Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 1 of 105 Chapter 7 Probability and Statistics

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 2 of 105 Outline 7.1 Visual Representations of Data 7.2 Frequency and Probability Distributions 7.3 Binomial Trials 7.4 The Mean 7.5 The Variance and Standard Deviation 7.6 The Normal Distribution 7.7 Normal Approximation to the Binomial Distribution

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 3 of Visual Representations of Data 1.Statistics 2.Frequency Table 3.Graphical Representations  Bar Chart, Pie Chart, and Histogram 4.Median and Quartiles 5.Box Plots 6.Interquartile Range and Five-Number Summary

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 4 of 105 Statistics Statistics is the branch of mathematics that deals with data: their collection, description, analysis, and use in prediction. Data can be presented in raw form or organized and displayed in tables or charts.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 5 of 105 Frequency Table A table like the one below is called a frequency table since it presents the frequency with which each response occurs.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 6 of 105 Bar Chart This graph shows the same data as the previous example as a bar chart.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 7 of 105 Pie Charts The pie chart consists of a circle subdivided into sectors, where each sector corresponds to a category. The area of each sector is proportional to the percentage of items in that category. This is accomplished by making the central angle of each sector equal to 360 times the percentage associated with the segment.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 8 of 105 Pie Charts (2) The pie chart of the data of the previous example is:

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 9 of 105 Histogram When the data is numeric data, then it can be represented by a histogram which is similar to a bar chart but there is no space between the bars.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 10 of 105 Example Frequency Table & Histogram The grades for the first quiz in a class of 25 students are (a) Organize the data into a frequency table. (b) Create a histogram for the data.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 11 of 105 Example Frequency Table & Histogram (a) GradeNumber

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 12 of 105 Example Frequency Table & Histogram (b)

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 13 of 105 Median and Quartiles The median of a set of numerical data is the data point that divides the bottom 50% of the data from the top 50%. To find the median of a set of N numbers, first arrange the numbers in increasing or decreasing order. The median is the middle number if N is odd and the average of the two middle numbers if N is even. The quartiles are the medians of the sets of data below and above the median.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 14 of 105 Example Median and Quartiles For the grade data given, (a) find the median; (b) find the quartiles. GradeNumber

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 15 of 105 Example Median and Quartiles (2) N = 25 so median is the 13 th grade: 7. There are 12 grades in the lower and upper halves. The upper quartile is the average of the 19 th and 20 th grade: Q 3 = (9 + 9)/2 = 9. The lower quartile is the average of the 6 th and 7 th grade: Q 1 = (5 + 5)/2 = 5. GradeNumber

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 16 of 105 Box Plot Graphing calculators can display a picture, called a box plot, that analyzes a set of data and shows not only the median, but also the quartiles, lowest data point (min) and largest data point (max).

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 17 of 105 Example Box Plot GradeNumber For the grade data given, find the box plot

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 18 of 105 Interquartile Range & Five-Number Summary The length of the rectangular part of the box plot, which is Q 3 - Q 1, is called the interquartile range. The five pieces of information, min, max, Q 2 = median, Q 1 and Q 3 are called the five-number summary.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 19 of 105 Example Box Plot For the grade data from the previous example, list the five- number summary and the interquartile range min = 0, Q 1 = 5, Q 2 = median = 7, Q 3 = 9, max = 10 Interquartile range is Q 3 - Q 1 = = 4.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 20 of 105  Bar charts, pie charts, histograms, and box plots help us turn raw data into visual forms that often allow us to see patterns in the data quickly.  The median of an ordered list of data is a number with the property that the same number of data items lie above it as below it. For an ordered list of N numbers, it is the middle number when N is odd, and the average of the two middle numbers when N is even. Summary Section Part 1

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 21 of 105  For an ordered list of data, the first quartile Q 1 is the median of the list of data items below the median, and the third quartile Q 3 is the median of the list of data items above the median. The difference of the third and first quartiles is called the interquartile range. The sequence of numbers consisting of the lowest number, Q 1, the median, Q 3, and the highest number is called the five-number summary. Summary Section Part 2

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 22 of Frequency and Probability Distributions 1.Frequency Distribution & Relative Frequency Distribution 2.Histogram of Probability Distribution 3.Probability of an Event in Histogram 4.Random Variable 5.Probability Distribution of a Random Variable

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 23 of 105 Frequency Distribution & Relative Frequency Distribution A table that includes every possible value of a statistical variable with its number of occurrences is called a frequency distribution. If instead of recording the number of occurrences, the proportion of occurrences are recorded, the table is called a relative frequency distribution.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 24 of 105 Example Distributions Two car dealerships provided a potential buyer sales data. Dealership A provided 1 year's worth of data and dealership B, 2 years' worth. Convert the following data to a relative frequency distribution.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 25 of 105 Example Distributions (2)

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 26 of 105 Example Distributions Put the relative frequency distributions of the previous example into a histogram. Note: The area of each rectangle equals the relative frequency for the data point.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 27 of 105 Histogram of Probability Distribution The histogram for a probability distribution is constructed in the same way as the histogram for a relative frequency distribution. Each outcome is represented on the number line, and above each outcome we erect a rectangle of width 1 and of height equal to the probability corresponding to that outcome.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 28 of 105 Example Probability Distribution Construct the histogram of the probability distribution for the experiment in which a coin is tossed five times and the number of occurrences of heads is recorded.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 29 of 105 Example Probability Distribution (2)

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 30 of 105 Probability of an Event in Histogram In a histogram of a probability distribution, the probability of an event E is the sum of the areas of the rectangles corresponding to the outcomes in E.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 31 of 105 Example Probability of an Event For the previous example, shade in the area that corresponds to the event "at least 3 heads." Area is shaded in blue.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 32 of 105 Random Variable Consider a theoretical experiment with numerical outcomes. Denote the outcome of the experiment by the letter X. Since the values of X are determined by the unpredictable random outcomes of the experiment, X is called a random variable.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 33 of 105 Random Variable (2) If k is one of the possible outcomes of the experiment, then we denote the probability of the outcome k by Pr(X = k). The probability distribution of X is a table listing the various values of X and their associated probabilities p i with p 1 + p 2 + … + p r = 1.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 34 of 105 Example Random Variable Consider an urn with 8 white balls and 2 green balls. A sample of three balls is chosen at random from the urn. Let X denote the number of green balls in the sample. Find the probability distribution of X.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 35 of 105 Example Random Variable (2) There are equally likely outcomes. X can be 0, 1, or 2.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 36 of 105 Example Random Variable & Distribution Let X denote the random variable defined as the sum of the upper faces appearing when two dice are thrown. Determine the probability distribution of X and draw its histogram.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 37 of 105 X = 5 X = 4 X = 3 X = 2 Example Random Variable & Distribution X = 6 X = 7 X = 8 X = 9 X = 10 X = 11 X = 12 The sample space is composed of 36 equally likely pairs. (1,1), (1,2), (1,3), (1,4), (1,5), (1,6) (2,1), (2,2), (2,3), (2,4), (2,5), (2,6) (3,1), (3,2), (3,3), (3,4), (3,5), (3,6) (4,1), (4,2), (4,3), (4,4), (4,5), (4,6) (5,1), (5,2), (5,3), (5,4), (5,5), (5,6) (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 38 of 105 Example Random Variable & Distribution

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 39 of 105  The probability distribution for a random variable can be displayed in a table or a histogram. With a histogram, the probability of an event is the sum of the areas of the rectangles corresponding to the outcomes in the event. Summary Section 7.2

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 40 of Binomial Trials 1.Binomial Trials: Success/Failure 2.Probability of k Successes

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 41 of 105 Binomial Trials: Success/Failure An experiment with just 2 outcomes is called a binomial trial (or Bernoulli trial.) One outcome is labeled a success and the other is labeled a failure. If p is the probability of success, the q = 1 - p is the probability of failure.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 42 of 105 Example Binomial Trials Examples of binomial trials: 1. Toss a coin and observe the outcome, heads or tails. 2. Administer a drug to a sick individual and classify the reaction as "effective" or "ineffective." 3. Manufacture a light bulb and classify it as "nondefective" or "defective."

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 43 of 105 Probability of k Successes If X is the number of "successes" in n independent trials, where in each trial the probability of a "success" is p, then for k = 0, 1, 2,…, n and q = 1 - p.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 44 of 105 Example Probability of k Successes Each time at bat the probability that a baseball player gets a hit is.300. He comes up to bat four times in a game. Assume that his times at bat are independent trials. Find the probability that he gets (a) exactly 2 hits and (b) at least 2 hits.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 45 of 105 Example Probability of k Successes (a) Each at-bat is considered an independent binomial trial. A "success" is a hit. So p =.300, q =.700 and n = 4. X is the number of hits in four at-bats. (a)(a)

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 46 of 105 Example Probability of k Successes (b) "At least two hits" means X > 2.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 47 of 105 Summary Section 7.3  If the probability of success in each trial of a binomial experiment is p, then the probability of k successes in n trials is where q = 1 – p.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 48 of The Mean 1.Population Versus Sample 2.Statistic Versus Parameter 3.Mean (Average) of a Sample 4.Mean (Average) of a Population 5.Expected Value 6.Expected Value of Binomial Trial

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 49 of 105 Population Versus Sample A population is a set of all elements about which information is desired. A sample is a subset of a population that is analyzed in an attempt to estimate certain properties of the entire population.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 50 of 105 Example Population Versus Sample A clothing manufacturer wants to know what style of jeans teens between 13 and 16 will buy. To help answer this question, 200 teens between 13 and 16 were surveyed. The population is all teens between 13 and 16. The sample is the 200 teens between 13 and 16 surveyed.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 51 of 105 Statistic Versus Parameter A numerical descriptive measurement made on a sample is called a statistic. Such a measurement made on a population is called a parameter of the population. Since we cannot usually have access to entire populations, we rely on our experimental results to obtain statistics, and we attempt to use the statistics to estimate the parameters of the population.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 52 of 105 Mean (Average) of a Sample Let an experiment have as outcomes the numbers x 1, x 2, …, x r with frequencies f 1, f 2,…, f r, respectively, so that f 1 + f 2 +…+ f r = n. Then the sample mean equals or

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 53 of 105 Mean (Average) of a Population If the population has x 1, x 2,…, x r with frequencies f 1, f 2,…, f r, respectively. Then the population mean equals or Note: Greek letters are used for parameters.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 54 of 105 Example Mean An ecologist observes the life expectancy of a certain species of deer held in captivity. The table shows the data observed on a population of 1000 deer. What is the mean life expectancy of this population?

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 55 of 105 Example Mean (2) The relative frequencies are given in the table.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 56 of 105 Expected Value The expected value of the random variable X which can take on the values x 1, x 2,…,x N with Pr(X = x 1 ) = p 1, Pr(X = x 2 ) = p 2,…, Pr(X = x N ) = p N is E(X) = x 1 p 1 + x 2 p 2 + …+ x N p N.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 57 of 105 Expected Value (2) The expected value of the random variable X is also called the mean of the probability distribution of X and is also designated by The expected value of a random variable is the center of the probability distribution in the sense that it is the balance point of the histogram.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 58 of 105 Example Expected Value Five coins are tossed and the number of heads observed. Find the expected value.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 59 of 105 Example Expected Value

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 60 of 105 Expected Value of Binomial Trial If X is a binomial random variable with parameters n and p, then E(X) = np.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 61 of 105 Example Expected Value Binomial Trial Five coins are tossed and the number of heads observed. Find the expected value. A "success" is a head and p =.5. The number of trials is n = 5. E(X) = np = 5(.5) = 2.5

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 62 of 105 Fair Game The expected value of a completely fair game is zero.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 63 of 105 Example Fair Game Two people play a dice game. A single die is thrown. If the outcome is 1 or 2, then A pays B $2. If the outcome is 3, 4, 5, or 6, then B pays A $4. What are the long-run expected winnings for A? X represents the payoff to A. Therefore, X is either -2 or 4.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 64 of 105 Example Fair Game (2) Pr(X = -2) = 2/6 = 1/3 Pr(X = 4) = 4/6 = 2/3 E(X) = -2(1/3) + 4(2/3) = 2 On average, A should expect to win $2 per play. If A paid B $4 on 1 and 2 but B paid A $2 on 3, 4, 5, and 6, then E(X) = 0 and the game would be fair.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 65 of 105 Summary Section 7.4  The sample mean of a sample of n numbers is the sum of the numbers divided by n.  The expected value of a random variable is the sum of the products of each outcome and its probability.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 66 of The Variance and Standard Deviation 1.Variance of Probability Distribution 2.Spread 3.Standard Deviation 4.Unbiased Estimate 5.Sample Variance and Standard Deviation 6.Alternative Definitions 7.Chebychev's Inequality

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 67 of 105 Variance of Probability Distribution Let X be a random variable with values x 1, x 2, …, x N and respective probabilities p 1, p 2,…, p N. The variance of the probability distribution is

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 68 of 105 Spread Roughly speaking, the variance measures the dispersal or spread of a distribution about its mean. The probability distribution whose histogram is drawn on the left has a smaller variance than that on the right.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 69 of 105 The standard deviation of probability distribution is Standard Deviation of Probability Distribution

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 70 of 105 Example Variance & Standard Deviation Compute the variance and the standard deviation for the population of scores on a five-question quiz in the table.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 71 of 105 Example Variance & Standard Deviation (2)

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 72 of 105 Unbiased Estimate If the average of a statistic, if that statistic were computed for each sample, equals the associated parameter for the population, then that statistic is said to be unbiased.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 73 of 105 Sample Variance and Standard Deviation The unbiased variance for a sample is The unbiased standard deviation for a sample is

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 74 of 105 Example Variance & Standard Deviation Compute the sample variance and standard deviation for the weekly sales of car dealership A.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 75 of 105 Example Variance & Standard Deviation (2)

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 76 of 105 Alternative Definitions Two alternative definitions for variance are and, for a binomial random variable with parameters n, p, and q,

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 77 of 105 Example Alternative Definition Find the variance when a fair coin is tossed 5 times and X is the number of heads.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 78 of 105 Chebychev's Inequality Chebychev's InequalitySuppose that a probability distribution with numerical outcomes has expected value and standard deviation Then the probability that a randomly chosen outcome lies between - c and + c is at least

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 79 of 105 Example Chebychev's Inequality A drug company sells bottles containing 100 capsules of penicillin. Due to bottling procedure, not every bottle contains exactly 100 capsules. Assume that the average number of capsules in a bottle is 100 and the standard deviation is 2. If the company ships 5000 bottles, estimate the number of bottles having between 95 and 105 capsules, inclusive.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 80 of 105 Example Chebychev's Inequality (2) The number of bottles containing between and capsules can be estimated using Chebychev's Inequality. On average, at least 84% will contain between 95 and 105 capsules.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 81 of 105 Summary Section Part 1  The variance of a random variable is the sum of the products of the square of each outcome's distance from the expected value and the outcome's probability. The variance of the random variable X can also be computed as E(X 2 ) - [E(X)] 2.  A binomial random variable with parameters n and p has expected value np and variance np(1 - p).

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 82 of 105 Summary Section Part 2  The square root of the variance is called the standard deviation.  Chebychev's Inequality states that the probability that an outcome of an experiment is within c units of the mean is at least, where is the standard deviation.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 83 of The Normal Distribution 1.Normal Curve 2.Normally Distributed Outcomes 3.Properties of Normal Curve 4.Standard Normal Curve 5.The Normal Distribution 6.Percentile 7.Probability for General Normal Distribution

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 84 of 105 Normal Curve The bell-shaped curve, as shown below, is call a normal curve.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 85 of 105 Normally Distributed Outcomes Examples of experiments that have normally distributed outcomes: 1. Choose an individual at random and observe his/her IQ. 2. Choose a 1-day-old infant and observe his/her weight. 3. Choose a leaf at random from a particular tree and observe its length.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 86 of 105 Properties of Normal Curve

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 87 of 105 Example Properties of Normal Curve A certain experiment has normally distributed outcomes with mean equal to 1. Shade the region corresponding to the probability that the outcome (a) lies between 1 and 3; (b) lies between 0 and 2; (c) is less than.5; (d) is greater than 2.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 88 of 105 Example Properties of Normal Curve (2)

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 89 of 105 Standard Normal Curve The equation of the normal curve is The standard normal curve has

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 90 of 105 The Normal Distribution A(z) is the area under the standard normal curve to the left of a normally distributed random variable z.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 91 of 105 Example The Normal Distribution Use the normal distribution table to determine the area corresponding to (a) z < -.5; (b) 1< z < 2; (c) z > 1.5.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 92 of 105 Example The Normal Distribution (2) (a) A(-.5) =.3085 (b) A(2) - A(1) = =.1359 (c) 1 - A(1.5) = =.0668

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 93 of 105 Percentile If a score S is the p th percentile of a normal distribution, then p% of all scores fall below S, and (100 - p)% of all scores fall above S. The p th percentile is written as z p.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 94 of 105 Example Percentile What is the 95 th percentile of the standard normal distribution? In the normal distribution, find the value of z such that A(z) =.95. A(1.65) =.9506 Therefore, z 95 = 1.65.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 95 of 105 Probability for General Normal Distribution If X is a random variable having a normal distribution with mean and standard deviation then where Z has the standard normal distribution and A(z) is the area under that distribution to the left of z.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 96 of 105 Example Probability Normal Distribution Find the 95 th percentile of infant birth weights if infant birth weights are normally distributed with = 7.75 and = 1.25 pounds. The value for the standard normal random variable is z 95 = Then x 95 = (1.65)(1.25) = 9.81 pounds.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 97 of 105 Summary Section Part 1  A normal curve is identified by its mean ( ) and its standard deviation ( ). The standard normal curve has = 0 and = 1. Areas of the region under the standard normal curve can be obtained with the aid of a table or graphing calculator.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 98 of 105 Summary Section Part 2  A random variable is said to be normally distributed if the probability that an outcome lies between a and b is the area of the region under a normal curve from x = a to x = b. After the numbers a and b are converted to standard deviations from the mean, the sought-after probability can be obtained as an area under the standard normal curve.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 99 of Normal Approximation to the Binomial Distribution 1.Normal Approximation

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 100 of 105 Suppose we perform a sequence of n binomial trials with probability of success p and probability of failure q = 1 - p and observe the number of successes. Then the histogram for the resulting probability distribution may be approximated by the normal curve with = np and Normal Approximation

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 101 of 105 Normal Approximation Binomial distribution with n = 40 and p =.3

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 102 of 105 Example Normal Approximation A plumbing-supplies manufacturer produces faucet washers that are packaged in boxes of 300. Quality control studies have shown that 2% of the washers are defective. What is the probability that more than 10 of the washers in a single box are defective?

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 103 of 105 Let X = the number of defective washers in a box. X is a binomial random variable with n = 300 and p =.02. We will use the approximating normal curve with  = 300(.02) = 6 and Since the right boundary of the X = 10 rectangle is 10.5, we are looking for Pr(X > 10.5). Example Normal Approximation (2)

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 104 of 105 The area of the region to the right of 1.85 is 1 - A(1.85) = = Therefore, 3.22% of the boxes should contain more than 10 defective washers. Example Normal Approximation (3)

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 105 of 105 Summary Section 7.7  Probabilities associated with a binomial random variable with parameters n and p can be approximated with a normal curve having  = np and