Goldstein/Schnieder/Lay: Finite Math & Its Applications, 9e 1 of 105 Chapter 7 Probability and...

Goldstein/Schnieder/Lay: Finite Math & Its Applications, 9e 1 of 105

Chapter 7

Probability and Statistics


Outline

7.1 Visual Representations of Data7.2 Frequency and Probability Distributions7.3 Binomial Trials7.4 The Mean7.5 The Variance and Standard Deviation7.6 The Normal Distribution7.7 Normal Approximation to the Binomial

Distribution


7.1 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


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.


Frequency Table

A table like the one below is called a frequency table since it presents the frequency with which each response occurs.


Bar Chart

This graph shows the same data as the previous example as a bar chart.


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.


Pie Charts (2)

The pie chart of the data of the previous example is:


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.


Example Frequency Table & Histogram

The grades for the first quiz in a class of 25 students are

8 7 6 10 5 10 7 1 8 0

10 5 9 3 8 6 10 4 9 10

7 0 9 5 8.

(a) Organize the data into a frequency table.

(b) Create a histogram for the data.


Example Frequency Table & Histogram (a)

Grade Number

10 5

9 3

8 4

7 3

6 2

5 3

4 1

3 1

2 0

1 1

0 2


Example Frequency Table & Histogram (b)


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.


Example Median and Quartiles

For the grade data given,

(a) find the median;

(b) find the quartiles.

Grade Number

10 5

9 3

8 4

7 3

6 2

5 3

4 1

3 1

2 0

1 1

0 2


Example Median and Quartiles (2)

N = 25 so median is the 13th grade: 7.There are 12 grades in the lower and upper halves. The upper quartile is the average of the 19th and 20th grade: Q3 = (9 + 9)/2 = 9.The lower quartile is the average of the 6th and 7th grade:Q1 = (5 + 5)/2 = 5.

Grade Number

10 5

9 3

8 4

7 3

6 2

5 3

4 1

3 1

2 0

1 1

0 2


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).


Example Box Plot

Grade Number

10 5

9 3

8 4

7 3

6 2

5 3

4 1

3 1

2 0

1 1

0 2

For the grade data given,

find the box plot.

0 5 7 9 10


Interquartile Range & Five-Number Summary

The length of the rectangular part of the box plot, which is Q3 - Q1, is called the interquartile range.

The five pieces of information, min, max, Q2 = median, Q1 and Q3 are called the five-number summary.


Example Box Plot

For the grade data from the previous example, list the five-number summary and the interquartile range.

0 5 7 9 10

min = 0, Q1 = 5, Q2 = median = 7, Q3 = 9, max = 10

Interquartile range is Q3 - Q1 = 9 - 5 = 4.


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 7.1 - Part 1


For an ordered list of data, the first quartile Q1 is the median of the list of data items below the median, and the third quartile Q3 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, Q1, the median, Q3, and the highest number is called the five-number summary.



7.2 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


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.


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.


Example Distributions (2)


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.


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.


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.


Example Probability Distribution (2)


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.


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.


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.


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 pi with p1 + p2 + … + pr = 1.


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.


Example Random Variable (2)

There are equally likely outcomes.

X can be 0, 1, or 2.

10120

3N

8 8 2

3 2 17 7Pr( 0) , Pr( 1)

10 1015 15

3 3

8 2

1 2 1Pr( 2)

10 15

3

X X

X


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.


X = 5X = 4X = 3X = 2


X = 6X = 7X = 8X = 9X = 10X = 11X = 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)


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


7.3 Binomial Trials

1. Binomial Trials: Success/Failure

2. Probability of k Successes


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.


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."


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.

Pr( ) k n knX k p q

k


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.


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) 2 4 24Pr( 2) .300 .700 .2646

2X


Example Probability of k Successes

(b) "At least two hits" means X > 2.

2 4 2 3 4 3 4 4 4

Pr( 2) Pr( 2) Pr( 3) Pr( 4)

4 4 4 = .300 .700 .300 .700 .300 .700

2 3 4

.3483

X X X X


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.

k n knp q

k


7.4 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

Goldstein/Schnieder/Lay: Finite Math & Its Applications 49

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.


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.


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.


Mean (Average) of a Sample

Let an experiment have as outcomes the numbers x1, x2, …, xr with frequencies f1, f2,…, fr, respectively, so that f1 + f2 +…+ fr = n. Then the sample mean equals

or

21 1 2 ,r rx f x f x fx

n

2

1 21 .r

r

f f fx x x x

n n n


Mean (Average) of a Population

If the population has x1, x2,…, xr with frequencies f1, f2,…, fr, respectively. Then the population mean equals

or

21 1 2 ,r rx f x f x f

N

2

1 21 .r

r

f f fx x x

N N N

Note: Greek letters are used for parameters.


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?


Example Mean (2)

The relative frequencies are given in the table.

1(0) 2(.06) 3 .18

4 .25 5 .20

6 .12 7 .05

8 .12 9 .02

4.87


Expected Value

The expected value of the random variable X which can take on the values x1, x2,…,xN with

Pr(X = x1) = p1, Pr(X = x2) = p2,…, Pr(X = xN) = pN

is

E(X) = x1p1 + x2p2+ …+ xNpN.


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.

.X


Example Expected Value

Five coins are tossed and the number of heads observed. Find the expected value.


Example Expected Value

5132 32

10 1032 32

5 132 32

0 1

2 3

4 5

2.5

X


Expected Value of Binomial Trial

If X is a binomial random variable with parameters n and p, then

E(X) = np.


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


Fair Game

The expected value of a completely fair game is zero.


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.


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.


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.


7.5 The Variance and Standard Deviation

1. Variance of Probability Distribution2. Spread 3. Standard Deviation4. Unbiased Estimate5. Sample Variance and Standard Deviation6. Alternative Definitions7. Chebychev's Inequality


Variance of Probability Distribution

Let X be a random variable with values x1, x2,

…, xN and respective probabilities p1, p2,…, pN. The variance of the probability distribution is

2 2 2

1 1 2 2variance .N Nx p x p x p


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.


The standard deviation of probability distribution is

2 2 221 1 2 2

2 2 2

1 1 2 22

,

or .

r r

r r

x p x p x p

x f x f x f

N

Standard Deviation of Probability Distribution

2 variance , where


Example Variance & Standard Deviation

Compute the variance and the standard deviation for the population of scores on a five-question quiz in the table.

0 4 1 9 2 6 3 14 4 18 5 93

60


Example Variance & Standard Deviation (2)

2 132Variance 2.2, 2.2 1.48

60


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.


Sample Variance and Standard Deviation

The unbiased variance for a sample is

The unbiased standard deviation for a sample is

2 2 2

1 1 2 22 .1

r rx x f x x f x x fs

n

2 .s s


Example Variance & Standard Deviation

Compute the sample variance and standard deviation for the weekly sales of car dealership A.


Example Variance & Standard Deviation (2)

2 2 22 151

2 2 2

2

[ 5 7.96 2 6 7.96 2 7 7.96 13

8 7.96 20 9 7.96 10 10 7.96 4

11 7.96 1] 1.45

1.45 1.20

s

s

5 2 6 2 7 13 8 20 9 10 10 4 11 1

52 7.96

x


Alternative Definitions

Two alternative definitions for variance are

and, for a binomial random variable with parameters n, p, and q,

2 2 2E X

2 .npq


Example Alternative Definition

Find the variance when a fair coin is tossed 5 times and X is the number of heads.

2 1 1 55

2 2 4


Chebychev's Inequality

Chebychev's Inequality Suppose 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

.

2

1 .c


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.


Example Chebychev's Inequality (2)

The number of bottles containing between 100 - 5 and 100 + 5 capsules can be estimated using Chebychev's Inequality.

2 2

100

2

5

21 1 .845

c

c

On average, at least 84% will contain between 95 and 105 capsules.



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).



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.

2

1 c


7.6 The Normal Distribution

1. Normal Curve2. Normally Distributed Outcomes3. Properties of Normal Curve4. Standard Normal Curve5. The Normal Distribution 6. Percentile7. Probability for General Normal Distribution


Normal Curve

The bell-shaped curve, as shown below, is call a normal curve.


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.


Properties of Normal Curve


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.


Example Properties of Normal Curve (2)


Standard Normal Curve

The equation of the normal curve is

21

21

2where 3.1416 and 2.7183.

x

y e

e

The standard normal curve has 0 and 1.


The Normal Distribution

A(z) is the area under the standard normal curve to the left of a normally distributed random variable z.


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.


Example The Normal Distribution (2)

(a) A(-.5) = .3085

(b) A(2) - A(1) = .9772 - .8413 = .1359

(c) 1 - A(1.5) = 1 - .9332 = .0668


Percentile

If a score S is the pth percentile of a normal distribution, then p% of all scores fall below S, and (100 - p)% of all scores fall above S. The pth percentile is written as zp.


Example Percentile

What is the 95th 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, z95 = 1.65.


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.

,

Pr( ) Pr

and Pr( ) Pr

a b b aa X b Z A A

x xX x Z A


Example Probability Normal Distribution

Find the 95th 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 z95 = 1.65.

Then x95 = 7.75 + (1.65)(1.25) 9.81 pounds.



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.



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.


7.7 Normal Approximation to the Binomial Distribution

1. Normal Approximation


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

.npq


Normal Approximation

Binomial distribution with n = 40 and p = .3


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?


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)

300 .02 .98 2.425.


The area of the region to the right of 1.85 is 1 - A(1.85) = 1 - .9678 = .0322.

Therefore, 3.22% of the boxes should contain more than 10 defective washers.

Example Normal Approximation (3)

10.5 61.85

2.425z


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 1 .np p

Goldstein/Schnieder/Lay: Finite Math & Its Applications, 9e 1 of 105 Chapter 7 Probability and...

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