classes taken The roll of a die A random variable is called Continuous if the outcomes cannot be listed because they occur over a range. Time to finish the exam Exact weight

Discrete or Continuous Classify the following as discrete or continuous: Length of the left thumb Number of children in a the family Number of devices in the house that connect to the Internet Sodium concentration in the bloodstream →Continuous →Discrete →Discrete →Continuous

Discrete Probability Distributions The most common way to display a pdf for discrete data is with a table. The probability distribution table always has two columns (or rows). The first, x, displays all the possible outcomes The second, P(x), displays the probabilities for these outcomes.

Examples of Probability Distribution Tables Die Roll Raffle Prize x P(x) 1 1/6 2 3 4 5 6 x P(x) 95 0.01 995 0.005 -5 0.985 The sum of all the probabilities must equal 1.

Examples of Probability Distribution Graphs

Examples of Probability Distribution Functions If 1/3 of the population is Hispanic and 20 randomly selected people from the population are chosen, then the probability distribution function for the number of Hispanics selected is Probability distribution functions can be quite complicated!

Continuous Data and Probability Distribution Functions Often represented as a curve. The area under the curve between two values of x represents the probability of x being between these two values. The total area under the curve must equal 1. The curve cannot lie below the x-axis.

Finding Probabilities for Uniform Distributions The curve above shows the probability distribution function for time to wait for a bus that comes every 12 minutes. Find the probability that you will wait between 5 and 10 minutes. Shade in the area. Find the area of the rectangle P(5 < x <10) = Base x Height = 5 x 0.8333 = 0.41655

6.2 The Normal Model Copyright © 2014 Pearson Education, Inc. All rights reserved

The Normal Model The Normal Model is a good model if: The distribution is unimodal. The distribution is approximately symmetric. The distribution is approximately bell shaped. The Normal Distribution is also called Gaussian.

Center and Spread of the Normal Distribution m stands for the center or mean of a distribution. s stands for the standard deviation of a distribution Note that the Greek letters m and s are used for distributions and x and s are used for sample data.

Notation and Area N(6,2) means the normal distribution with mean m = 6 and standard deviation s = 2. The area under the normal curve, above the x-axis, and to the left of x = 4 represents P(x < 4). P(x < 4) = P(x ≤ 4) for a continuous variable.

Probability and StatCrunch Draw a rough sketch of the normal curve along with the mean and area to be found. Go to Stat→ Calculators→ Normal. Type in the mean and Choose <= or >=, and the Std. Dev. Type in the value of x. Click on “Compute”.

Using StatCrunch for Probability

Example: Baby Seals Research has shown that the mean length of a newborn Pacific harbor seal is 29.5 in. and that s = 1.2 in. Suppose that the lengths follow the Normal model. Find the probability that a randomly selected pup will be more than 32 in. P(x > 32) ≈ 0.019

Finding the Probability Between Two Values Since StatCrunch can only handle probabilities involving “≤” and “≥”, use geometry to find the probability of x falling between two values. P(3 < x < 6) = P(x < 6) – P(x < 3) ≈ 0.9772 – 0.1587 = 0.8185

Finding Probabilities from Percentiles Newborn seals in the bottom 10th percentile will probably not survive. Given that new born seals are N(29.5,1.2), find the 10th percentile. With StatCrunch put in 0.1 for the probability and click on Compute. The 10th percentile is about 27.96 in.

The Normal Model and the Empirical Rule The Empirical Rule told us that if a distribution is approximately normal, then 68% of the data will fall within 1 standard deviation of the mean, 95% within 2, and 99.7% within 3. If the distribution is exactly normal, then these numbers are just the corresponding areas under the normal curve.

6.3 The Binomial Model Copyright © 2014 Pearson Education, Inc. All rights reserved

The Binomial Model The Binomial Model applies if: There are a fixed number of trials. Only two outcomes are possible for each trial: Yes or No, Success or Failure, Heads or Tails, etc. The probability of success, p, is the same for each trial. The trials are independent.

Binomial or Not? 40 randomly selected college students were asked if they selected their major in order to get a good job. Binomial 35 randomly selected Americans were asked what country their mothers were born. Not Binomial, more than two possible answers per trial. To estimate the probability that students will pass an exam, the professor records a study group’s success on the exam. Not Binomial, since the outcomes are not independent.

Surveys and Independence The Binomial Model may be used if the respondents of a survey are selected with replacement. If the selection is done without replacement and the population size is at least 10 times larger than the sample size, then we may still use the Binomial Model as an approximation.

Visualizing the Binomial Distribution If n is large and p is close to 0.5, then the binomial distribution is approximately normal

Words and Inequalities Notice that “Less Than” and “At Least” are complements and “More Than” and “At Most” are Complements. Exactly Less Than At Least More Than At Most → = → < → => → > → <=

Finding a Binomial Probability 12% of all US women will eventually develop breast cancer. If 30 women are randomly selected, what is the probability that exactly 4 of them will eventually develop breast cancer? b(30,0.12,4) = ? StatCrunch: Stat→Calculators→Binomial n = 30, p = 0.12, P(x = 4) The probability that exactly 4 of the 30 women will develop breast cancer is about 0.20.

Finding a Binomial Probability 14% of all clothing bought online is returned. If an online retailer sells 35 items of clothing, what is the probability that: At least 5 will be returned? n = 35, p = .14, P(x => 5) ≈ 0.55 Fewer than 7 will be returned? n = 35, p = .14, P(x < 7) ≈ 0.79 More than 6 will be returned? n = 35, p = .14, P(x > 6) ≈ 0.21 At most 4 will be returned? n = 35, p = .14, P(x <= 4) ≈ 0.45

The Expected Value If we roll a six sided die 30 times then we would expect to roll a two 30 x 1/6 = 5 times. For a Binomial Distribution, m = np is called the mean or the expected value. A Binomial Distribution with n trials and probability of success p has standard deviation

The Standard Deviation On any particular day, there is a 6% chance of a fatal accident in the city. Find the mean and standard deviation for the number of accidents in a (365 day) year. m = np = 365 x 0.06 = 21.9 We expect about 22 fatal accidents per year give or take four or five accidents.

Chapter 6 Case Study Copyright © 2014 Pearson Education, Inc. All rights reserved

Too Heavy or Not? McDonald’s claims that its ice cream cones weigh 3.18 ounces. However, one of the authors bought five cones and found that all five weighed more than that. Is this surprising? Assume the ice cream weights are normally distributed or at least symmetric about the mean.

McDonald’s claims that its ice cream cones weigh 3. 18 ounces McDonald’s claims that its ice cream cones weigh 3.18 ounces. However, one of the authors bought five cones and found that all five weighed more than that. Is this surprising? n = 5, p = 0.5 P(x = 5) ≈ 0.031 If the mean is 3.18 ounces as McDonald’s claims, then there is only a 3.1% chance that out of 5 randomly chosen cones, they will all weigh above 3.18 ounces. With such a small probability, suspicion is raised.

Chapter 6 Guided Exercise 1 Copyright © 2014 Pearson Education, Inc. All rights reserved

SAT Scores According to data from the College Board, the mean quantitative SAT score for female college-bound high school seniors in 2009 was 500. SAT scores are approximately Normally distributed with a population standard deviation of 100. What percentage of the female college-bound high school seniors had scores above 675?

N(500,100) To find the z-score for 675, subtract the mean and divide by the standard deviation. Report the z-score.

N(500,100) Refer to the Normal curve. Explain why the SAT score of 500 is right below the z-score of 0. The dots on the axis mark the location of z-scores that are integers from -3 to 3. The mean is always at the center of the Normal curve. The mean SAT score is 500 and the mean z-score is 0 by definition.

N(500,100) Carefully sketch a copy of the curve. Pencil in the SAT scores of 200, 300, 400, 600, and 700 in the correct places. Notice that the mean is 500 and the standard deviation is 100. The z-scores are -3, -2, -1, and 1.

N(500,100) Draw a vertical line through the curve at 675. Just above the 675 (indicated on the graph with “???”). Put in the corresponding z-score. We want to find what percentage of students had scores above 675. Shade the area to the right of this boundary, because numbers to the right are larger.

N(500,100) Use StatCrunch to find the area to the right of this z-score P(z > 1.75) ≈ 0.04

N(500,100) P(z > 1.75) ≈ 0.04 Finally, write a sentence telling what you found. About 4% of the female college-bound high school seniors had scores above 675.

Chapter 6 Guided Exercise 2 Copyright © 2014 Pearson Education, Inc. All rights reserved

SAT Scores According to data from the College Board, the mean quantitative SAT score for female college-bound high school seniors in 2009 was 500. SAT scores are approximately Normally distributed with a population standard deviation of 100. A scholarship committee wants to give awards to college-bound women who score at the 96th percentile or above on the SAT. What score does an applicant need?

SAT Scores: N(500,100) A scholarship committee wants to give awards to college-bound women who score at the 96th percentile or above on the SAT. What score does an applicant need? Will the SAT test score be above the mean or below it? Explain. The 96th percentile is the score such that 96% of all scores are at or below this score. The 50th percentile is the mean, so the 96th percentile is above the mean.

SAT Scores: N(500,100) Label the curve with integer z-scores. The dots represent the position of integer z-scores from -3 to 3.

SAT Scores: N(500,100) Use StatCrunch to find the z-score that has area to the left 0.96. z ≈ 1.75

SAT Scores: N(500,100) , z ≈ 1.75 Add that z-score to the sketch and draw a vertical line above it through the curve. Shade the left side because the area to the left is what is given.

SAT Scores: N(500,100) , z ≈ 1.75 Find the SAT score that corresponds to the z-score. The score should be z standard deviations above the mean, so x = m + zs = 500 + (1.75)(100) = 675

SAT Scores: N(500,100) , z ≈ 1.75, x = 675

SAT Scores: N(500,100) , z ≈ 1.75, x = 675 Finally, write a sentence stating what you found. The applicant needs a score of at least 675 to receive the scholarship.