Lecture 8 The Binomial Distribution Probability Distributions: Normal and Binomial 1 2 Binomial Distribution >A binomial experiment possesses the following properties. The experiment consists of a fixed number (n) of trials The result of each trial can be classified into one of two categories: success or failure The probability (π) of a success remains constant for each trial Each trial of the experiment is independent of the other trials 3 Binomial Distribution APPROACHES TO SOLVING: >FORMULA >TABLES >APPROXIMATION USING THE NORMAL DISTRIBUTION - later 4
Formula Approach Notation n! x) ( ) p ( ) ( ) x q (n P x = x! n x! Where: P(x) = the probability of x successes in n trials n = the number of trials p = the probability of success on any one trial q = (1 - p) C n x = x! n = x x! n! ( n x)! n! ( ) n x! > You may be familiar with one of the notations at the left > It is the number of ways of choosing x objects from a total of x objects > The notation x! means the product of all the numbers from 1 to x 5 6 >80% of transactions handled by CSRs could normally be handled by ATMs or over the phone. During a slow period in the afternoon, 5 customers were helped by CSRs. >What is the probability that all could have been served without CSRs? >What is the probability that 3 needed CSR intervention? Formula Solution > p = 0.80, i.e. Prob. of not needing CSR >n = 5 > x = 5, x = 5-3 = 2 p(5) 5 5 0 5 2 3 = C ( 0.8) ( 0.2) p(2) = C 5 2( 0.8) ( 0.2) 5! 5 0 5! 2 3 = ( 0.8) ( 0.2) = ( 0.8) ( 0.2) 5! 0! 2! 3! = ( 10)( 0.64)( 0.008) = 0.32768 = 32.768% = 0.0512 = 5.12% 7 8
Binomial Tables > The tables in text Appendix B (pages 731-737) give binomial probabilities for n = 1 through 15, 20 and 25 trials and a range of probabilities > Below, is a table for n=5 n = 5 π x 0.05 0.10 0.15 0.20 0.25 0.30 0.40 0.50 0.60 0.70 0.75 0.80 0.90 0.95 0 0.774 0.590 0.444 0.328 0.237 0.168 0.078 0.031 0.010 0.002 0.001 0.000 0.000 0.000 1 0.204 0.328 0.392 0.410 0.396 0.360 0.259 0.156 0.077 0.028 0.015 0.006 0.000 0.000 2 0.021 0.073 0.138 0.205 0.264 0.309 0.346 0.313 0.230 0.132 0.088 0.051 0.008 0.001 3 0.001 0.008 0.024 0.051 0.088 0.132 0.230 0.313 0.346 0.309 0.264 0.205 0.073 0.021 4 0.000 0.000 0.002 0.006 0.015 0.028 0.077 0.156 0.259 0.360 0.396 0.410 0.328 0.204 5 0.000 0.000 0.000 0.000 0.001 0.002 0.010 0.031 0.078 0.168 0.237 0.328 0.590 0.774 n = 5 π x 0.05 0.10 0.15 0.20 0.25 0.30 0.40 0.50 0.60 0.70 0.75 0.80 0.90 0.95 0 0.774 0.590 0.444 0.328 0.237 0.168 0.078 0.031 0.010 0.002 0.001 0.000 0.000 0.000 1 0.204 0.328 0.392 0.410 0.396 0.360 0.259 0.156 0.077 0.028 0.015 0.006 0.000 0.000 2 0.021 0.073 0.138 0.205 0.264 0.309 0.346 0.313 0.230 0.132 0.088 0.051 0.008 0.001 3 0.001 0.008 0.024 0.051 0.088 0.132 0.230 0.313 0.346 0.309 0.264 0.205 0.073 0.021 4 0.000 0.000 0.002 0.006 0.015 0.028 0.077 0.156 0.259 0.360 0.396 0.410 0.328 0.204 5 0.000 0.000 0.000 0.000 0.001 0.002 0.010 0.031 0.078 0.168 0.237 0.328 0.590 0.774 P(X=2) P(X=5) 9 10 Binomial Expected Value, Variance & Standard Deviation >Expected Value: E(x) = µ = np >Variance: Var(x) = σ 2 = npq >Standard Deviation SD(x) = σ = npq Binomial Recognition >Each result is binary, i.e. a success or failure, good or bad, yes or no DISCRETE NO FRACTIONAL VALUES >Maximum = n, Minimum =0 >The probability of any result remains constant unlike lotto, once a number is taken, it cannot be chosen again >Each result is independent 11 12
Binomial Exercise >The probability of a statistics student doing their assignment more than 24 hours before it is due is 25%. If ten students are chosen at random the day before it is due, what is the probability that none have done it? 2 have done it? more than 3 have done it? less than 5 have done it? 13 x 0.25 0 0.056 1 0.188 2 0.282 3 0.250 4 0.146 5 0.058 6 0.016 7 0.003 8 0.000 9 0.000 10 0.000 Tables Approach P(X=0) = 0.056 P(X=2) = 0.282 P(X>3) = 0.223 P(X<5) = 0.922 14 P(X = 0) = P(X = 2) = Using the Formula 10! 0! ( )( ) ( ) 0 0.25 ( 0.75 ) 10! 10! 2! 2 = 0.0563 ( )( ) ( 0.25) ( 0.75) = 8! 0. 2816 > For P(X>3) and P(X<5) we would need to use the formula many times, which is generally not practical 8 10 >Four out of five dentists recommend Crest. If you survey 10 dentists, what is the probability that: More than 4 out of 5 recommend Crest? Less than 4 out of 5 recommend Crest? Exactly 4 out of 5 recommend Crest? 15 16
x 0.80 0 0.000 1 0.000 2 0.000 3 0.001 4 0.006 5 0.026 6 0.088 Table 7 0.201 0.322 - Less than 4 out of 5 8 0.302 0.302 - Exactly 4 out of 5 9 0.268 0.375 - More than 4 out of 5 10 0.107 >A student believes that she will be accepted to 80% of the universities to which she applies. What is the probability not being accepted to any university if she applies to 4 universities? >P(X=0) = (0.2) 4 = 0.0016 17 18 Normal Distribution Normal Distribution >We have seen the normal distribution several times before >Often referred to as the Bell Curve >Occurs often in nature 19 20
Normal Distribution Normal Properties Probability Density >All Normal Distributions are simply scaled versions of each other >Changing the mean shifts the distribution right or left >Changing the standard deviation makes it flatter or narrower -4-3 -2-1 0 1 2 3 4 Z Value- Standard Deviations from Mean 21 22 Standard Normal > All normals can be converted to the standard normal (mean, 0; standard deviation, 1) by subtracting the mean from each value and dividing by the standard deviation Z = X - µ σ x x Standard Normal >This transformation results in Z being distributed with a mean of 0 and a standard deviation of 1 >Z simply measures a distance: the number of standard deviations above or below 0 23 24
Standard Normal >Thus, for any value or observation, its Z-value lets us know how many standard deviations it is above or below zero >This is the simplicity of the normal distribution: All normally distributed data can be converted to a standard normal Standard Normal >This means that there is only one table that we need to reference and learn >For any normally distributed variable, the distribution of its probabilities are identical to all other normally distributed data or variable - both can be converted into the standard normal 25 26 Understanding The Normal >The Normal Distribution is SYMMETRICAL about its mean half above & half below >The normal is infinite in both directions There are very small probabilities of extreme values Understanding The Normal >The probability of a value falling within a specified number of standard deviations is given by the well known EMPIRICAL RULE that we looked at in Lecture 4 27 28
Empirical Rule Normal Distribution Number of Std. Dev. Proportion of Distribution Approximately Exactly ±1 2/3 68.4% ±2 95% 19 out of 20 95.5% ±3 Substantially all 99.7% Probability Density 0.15% Tail 2.5% Tail 2/3 95% 99.7% 2.5% Tail 0.15% Tail -4-3 -2-1 0 1 2 3 4 29 Z Value- Standard Deviations from Mean 30 Normal Distribution APPROACHES TO SOLVING: >TABLES >COMPUTER - not covered in this course Normal Table >The normal table is in text Appendix D on page 743 >Only half the table is given, since the Normal Distribution is symmetrical >The values given is the P(0< z ), the area between Z=0 and Z=z 31 32
Normal Table (Part) Second Decimal Place z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359 0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753 0.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141 0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517 0.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879 0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224 0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549 0.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852 0.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133 0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389 P(Z< 0.98) In Class Exercise >A statistics midterm had an average of 75 and a standard deviation of 10. >What is the probability that a student chosen at random received: At least B+? Less than C+? B+? B or B+? C or C+? 33 34 Question Normal Standardized At least B+ P(X>75) P(Z>0.0) Less than C+ P(X<65) P(Z<-1.0) B+ P(75<X<80) P(0<Z<0.5) B or B+ P(70<X<80) P(-0.5<Z<0.5) C or C+ P(60<X<70) P(-1.5<Z<-0.5) Question Standardized From Table At least B+ P(Z>0.0) 0.5-0 = 0.5 Less than C+ P(Z<-1.0) =P(Z>1.0) 0.5-0.3413 = 0.1587 B+ P(0<Z<0.5) 0.1915 B or B+ P(-0.5<Z<0.5) =P(Z<0.5)x2 0.1915x2 = 0.3830 C or C+ P(-1.5<Z<-0.5) =P(0.5<Z<1.5) 0.4332-0.1915 = 0.2417 35 36
YOU LEARN STATISTICS BY DOING STATISTICS 37