ECON1310 Quantitative Economic and Business Analysis A

Transcription

1 ECON1310 Quantitative Economic and Business Analysis A Topic 1 Descriptive Statistics 1 Main points - Statistics descriptive collecting/presenting data; inferential drawing conclusions from - Data types qualitative: nominal, ordinal; quantitative: discrete (counting), continuous (measuring) - Level of measurement nominal (name), ordinal (rank), interval (gaps comparable), ratio (true zero) - Variable characteristic, population all members, census survey pop, subset of pop - Reasons for sampling cheaper, efficient, practical, reduces destruction, difficult ask whole pop - Sampling errors sampling: unavoidable, reduced through altering size; non-sampling, avoidable, because response/recording/processing errors - Sampling methods SRS: simple, every item equal chance selected; systematic: every kth person; stratified: homogenous strata, then SRS; cluster: heterogeneous SRS Objective of sampling - Estimate population parameters (eg. means, proportions, variances) using info contained in - Control quantity of info contained in by controlling amount data collected (eg. size) and design (eg. simple random sampling) o Good makes more accurate What is statistics? - Mathematics dealing with collection, presentation, analysis, interpretation of data - Logical thinking using data to estimate relationships between variables Two branches - Descriptive (First 4 lectures) collecting, describing, presenting data o Ask information, then explain it - Inferential (Lecture 5 onwards) drawing conclusions/making decisions concerning population based on collected Data sources - Primary data collected for specific purpose - Secondary data already available, collected for other reasons from govt departments, industry associations, universities, commercial researchers Types of data - Qualitative/categorical non-numerical data that are frequency counts of categories from variables

2 o Data is simply identified/name, has no numerical meaning o Two types: Nominal uses labels to classify Eg. Marital status 1 for married, 2 for single etc. Ordinal natural ordering to rank units, diffs between numbers/levels not same Eg. difference between happy/very happy person not equal to difference between sad/very sad person - Quantitative/numerical date takes numerical values from counting/measurements o Natural order, numbers represent a quantity o Two types discrete, continuous Discrete counting processes, values are whole numbers, no decimals Eg. number of siblings, number of times tails appears when tossing coin Continuous taking measurements, values can have decimals Eg. time, weight, height Levels of data measurement - Four levels, ranked from lowest to highest measurement level o Characterised by properties - 1. Nominal only classification o Number merely a name, used to sort/distinguish objects; ordering of categories is arbitrary o Eg. gender, ethnicity, political affiliation, marital status - 2. Ordinal classification and order o Number used to indicate rank, relative magnitude of numbers meaningful o Diff between numbers not comparable, provides no info about difference between points o Eg. Ranking of 1 st, 2 nd, 3 rd, 4 th tells nothing about how much better 1 st is than 4 th o Eg. team rankings, exam grades, development stages - 3. Interval classification, order, equal intervals o Gaps between numbers are comparable, have same meaning regardless of location on scale o Ability to meaningfully add/subtract data o Zero has no meaning arbitrary figure, matter of convenience, not natural/fixed data point o Eg. temperature each degree has same value, but 0 doesn t mean absence of temp o Eg. IQ scores, standardised testing, temperature, magnitude estimation scales

3 - 4. Ratio classification, order, equal intervals, true zero o Highest form data measurement, most familiar o Differences between data interpretable o Natural zero absence of the property being measured o Real method important for maths/physics o Eg. weight, height, bank balance, number of bedrooms in house, mass, length, time Classify by type of data - Hair colour qualitative, nominal - Blood type qualitative, nominal - Age quantitative, discrete (year by year), continuous (years, months, days, hours, minutes) - Height quantitative, continuous Classify by level of measurement - 3 rd fastest in race ordinal ranking, but doesn t tell difference between numbers - Number of cows ratio differences between data interpretable, may have 0 cows - Political affiliation nominal purely assigning arbitrary numerical name, sorting function - Temperature interval gaps comparable but no true zero Basic definitions - Variable characteristic, expected to differ from one person to another - Population collection of all members of group being investigated, general/broad definition - Census survey of entire population - Sample subset of units in population - Parameter - descriptive measure of population, represented by Greek letters o μ for mean, σ for standard deviation, σ² for population variance - Statistic descriptive measure of, represented by Roman letters o X for mean, S² for variance Objective of sampling - Estimate population parameters (eg. means, proportions, variances) using info contained in - Control quantity of info contained in by controlling amount data collected (eg. size) and design (eg. simple random sampling) o Good makes more accurate Factors influencing size - Resources (time, money) available - Amount of error that can be tolerated - Amount of variation in population o Homogenous lots of people share similar traits, can ask less o Heterogeneous many people different traits (eg. gender, age,) ask more representative - Sampling method - Confidence level how confident in data? Want less error ask more people - Population size ask more, more representative

4 Reasons for sampling - Studying behaviour of, get good idea of population behaviour; not perfect but good enough for decision making - 5 main reasons: o 1. Less costly o 2. Less time consuming o 3. More practical than census, possible gather more details o 4. If survey population, possible no product left (destruction) o 5. Possible population can t be accessed, difficult ask all people Statistical inference - Using can never estimate with certainty; include measure of reliability o Confidence and Significance level Errors in sampling - 2 types: sampling, non-sampling o Sampling difference between characteristic value in population parameter and value in statistic Sampling Error = Parameter Statistic Eg. Population mean age = 21, Sample (10 people) mean age = 30 Unavoidable consequence of only observing of elements Reduced by increasing size/altering design o Non-sampling: Faulty sampling frame (listing/source of population) Eg. Address book doesn t have everyone s details, not whole population Non-response persons in don t respond Response/recording/processing errors Sampling design and methods - Random v Non-random sampling o Random every population unit has known probability of being selected in Apply probability theory inferences/conclusions made about population More accurate/reliable than non-random - Random: o Simple Random Sampling (SRS) Every item from sampling frame has equal chance being selected, lottery, raffle o Systematic Sampling chosen from ordered list of population, with starting point chosen randomly and each successive member selected systematically Random every kth person selected from population Process:

5 1. Define sampling frame listing of population (N) 2. Define size (n) 3. Calculate k k = N/n 4. Randomly select starting point between 1 and k 5. Select every kth individual Advantages: Convenience Sampling evenly distributed across frame knowledgeable person easily determine whether sampling plan followed accurately Disadvantages subject to data periodicity, sampling would be nonrandom/cyclical o Stratified Process: Population divided into non-overlapping subpopulations called strata Researcher extracts SRS from each of strata Advantages reduces potential for sampling error Disadvantages costly each unit assigned to stratum before random selection Info usually secondary Strata should internally be homogenous and externally contrast with each other Proportionate % from each stratum proportionate to % of strata in population o Cluster Pop divided into non-overlapping areas/clusters, but clusters are heterogeneous Each cluster has many elements, a microcosm of population After choosing clusters, researchers randomly select from clusters Two-stage sampling sometimes cluster too large second set of clusters taken from each original cluster Eg. break Australia into states, then into suburbs Advantages: Convenience, cost clustering reduces distance to d elements Disadvantages: If cluster elements similar, may be less efficient than SRS Costs greater than SRS - Non-random not suitable for inferential statistics methods o Convenience elements for selected for researchers convenience Choose elements readily available, nearby Sample less variable, not representative o Judgment attempt obtain representative, resulting saved time/money Can t calculate sampling error