Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-1 2nd Lesson Probability and Sampling Distributions

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-2 Probability Distributions Continuous Probability Distributions Binomial Hypergeometric Poisson Probability Distributions Discrete Probability Distributions Normal Chi Square Fisher MultinomialStudent-t

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-3 A discrete random variable is a variable that can assume only a countable number of values Many possible outcomes: number of complaints per day number of TV’s in a household number of rings before the phone is answered Only two possible outcomes: gender: male or female defective: yes or no spreads peanut butter first vs. spreads jelly first Discrete Probability Distributions

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-4 Continuous Probability Distributions A continuous random variable is a variable that can assume any value on a continuum (can assume an uncountable number of values) thickness of an item time required to complete a task temperature of a solution height, in inches These can potentially take on any value, depending only on the ability to measure accurately.

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-5 The Binomial Distribution Binomial Hypergeometric Poisson Probability Distributions Discrete Probability Distributions Multinomial

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-6 The Binomial Distribution Characteristics of the Binomial Distribution: A trial has only two possible outcomes – “success” or “failure” There is a fixed number, n, of identical trials The trials of the experiment are independent of each other The probability of a success, p, remains constant from trial to trial If p represents the probability of a success, then (1-p) = q is the probability of a failure

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-7 Binomial Distribution Settings A manufacturing plant labels items as either defective or acceptable A firm bidding for a contract will either get the contract or not A marketing research firm receives survey responses of “yes I will buy” or “no I will not” New job applicants either accept the offer or reject it

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-8 Counting Rule for Combinations A combination is an outcome of an experiment where x objects are selected from a group of n objects where: n! =n(n - 1)(n - 2)... (2)(1) x! = x(x - 1)(x - 2)... (2)(1) 0! = 1 (by definition)

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-9 P(x) = probability of x successes in n trials, with probability of success p on each trial x = number of ‘successes’ in sample, (x = 0, 1, 2,..., n) p = probability of “success” per trial q = probability of “failure” = (1 – p) n = number of trials (sample size) P(x) n x ! nx pq x n x ! ()!    Example: Flip a coin four times, let x = # heads: n = 4 p = 0.5 q = (1 -.5) =.5 x = 0, 1, 2, 3, 4 Binomial Distribution Formula

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-10 n = 5 p = 0.1 n = 5 p = 0.5 Mean X P(X) X P(X) 0 Binomial Distribution The shape of the binomial distribution depends on the values of p and n Here, n = 5 and p =.1 Here, n = 5 and p =.5

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-11 Binomial Distribution Characteristics Mean Variance and Standard Deviation Wheren = sample size p = probability of success q = (1 – p) = probability of failure

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-12 n = 5 p = 0.1 n = 5 p = 0.5 Mean X P(X) X P(X) 0 Binomial Characteristics Examples

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-13 Using Binomial Tables n = 10 xp=.15p=.20p=.25p=.30p=.35p=.40p=.45p= p=.85p=.80p=.75p=.70p=.65p=.60p=.55p=.50x Examples: n = 10, p =.35, x = 3: P(x = 3|n =10, p =.35) =.2522 n = 10, p =.75, x = 2: P(x = 2|n =10, p =.75) =.0004

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-14 The Multinomial Distribution Binomial Hypergeometric Poisson Probability Distributions Discrete Probability Distributions Multinomial

The Multinomial Distribution Percobaan binomial menjadi multinomial jika tiap trial/usaha menghasilkan lebih dari 2 kemungkinan untuk muncul. Bila suatu usaha tertentu dapat menghasilkan k macam hasil E 1, E 2, …E k dengan peluang p 1, p 2,..p k, maka distribusi peluang peubah acak X 1, X 2, …X k yang menyatakan banyak terjadinya E 1, E 2, …E k dalam n usaha yg independent ialah: Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-15

The Multinomial Distribution Contoh: Dua dadu dilemparkan 6 kali, Berapa peluang mendapat jumlah 7 atau 11 muncul 2 kali, sepasang bilangan yang sama satu kali, dan kombinasi lainnya 3 kali? Jawab: E ₁ = Kejadian jumlah 7 atau 11 muncul E₂ = Kejadian sepasang bilangan sama muncul E₃ = Kejadian kombinasi lainnya Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-16

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-17 The Poisson Distribution Binomial Hypergeometric Poisson Probability Distributions Discrete Probability Distributions Multinomial

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-18 The Poisson Distribution Characteristics of the Poisson Distribution: The outcomes of interest are rare relative to the possible outcomes The average number of outcomes of interest per time or space interval is The number of outcomes of interest are random, and the occurrence of one outcome does not influence the chances of another outcome of interest The probability of that an outcome of interest occurs in a given segment is the same for all segments

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-19 Poisson Distribution Formula where: t = size of the segment of interest x = number of successes in segment of interest = expected number of successes in a segment of unit size e = base of the natural logarithm system ( )

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-20 Poisson Distribution Characteristics Mean Variance and Standard Deviation where = number of successes in a segment of unit size t = the size of the segment of interest

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-21 Using Poisson Tables X t Example: Find P(x = 2) if =.05 and t = 100

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-22 Graph of Poisson Probabilities X t = P(x = 2) =.0758 Graphically: =.05 and t = 100

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-23 Poisson Distribution Shape The shape of the Poisson Distribution depends on the parameters and t: t = 0.50 t = 3.0

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-24 The Hypergeometric Distribution Binomial Hypergeometric Poisson Probability Distributions Discrete Probability Distributions Multinomial

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-25 The Hypergeometric Distribution “n” trials in a sample taken from a finite population of size N Sample taken without replacement Trials are dependent Concerned with finding the probability of “x” successes in the sample where there are “X” successes in the population

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-26 Hypergeometric Distribution Formula. Where N = Population size X = number of successes in the population n = sample size x = number of successes in the sample n – x = number of failures in the sample (Two possible outcomes per trial)

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-27 Hypergeometric Distribution Formula ■Example: 3 Light bulbs were selected from 10. Of the 10 there were 4 defective. What is the probability that 2 of the 3 selected are defective? N = 10n = 3 X = 4 x = 2

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-28 The Normal Distribution Continuous Probability Distributions Normal Chi Square Fisher Student-t

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-29 The Normal Distribution ‘ Bell Shaped’ Symmetrical Mean, Median and Mode are Equal Location is determined by the mean, μ Spread is determined by the standard deviation, σ The random variable has an infinite theoretical range: +  to   Mean = Median = Mode x f(x) μ σ

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-30 By varying the parameters μ and σ, we obtain different normal distributions Many Normal Distributions

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-31 The Normal Distribution Shape x f(x) μ σ Changing μ shifts the distribution left or right. Changing σ increases or decreases the spread.

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-32 Finding Normal Probabilities Probability is the area under the curve! ab x f(x) Paxb( )  Probability is measured by the area under the curve

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-33 f(x) x μ Probability as Area Under the Curve 0.5 The total area under the curve is 1.0, and the curve is symmetric, so half is above the mean, half is below

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-34 Empirical Rules μ ± 1 σ  encloses about 68% of x’s  f(x) x μ μ  σμ  σ What can we say about the distribution of values around the mean? There are some general rules: σσ 68.26%

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-35 The Empirical Rule μ ± 2σ covers about 95% of x’s μ ± 3σ covers about 99.7% of x’s xμ 2σ2σ2σ2σ xμ 3σ3σ3σ3σ 95.44%99.72% (continued)

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-36 Importance of the Rule If a value is about 2 or more standard deviations away from the mean in a normal distribution, then it is far from the mean The chance that a value that far or farther away from the mean is highly unlikely, given that particular mean and standard deviation

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-37 The Standard Normal Distribution Also known as the “z” distribution Mean is defined to be 0 Standard Deviation is 1 z f(z) 0 1 Values above the mean have positive z-values, values below the mean have negative z-values

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-38 The Standard Normal Any normal distribution (with any mean and standard deviation combination) can be transformed into the standard normal distribution (z) Need to transform x units into z units

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-39 Translation to the Standard Normal Distribution Translate from x to the standard normal (the “z” distribution) by subtracting the mean of x and dividing by its standard deviation:

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-40 Example If x is distributed normally with mean of 100 and standard deviation of 50, the z value for x = 250 is This says that x = 250 is three standard deviations (3 increments of 50 units) above the mean of 100.

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-41 Comparing x and z units z x Note that the distribution is the same, only the scale has changed. We can express the problem in original units (x) or in standardized units (z) μ = 100 σ = 50

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-42 The Standard Normal Table The Standard Normal table in the textbook (Appendix D) gives the probability from the mean (zero) up to a desired value for z z Example: P(0 < z < 2.00) =.4772

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-43 The Standard Normal Table The value within the table gives the probability from z = 0 up to the desired z value P(0 < z < 2.00) =.4772 The row shows the value of z to the first decimal point The column gives the value of z to the second decimal point (continued)

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-44 General Procedure for Finding Probabilities Draw the normal curve for the problem in terms of x Translate x-values to z-values Use the Standard Normal Table To find P(a < x < b) when x is distributed normally:

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-45 Z Table example Suppose x is normal with mean 8.0 and standard deviation 5.0. Find P(8 < x < 8.6) P(8 < x < 8.6) = P(0 < z < 0.12) Z x8.6 8 Calculate z-values:

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-46 Z Table example Suppose x is normal with mean 8.0 and standard deviation 5.0. Find P(8 < x < 8.6) P(0 < z < 0.12) z x P(8 < x < 8.6)  = 8  = 5  = 0  = 1 (continued)

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-47 Z 0.12 z Solution: Finding P(0 < z < 0.12) Standard Normal Probability Table (Portion) 0.00 = P(0 < z < 0.12) P(8 < x < 8.6)

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-48 Finding Normal Probabilities Suppose x is normal with mean 8.0 and standard deviation 5.0. Now Find P(x < 8.6) Z

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-49 Finding Normal Probabilities Suppose x is normal with mean 8.0 and standard deviation 5.0. Now Find P(x < 8.6) (continued) Z P(x < 8.6) = P(z < 0.12) = P(z < 0) + P(0 < z < 0.12) = =.5478

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-50 Upper Tail Probabilities Suppose x is normal with mean 8.0 and standard deviation 5.0. Now Find P(x > 8.6) Z

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-51 Now Find P(x > 8.6)… (continued) Z Z =.4522 P(x > 8.6) = P(z > 0.12) = P(z > 0) - P(0 < z < 0.12) = =.4522 Upper Tail Probabilities

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-52 Lower Tail Probabilities Suppose x is normal with mean 8.0 and standard deviation 5.0. Now Find P(7.4 < x < 8) Z

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-53 Lower Tail Probabilities Now Find P(7.4 < x < 8)… Z The Normal distribution is symmetric, so we use the same table even if z-values are negative: P(7.4 < x < 8) = P(-0.12 < z < 0) =.0478 (continued).0478

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-54 The Student-t Distribution Continuous Probability Distributions Normal Chi kuadrat Fisher Student-t

The Student-t Distribution Jarang sekali diketahui varian suatu populasi Untuk sampel n ≥30, taksiran dapat diperoleh dari Efeknya : jika sampel kecil maka distribusi peubah acak menyimpang jauh dari distribusi normal baku sehingga perlu digunakan distribusi T Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-55

The Student-t Distribution Jika v besar (v ≥30) maka grafik f(t) mendekati kurva normal standard. Bentuk Distribusi student -t Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-56

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-57 The Chi Square Distribution Continuous Probability Distributions Normal Chi Square Fisher Student-t

The Chi Square Distribution Beberapa manfaat dari distribusi Chi-Kuadrat antara lain: Menguji perbedaan secara signifikan antara frekuensi yang diamati dengan frekuensi teoritis menguji kebebasan antar faktor dari data dalam daftar kontingensi menguji kedekatan data sampel dengan suatu fungsi distribusi seperti binomial, Poisson, atau normal. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-58

The Chi Square Distribution Peubah acak kontinu X berdistribusi chi-square dengan derajat kebebasan v, bila fungsi padatnya diberikan oleh, untuk x lainnya Dengan v bilangan bulat positif Rataan dan varians distribusi chi-square Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-59

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-60 The Fisher Distribution Continuous Probability Distributions Normal Chi Square Fisher Student-t

The Fisher Distribution Misalkan U dan V dua peubah acak bebas masing-masing berdistribusi chi-square dengan derajat bebas v ₁ dan v₂. Maka distribusi peubah acak Diberikan oleh untuk x lainnya Ini dikenal dengan nama distribusi-F Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-61

The Fisher Distribution Tulislah untuk dengan derajat kebebasan dan, maka Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-62

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 6-63 Sampling Sample Statistics are used to estimate Population Parameters ex: X is an estimate of the population mean, μ Problems: Different samples provide different estimates of the population parameter Sample results have potential variability, thus sampling error exits

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 6-64 Sampling Error Sampling Error: The difference between a value (a statistic) computed from a sample and the corresponding value (a parameter) computed from a population Example: (for the mean) where:

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 6-65 Review Population mean:Sample Mean: where: μ = Population mean x = sample mean x i = Values in the population or sample N = Population size n = sample size

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 6-66 Example If the population mean is μ = 98.6 degrees and a sample of n = 5 temperatures yields a sample mean of = 99.2 degrees, then the sampling error is

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 6-67 Sampling Errors Different samples will yield different sampling errors The sampling error may be positive or negative ( may be greater than or less than μ) The expected sampling error decreases as the sample size increases

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 6-68 Sampling Distribution A sampling distribution is a distribution of the possible values of a statistic for a given size sample selected from a population

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 6-69 Developing a Sampling Distribution Assume there is a population … Population size N=4 Random variable, x, is age of individuals Values of x: 18, 20, 22, 24 (years) A B C D

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap A B C D Uniform Distribution P(x) x (continued) Summary Measures for the Population Distribution: Developing a Sampling Distribution

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap possible samples (sampling with replacement) Now consider all possible samples of size n=2 (continued) Developing a Sampling Distribution 16 Sample Means

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 6-72 Sampling Distribution of All Sample Means P(x) x Sample Means Distribution 16 Sample Means _ Developing a Sampling Distribution (continued) (no longer uniform)

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 6-73 Summary Measures of this Sampling Distribution: Developing a Sampling Distribution (continued)

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 6-74 Comparing the Population with its Sampling Distribution P(x) x A B C D Population N = 4 P(x) x _ Sample Means Distribution n = 2

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 6-75 If the Population is Normal (THEOREM 6-1) If a population is normal with mean μ and standard deviation σ, the sampling distribution of is also normally distributed with and

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 6-76 z-value for Sampling Distribution of x Z-value for the sampling distribution of : where:= sample mean = population mean = population standard deviation n = sample size

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 6-77 Finite Population Correction Apply the Finite Population Correction if: the sample is large relative to the population (n is greater than 5% of N) and… Sampling is without replacement Then

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 6-78 Normal Population Distribution Normal Sampling Distribution (has the same mean) Sampling Distribution Properties (i.e. is unbiased )

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 6-79 Sampling Distribution Properties For sampling with replacement: As n increases, decreases Larger sample size Smaller sample size (continued)

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 6-80 If the Population is not Normal We can apply the Central Limit Theorem: Even if the population is not normal, …sample means from the population will be approximately normal as long as the sample size is large enough …and the sampling distribution will have and

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 6-81 n↑n↑ Central Limit Theorem As the sample size gets large enough… the sampling distribution becomes almost normal regardless of shape of population

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 6-82 Population Distribution Sampling Distribution (becomes normal as n increases) Central Tendency Variation (Sampling with replacement) Larger sample size Smaller sample size If the Population is not Normal (continued) Sampling distribution properties:

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 6-83 How Large is Large Enough? For most distributions, n > 30 will give a sampling distribution that is nearly normal For fairly symmetric distributions, n > 15 For normal population distributions, the sampling distribution of the mean is always normally distributed

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 6-84 Example Suppose a population has mean μ = 8 and standard deviation σ = 3. Suppose a random sample of size n = 36 is selected. What is the probability that the sample mean is between 7.8 and 8.2?

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 6-85 Example Solution: Even if the population is not normally distributed, the central limit theorem can be used (n > 30) … so the sampling distribution of is approximately normal … with mean = 8 …and standard deviation (continued)

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 6-86 Example Solution (continued): (continued) z Sampling Distribution Standard Normal Distribution Population Distribution ? ? ? ? ? ? ?? ? ? ? ? SampleStandardize

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 6-87 Population Proportions, p p = the proportion of population having some characteristic Sample proportion ( p ) provides an estimate of p: If two outcomes, p has a binomial distribution

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 6-88 Sampling Distribution of p Approximated by a normal distribution if: where and (where p = population proportion) Sampling Distribution P( p ) p

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 6-89 z-Value for Proportions If sampling is without replacement and n is greater than 5% of the population size, then must use the finite population correction factor: Standardize p to a z value with the formula:

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 6-90 Example If the true proportion of voters who support Proposition A is p =.4, what is the probability that a sample of size 200 yields a sample proportion between.40 and.45? i.e.: if p =.4 and n = 200, what is P(.40 ≤ p ≤.45) ?

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 6-91 Example if p =.4 and n = 200, what is P(.40 ≤ p ≤.45) ? (continued) Find : Convert to standard normal:

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 6-92 Example z Standardize Sampling Distribution Standardized Normal Distribution if p =.4 and n = 200, what is P(.40 ≤ p ≤.45) ? (continued) Use standard normal table: P(0 ≤ z ≤ 1.44) = p