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# Study Guide For Midterm SCO 2550

U of M

GPA 3.76

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This 20 page Study Guide was uploaded by Madison Rude on Friday February 27, 2015. The Study Guide belongs to SCO 2550 at University of Minnesota taught by Steven Huchendorf in Spring2015. Since its upload, it has received 203 views. For similar materials see Business Statistics in Economcs at University of Minnesota.

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Date Created: 02/27/15

CH 1 Notes Week 1 12015 11 The Science of Statistics Statistics science of data collecting classifying summarizing organizing analyzing amp interpreting numerical and categorical data 12 Types of Statistical Applications in Business Sampling selecting which sets of data or phenomenon you wish to anaylyze interpret Statistics 0 Describing sets of data 0 Drawing conclusions Descriptive Statistics uses graphical numerical methods to explore data look for patterns summarize etc and present information articulately Inferential Statistics uses sample data to estimate make generalization about data 13 Fundamental Elements of Statistics Experimental Unit object we collect data from Population a set of units we want to study Variables the 1 characteristics traits we focus on in the population Measurement assigning numbers to a variable Census measuring a variable for every unit in the population Sample a subset within the population Statistical Inference estimate generalize about a population base on a sample Reliability how good the inference is o Inferences are usually based on PART of the population due to resource constraints Error of Estimation EXAMPLE difference in age between the whole population and the sample population Measure of Reliability a number describing the degree of uncertainty about an inference 4 Elements of Statistical inference 1 Population sample 2 Variables 3 Graphs Tables Summary Tools 4 Identify patters in data EX Reliability is usually stated with ix 51 i 5 28 r 2 14 Processes Process a data operation that creates output from input EX From processes we get production or manufactuing Raw material gt product Parts r whole units Black Box a process with unknown unspecified actions INPUT gt TranSfOFmation gt 152 148 121 etc n A Measurement Process used to assign numbers to variables when output isn39t numerical EX Using a scale measurement process to weigh a final product car Sample the output produced by a process 15 Types of Data Quantitative Data measurements recorded on a natural numerical scale EX Temperature unemployment rate scores number of people Qualitative Data measurements that can39t be measured on a numerical scale they can only be classified into categories EX Party affiliation status good not good size ranking scale 16 Collecting Data Sampling and Related Issues We can obtain data in 3 ways 1 From a published source 2 From a designed experiment 3 From an observational study survey Published Source a book newspaper journal etc Designed Experiment researcher strictly controls the units in the study Observational Study observing units in their natural habitats and recording the variables of interest researcher makes no effort to control the experiment Representative Sample exhibits characteristics typical of those processed by the population of interest Simple Random Sample a sample collected in a way where every different sample has an equal chance of selection Random Number Generator used to create random samples Other Random Sampling Designs Stratified random sampling Cluster sampling Systematic Sampling Randomized response sampling stratified random sampling is used when populations can be divided into a few categories strata Systematic Sampling systematically selecting every nth unit from the list of all the units EX Every 6th customer Randomized Response Sampling used to elicit an honest response on a sensitive question Selection Bias when some samples in the population are less likely to be selected than others Nonresponse Bias when a bias exists because data wasn39t collected on all units in the sample Measurement Error inaccuracies in values of data because of improper ambiguous leading question methods 17 Critical Thinking with Statistics Statistical Thinking being able to rationalize make sense of data amp inferences EX Variation does exist in population inference CH 2 Notes Week 2 12615 23 Numerical Measures of Central Tendency Numerical Descriptive Measures measures made to assign descriptive numbers to a sample so we can make inferences about it Central Tendency data tends to cluster around certain values IA 4 Center Spread Variability spread of data Mean of a Sample average of the sample denoted by X x bar Formula for Sample Mean 2 X1 i I V Population Mean the mean of the entire population rather than just a sample we use p mu to denote this Data Accuracy Depends on 2 Factors 1 size of the sample gt bigger does better 2 Variability spread of data gt more variability less accuracy Median middle number in the data set when it is arranged in ascending descending order Most valuable with large data sets Symbol m amp n for the population median Calculating a Sample Median m arrange n measurements from smallest to largest 1 If n is odd m is the middle number 2 If n is even m is the mean of the 2 middle numbers Sometimes median is a better central tendency predictor because it is less sensitive to outliers large small data Skewed when one tail of a distribution is more extreme than the other Rightward skewness means the mean is greater than the median Leftward skewness means the mean is less than the median Mode measurement that occurs most frequently in the data set Modal Class a class interval with the largest relative frequency Define mode as the midpoint of the modal class 24 Numerical Measures of Variability Measure of Variability also called the quotspreadquot of the data This is less reliable than this Range smallest value subtracted from largest value Sample Variance Sample Variance Sample Standard Deviation 32 2o n2 m n 1 Sample Standard Deviation this can also be denoted as s s2 Population Variance o2 sigma squared pop Varianceopop Standard deviation 02 2xlu2 N 21 Class and Frequency Class categories qualitative data fall into Class Frequency number of times an observation in a data set falls into a class Class Relative Frequency class frequency divided by total observations CRF f n Class Percentage class relative frequency divided by 100 25 Using the Mean and Standard Deviation to Describe Data Empirical Rule of data falling inside x number of standard deviations 1 standard deviation 68 2 standard deviations 95 3 standard deviations 997 26 Numerical Measures of Relative Standing Measure of Relative Standing descriptive measures to describe the relationship of a measurement to the rest of the data Percentile Ranking Score EX Company A has sales in the 90th percentile compared to other companies 90 of companies rank lower 10 rank higher Sample zscore for a measurement in the sample x is xi x S Population zscore for a measurement in the population x is x M 039 27 Methods for Detecting Outliers Box Plots amp Zscores Z Outliers 1 Observed recorded entered into computer incorrectly 2 Comes from a different population 3 Represents a chance occurance amp z scores help identify outliers IQR interquartile range Qu Qi I Qu upper quartile 75th percentile I Qi lower quartile 25th percentile 210 Distorting the Truth with Descriptive Statistics Look out for Changing the vertical amp horizontal axis intervals to visually manipulate data Creative titles A width distortion in bar graphs Beginning the vertical axis at a value other than the origin Chapter 3 Notes 222015 31 Events Sample Spaces amp Probability Observation the result we see amp record Experiment the process of making observations that leads to a single outcome that can t be predicted with certainty Sample Point most basic outcome of an experiment Sample Space collection of all sample points in an experiment Probability Rules for Sample Points 1 Must lie between 0 and 1 2 Probabilities of all sample points in a sample space MUST add to 1 Event specific collection of sample points Sample Event contains only a single sample point Compound Event contains two or more sample points Probability of an Event calculated by summing all probabilities of the sample points in the sample space for x Steps for Calculating Probabilities of Events 1 Define the experiment 2 List all sample points 3 Assign probabilities to sample points 4 Determine collection of sample events contained in event of interest 5 Sum the sample point probabilities to get event probability Combinatorial Math a branch of mathematics used to develop counting rules Combinations Rule Sample of n elements to be drawn from r elements this rule only works if you re not replacing the items before choosing new ones 32 Unions amp Intersections Union A or B or both occur on a single performance of the experiment A union B all sample points belonging to A or B or Both Intersections event that occurs if both A amp B occur on a single performance of the experiment A intersect B all sample points belonging to both A amp B A U B A intersect B Complementary Events the complement of event A is all the sample points that are NOT in event A subtract A from 1 to get the complement of A subtract the complement of A from 1 to get A 34 The Additive Rule amp Mutually Exclusive Events Additive Rule of Probability The probability of the union of events A amp B the sum of probabilities ofA amp B minus the probability of the intersection of events A amp B PA or B PA PB PA and B OR PA U B PA PB PA m B Mutually Exclusive Events when A amp B never intersect Probability of Union of 2 Mutually Exclusive Events The probability of mutually exclusive events A amp B the sum of probabilities of A amp B PAUB PA PB 34 Conditional Probability Conditional Probability when we have additional knowledge about the likelihood of an event PAIB the probability of event A given that B occurs P A n n PAIB PB 36 The Multiplicative Rule amp Independent Events Multiplicative Rule of Probability PA intersect B PAPBIA g PA intersect B PBPAIB Independent Events if the occurrence of B doesn t alter the probability that A has occurred PAIB PA PBIA PB Probability of Intersection of 2 Independent Events If events A amp B are independent the probability of PA intersect B PAPB Chapter 4 Notes 292015 Random Variable a variable that assumes numerical values associated with the random outcomes of an experiment where only one numerical value is assigned to each sample point Discrete Random Variables random variables that can assume a countable number of values Continuous Random Variable random variables that can assume values corresponding to any points contained in one or more intervals EX Discrete 1 of sales in a week x 0123 2 of errors 3 of customers Continuous 1 length of time between arrivals and departures 2 weight of food item bought in a supermarket Probability Distribution for a discrete random variable is a graph table formula that specifies the probability associated with each possible value the random variable could assume gtquotpx must be greater than zero for all values of X the sum of px must 1 the probability distribution for a random variable is a theoretical model for a relative frequency distribution of a population Mean Expected Value Of a discrete random variable X is u Tl lll L I expected value is a measure of central tendency o u is the mean value of X in an infinite number of experimental repetitions o the value of x occur in equivalent proportions to the probabilities of x Population Variance because X is a random variable so is X3102 we can find the mean value of X3102 using this equation E x 2 206 L02 2906 this is the expected value of the squared distance from the mean g 02 Ex 102 Standard Deviation of Discrete Random Variables a J no mzmx 43 The Binomial Distributions Binomial Random Variables when there are 2 outcomes yes no pass fail heads tails Characteristics of a Binomial Experiment 1 Consists of n identical trials 2 Only 2 possible outcomes of each trial 3 The probability of S remains the same from trial to trial It is denoted by p the probability of F is donated by q note that q 1p 4 Trials are independent 5 The binomial random variable X is the number of S s in n trials The Binomial Probability Distribution I N X P 1 1 p p probability of a success on a single trial q 1p n number of trials X number of successes in n trials 72 r n I 39 7 nX number of failures in n trials 2 where r X Mean Variance amp Standard Deviation for a Binomial Random Variable Mean a np Variance 02 Standard Deviation a 1npq 46 The Normal Distribution Bell Curve normal distribution Standard Normal Distribution a normal distribution with a 0 and a 1 A random variable with a standard normal distribution denoted by the symbol Z is called a standard normal random variable the area under the curve of a standard normal probability distribution 2 1 To convert a normal random variable we must first convert X to a z score Z is a standard normal random variable Steps to Finding Probability Corresponding to a Normal Random Variable 1 sketch normal distribution amp indicate mean and random variable x 2 convert boundary of shaded area from x values to standard normal random variable x a 2 values usmg form 2 T 3 use Table II to find areas corresponding to the 2 values Chapter 5 Notes 21615 51 The Concept of Sampling Distribution Neither sample mean 39lt nor same median m will always fall closer to u Sampling Distribution calculated from a sample of n measurements in a sample statistics and it the probability distribution of the statistic 9WD The sampling distribution tells us about the behavior of 39lt in repeated sample Consists of ALL possible values of the sample statistic calculated from all possible samples of a specific size n drawn from the population From this we need to make an inference about ALL items of interest in the population Steps to find a sample distribution Get all possible samples of size n from the population Calculated sample statistic 39lt for each sample Put all lt s into a distribution Describe the behavior of lt s 3 THINGS TO KNOW ABOUT ANY DISTRIBUTION Remember LOVS Location Variability Shape Central Limit Theorem The mean of the sampling distribution lt must the mean of the population a The standard deviation of sampling distribution lt must the standard deviation of population divided by the square root of sample size n 0 i xE This tells us the average error when using any one X to estimate the true population mean Also called the quotstandard error of the mean The shape of the sampling distribution 39lt o For a known a the shape of the sampling distribution will be a normal distribution 0 Normal if population normal with a known a o If the population is normal with an unknown 0 with a small n then use the tdistribution with n l degrees of freedom Sample Proportion p of the sample statistic use the same steps as finding 39 Central Limit Theorem The population proportion must equal the expected sample statistic proportion for all n 0 p is an unbiased estimator for p 0 shape of the sampling distribution of the statistic p is a normal distribution for sufficient n 0 both mo and nqhat must be greater than 15 52 Properties of a Sampling Distributions Unbiasedness and Minimum Variance Point Estimator of a population parameter is a rule or formula that tells us how to use the sample data to calculate a single that can be used as an estimate of population parameter EX Sample mean X is an estimator of the population mean Sample variance 52 is an estimator of population variance many different point estimators can be used to estimate the same parameter By estimating the sampling distribution we can calculate the error of estimation Unbiased Statistic when the mean of the sampling distribution what it is intended to estimate Biased Statistic mean of a sampling distribution does not the parameter it is intended to estimate biased stats underestimate or overestimate the parameter such as sample variance When inferring a population parameter use the sample stat with a distribution that is unbiased amp has small standard deviation 53 Sampling Distribution of a Sample Mean amp Central Limit Theorem x is the minimum variance unbiased estimator ofp standard deviation of 0 Theorem 51 if a random sample of n observations is selected from a population with a normal distribution the sampling distribution of x will be a normal distribution Theorem 52 Central Limit Theorem Random sample of n observations selected from a population with mean u amp standard deviation 0 When n is sufficiently large sampling distribution of x will be approximately a normal distribution with mean pi p and standard deviation 02 Larger sample size the better the normal approximation to sampling distribution of x For MOST sample sizes of n 2 30 will suffice for the normal approximation to be reasonable Standard deviation of the sampling distribution decreases as the sample size increases 0 Larger sample more accuracy 54 The Sampling Distribution of the Sample Proportion Sample Proportion p is a good estimator of the population proportion p Sampling Distribution of p 1 Mean of the sample distribution to the true binomial proportion p 2 Standard deviation of the sampling distribution is approximately normal sample is considered large if m6 and nqhat are greater than 15 Chapter 6 Notes 22315 61 Identifying and Estimating the Target Parameter Target Parameter the unknown population parameter we are interested in estimating 0 With quantitative data you re looking for mean or variance 0 With qualitative data binomial you re looking for binomial proportion of success Point Estimator ruleformula that tells us how to use sample data to calculate a single number we can use to estimate the target parameter Confidence Interval formula that tells us how to use sample data to calculate an interval that estimates the target parameter 62 Confidence Interval for a Population Mean Normal 2 statistic Confidence Coefficient probability that a randomly selected confidence interval encloses the population parameter Confidence Level the confidence coefficient expressed by a percentage 0 To choose a confidence level other than 95 we find a different area under the curve zaz is the zvalue where lies to the right of z 0 confidence interval with a confidence coefficient 1 a is Riz 0z 2 a If we have a known a XiZ s If we have an unknown 0 XiZ 63 Confidence Interval for a Population Mean Student s tstatistic Problems with small samples 1 The shape of the distribution depends on the population that is sampled we can t assume normality Solution if the sampled population is approximately normal so is the sample 2 The sample standard deviation 5 is a poor approximation for a in small samples Solution instead ofz P use I P 02 ax sxE of sample statistic for small samples which is the standard deviation The only difference between a zstat and a tstat is that the t stat is more variable The variability of t depends on sample size We express this dependence by saying the t stat has n l degrees of freedom 64 LargeSample Confidence Interval for a Population Proportion Properties of the Sampling Distribution of p p is an unbiased estimator of p 2 standard deviation of the sampling distribution of p is Wlpqn that is 03916 xpqn for large samples the sampling distribution of p is approximately normal Large Sample Confidence Interval of p iz 0x Which I5 iZe 2961n 2 2 but you can replace p and q with phat and q hat pxn 65 Determining the Sample Size Sample Size Determination for 1001a confidence interval for H To estimate u with a margin for error mfe and with 1001a confidence the required sample size n is found as follows 2 W The solution for n is given by ac 2 n mfg 0 is usually unknown we can estimate with the s from a prior sample g we can find the range R of observations in the population with o R4 and round up Here are the sample equations adjusted to find a sample proportion 2 W W

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