QMB Test 1 Study Guide Quantitative Business Methods Discrete Probability Distributions Discrete Data Values are typically whole numbers integers o Counted not measured Ex Number of cars Continuous Data Any value is possible depending on ability to measure accurately o Fractional values are possible Ex Thickness of an item height of item in inches Probability Distributions o Discrete Probability Model The height of the bar represents the probability There are gaps o Continuous Probability Model No Gaps Mathematical formula Normal Bell shaped curve Probability measured by area under a curve Discrete Random Variables Possible outcomes when conduction an experiment typically whole numbers finite number Continuous random variable infinite number Rules for Discrete Probability Distributions Listing of all possible outcomes of an experiment for a discrete random variable Relative frequency of each outcome or probability must be mutually exclusive o Two things can t happen at the same time Probability of each outcome must be between zero and one Sum of all probabilities in the distribution must equal The Mean of Discrete Probability Distributions weighted average of outcomes of random variables o Also known as the expected value E X The Parameter true value represented o Example is the mean Calculated by using every value in population The Variance measure of the spread of individual values around the mean of data set o The standard deviation is the square root of variance o If the variance equals zero all numbers are the same and there is no variance spread Expected Monetary value EMV Expressed in terms of dollars o Represents long term average Example Random Variable 1 2 3 4 1 Find the mean Probability 0 18 0 25 0 35 0 22 1 0 18 2 0 25 3 0 35 4 0 22 2 61 likely number to occur is between 2 and 3 2 Find the variance 1 2 16 2 0 18 2 2 16 2 0 25 3 2 16 2 0 35 4 2 61 2 0 22 1 0379 Binomial Distributions Binomial Distribution o Characteristics Consists of a fixed number of trials denoted n Each trial only has two possibilities either success or failure Success outcome of interest Failure not the outcome of interest Probability of success p and the probability of failure q are constant throughout the experiment Each trial is independent of the other trials in the experiment o Examples Survey response with only yes or no questions Flipping a coin Ex Thickness of an item height of item in inches o Formula Properties P Q 1 1 P Q 1 Q P P X N n n x x X pxqn x N number of trials X number of successes Q probability of failure P probability of success The mean and standard deviation o Mean np o Standard deviation npq o Fractional values are possible Example Claim 6 asked make a purchase Success make a purchase P 0 06 Q 1 0 06 94 N 15 1 P x 0 15 15 0 0 06 0 0 94 15 3953 2 P x 3 P x 0 P x 1 P x 2 a P 0 15 15 15 0 0 06 0 0 94 15 3953 b P 1 14 15 14 1 0 06 0 94 14 3785 c P 2 13 15 13 2 0 06 2 0 94 13 1691 d 3953 3785 1691 9429 3 P x 1 P x 2 P x 3 P x 15 a P x 1 Doesn t inclue P X 0 or P X 1 b P 0 15 15 15 0 0 06 0 0 94 15 3953 c P 1 14 15 14 1 0 06 0 94 14 3785 d 1 3953 3785 02262 Continuous Probability Distributions o Continuous random variable are outcomes that take on numerical value equal to zero The probability of one specific value occurring is theoretically Based on intervals not individual values Probability is represented by area under the probability distribution continuous probability distribution can have a variety of different shapes o Normal Distributions Characteristics Symmetrical around the mean Bell Shaped Values closest to the mean are more likely to occur Area left of mean 0 5 same with the right Distribution mean and standard deviation describe shape o Change in mean shifts distribution left or right o Change in standard deviation increases or decreases the spread o Probability Density Function mathematical description of a probability distribution Represents the relative distribution of frequency of a continuous random variable Any normal distribution can be transformed into standard normal distribution Transform x units to z units X continuous data value of interest O distribution s standard deviation Distribution s mean o The distribution X and Z are the same only the scale has changed X original units Z standardized units o When using z score table table provides the cumulative area under the standard normal curve that lies left of the z score For the upper tail need to subtract the number from the z table from 1 Continuous Probability Distributions Continued Example o Set average daily rate at the 80th percentile Mean 237 22 standard deviation 21 45 To figure out what x equals for it to be the 80th percentile must use the formula x z o Z for z use the table that corresponds with percentile 800 on the inside of the table 84 o X 237 22 84 21 45 o X 237 22 18 018 o X 255 24 Example 2 mean o Determine interval that covers one standard deviation around the Mean 692 standard deviation 50 To figure out what lies within one standard deviation of mean use mean standard deviation o 692 50 642 742 To figure out what lies within one standard deviation of mean use mean 2 standard deviation o 692 2 50 692 792 To figure out what lies within one standard deviation of mean use mean 3 standard deviation o 692 3 50 542 842 Empirical Rule One standard deviation covers 68 Two standard deviations covers 95 of all the data of all the data Three standard deviations covers 99 7 of all the data Normal distribution approximation can be used when sample size is large enough so that np 5 and nq 5 represents distribution is roughly bell shaped o To find out the minimum required sample size 5 smaller number of p or q Example p 9 q 1 conditions Example p 99 q 01 conditions o 5 1 50 required as the sample size to satisfy o 5 01 500 required as the sample size to satisfy Example 3 o Suppose 15 of people are left handed what is the probability of finding exactly 9 left handed people in a random sample of 50 First thing to do is make sure that problem satisfies condition of np 5 and nq 5 P 15 q 85 n 50 x 9 o Mean np 15 50 7 5 o Standard deviation npq 0 15 0 85 50 2 525 For this problem use continuity correction factor 0 5 p 8 5 x 9 5 1298 Rules for adding and subtracting for approximation continuity …