Mathematics in Modern World, Lecture notes of Mathematics

Download Mathematics in Modern World and more Lecture notes Mathematics in PDF only on Docsity! Mathematics in the Modern World GEED 10053 WEEK 6-8 Statistics and Data Management Data Gathering and Sampling Techniques Data Presentation Descriptive Measures Statistics P1 Prof. C. Equiza Topic Outline v Steps in Statistical Investigation v Sampling Techniques, Sample Size Considerations, Methods of Data Collection v Levels of Measurement v Presentations of Data, Textual, Tabular and v Graphical Presentations: Graphs, Charts, Time Series Plots Scope of Statistics 1. Statistics and INDUSTRY. Statistics is widely used in many industries. In industries, control charts are widely used to maintain a certain quality level. In production engineering, to find whether the product is conforming to specifications or not, statistical tools, namely inspection plans, control charts, etc., are of extreme importance. In inspection plans we have to resort to some kind of sampling - a very important aspect of Statistics. 2. Statistics and Commerce. Statistics are lifeblood of successful commerce. Any businessman cannot afford to either by under stocking or having overstock of his goods. In the beginning he estimates the demand for his goods and then takes steps to adjust with his output or purchases. Thus statistics is indispensable in business and commerce. 3. Statistics and Economics. Statistical methods are useful in measuring numerical changes in complex groups and interpreting collective phenomenon. Nowadays the uses of statistics are abundantly made in any economic study. Both in economic theory and practice, statistical methods play an important role. 4. Statistics and EDUCATION. Statistics is widely used in education. Research has become a common feature in all branches of activities. Statistics is necessary for the formulation of policies to start new course, consideration of facilities available for new courses etc. There are many people engaged in research work to test the past knowledge and evolve new knowledge. These are possible only through statistics. 5. Statistics and Planning. Statistics is indispensable in planning. In the modern world, which can be termed as the “world of planning”, almost all the organizations in the government are seeking the help of planning for efficient working, for the formulation of policy decisions and execution of the same. In order to achieve the above goals, the statistical data relating to production, consumption, demand, supply, prices, investments, income expenditure etc and various advanced statistical techniques for processing, analyzing and interpreting such complex data are of importance. 6. Statistics and Medicine. In Medical sciences, statistical tools are widely used. In order to test the efficiency of a new drug or medicine, t - test is used or to compare the efficiency of two drugs or two medicines, t-test for the two samples is used. More and more applications of statistics are at present used in clinical investigation. 7. Statistics and Modern Applications. Recent developments in the fields of computer technology and information technology have enabled statistics to integrate their models and thus make statistics a part of decision making procedures of many organizations. There are so many software packages available for solving design of experiments, forecasting simulation problems etc. Limitation of Statistics 1. Statistics is not SUITABLE to the STUDY of QUALITATIVE phenomenon. Since statistics is basically a science and deals with a set of numerical data, it is applicable to the study of only these subjects of enquiry, which can be expressed in terms of quantitative measurements. As a matter of fact, qualitative phenomenon like honesty, poverty, beauty, intelligence etc, cannot be expressed numerically and any statistical analysis cannot be directly applied on these qualitative phenomenon. 2. Statistics does not STUDY INDIVIDUALS. Statistics does not give any specific importance to the individual items; in fact it deals with an aggregate of objects. Individual items, when they are taken individually do not constitute any statistical data and do not serve any purpose for any statistical enquiry. 3. Statistical laws are not exact. It is well known that mathematical and physical sciences are exact. But statistical laws are not exact and statistical laws are only approximations. Statistical conclusions are not universally true. They are true only on an average. Steps in Statistical Investigation 1. Defining the problem a) Identify a specific problem. b) Define the scope and limitations, assumptions to be made, and expected outcomes. 2. Collection of data a) Make sure to collect the data properly. b) Incomplete, fabricated, outdated, and inaccurate data are useless. 3. SUMMArization and TABULATION of data a) This refers to organization of data in text, tables, graphs and charts, so that logical conclusion can be derived from them. b) Explore the data to obtain additional insight that could contribute to the study. 4. Analysis of data a) This pertains to the process of deriving from the given data relevant information from which numerical descriptions can be formulated. b) Summarized data must be examined so that insights and meaningful information ca be produced to support decision-making or solutions to the question or problem at hand. 5. Interpretation of data and RESULTS a) Refers to the task of drawing conclusions from the analyzed data. b) Results must be able to answer the research problem and give recommendations. 6. Presentation of the RESULT a) Present all pertinent results in a clear and concise manner. b) Use appropriate form of media to present results. Sampling and Sampling Techniques Sampling refers to the process of obtaining samples from the population. Sampling maybe categorized as either probability sampling or non-probability sampling. Probability sampling, also referred to as random sampling, is the method of sampling in which every member of the population have equal chance of being selected as sample; otherwise, it is considered as non-probability sampling. We should note that in able to properly use the techniques of statistical inference, probability sampling must be used to obtain samples. Sample Size Consideration The sample size is typically denoted by n and it is always a positive integer. No exact sample size can be mentioned here and it can vary in different research settings. However, all else being equal, large sized sample leads to increased precision in estimates of various properties of the population. To determine the sample size we can apply one of the following methods: 1. Slovin’s FoRMUla. Slovin’s formula is used to calculate the sample size n given the population size and a margin of error E. It is a formula use to estimate sampling size of a random sample from a given population. We can compute N n = ; 1 + NE2 where N is the population size. Example: A researcher plans to conduct a survey about food preference of BS Stat students. If the population of students is 1000, use the Slovin’s formula to find the sample size if the margin of error is 5%. Solution. Using the Slovin’s formula, we get 1000 n = 1 + 1000(0:05)2 ≈ 285:71: Therefore, the researcher needs to survey 286 BS Stat Students. (𝑍𝛼 /2)2σ2 n = E2 ; 2. MINIMUM Sample Size for Estimating a POPULATION Mean. The estimated minimum sample size n needed to estimate a population mean — to within E units at 100(1 − 𝛼)% confidence is where σ is the known population standard deviation, E is the margin of error and 𝑍! /2 is a value which can be obtained in the z -table. Example: Suppose we want to know the average age of STEM students. We would like to be 99% confident about our results. From previous study, we know that the standard deviation for the population is 1.3. How many students should be chosen for a survey if the margin of error is 0.2. Solution. Find 𝑍" /2 by looking at the z -table. The closest z -score for 0.005 in the z -table is 2.58. Thus, σ = (1 − 0:99) = 0:01 =⇒ 𝑍! /2 = z0:005 (2.58)2(1.3)2 n = (0:2)2 ≈ 281.23. which we round up to 282, since it is impossible to take a fractional observation. We need a 282 STEM students as a sample for our study. 2 : 1. (𝑍𝛼 /2)2 p̂(1 − p̂) E n = 3. MINIMUM Sample Size for Estimating a POPULATION Proportion The estimated minimum sample size n needed to estimate a population proportion p to within E at 100(1 − 𝛼)% confidence is This is also called the Cochran FoRMULA. The dilemma here is that the formula for estimating how large a sample to take contains the number p̂ , which we know only after we have taken the sample. There are two ways out of this dilemma. v First, typically the researcher will have some idea as to the value of the population proportion p, hence of what the sample proportion p̂ is likely to be. For example, if last month 37% of all voters thought that state taxes are too high, then it is likely that the proportion with that opinion this month will not be dramatically different, and we would use the value 0.37 for p̂ in the formula. v The second approach to resolving the dilemma is simply to replace p̂ in the formula by 0.5. This is because if p̂ is large then 1 − p̂ is small, and vice versa, which limits their product to a maximum value of 0.25, which occurs when p̂ = 0.5. This is called the most conservative estimate, since it gives the largest possible estimate of n. 1. Survey Method. The survey is a method of collecting data on the variable of interest by asking people questions. This may be done, by interview or by using questionnaires. 2. Observation. Observation is a method of obtaining data or information by using our primary senses. 3. Experiment. Experiment is a method of collecting data where there is direct human intervention on the conditions that may affect the values of the variable of interest. Methods of Data Collection Levels of Measurement 1. The nominal level of measUrement classifies data into mutually exclusive (non-overlapping) categories in which no order or ranking can be imposed on the data. Example: Gender (male, female), Zip Code, Color, Nationality, Political affiliation, Religious affiliation. 2. The ordinal level of MEASUREMENT classifies data into categories that can be ranked; however, precise differences between the ranks do not exist. Example: Grade(A,B,C,D,F), Rating Scale/Likert scale, Ranking of tennis players, Judging (First place, second place, etc. 3. The interval level of MEASUREMENT ranks data, and precise differences between units of measure do exist; however, there is no meaningful zero. Example: Temperature, IQ, SAT score 4. The ratio level of MEASUREMENT possesses all the characteristics of interval measurement, and there exists a true zero. In addition, true ratios exist when the same variable is measured on two different members of the population Example: Height, Weight, volume, Time, Salary, Age The following is an example of a Two-Way Table. Table 2 NUMBer of STUDENTS Enrolled for the Last 6 Years When GROUPed According to Sex Sex Year 2012 2013 2014 2015 2016 2017 Total Male 5560 6095 7386 8056 7945 6451 41493 Female 7890 7105 8003 8734 10955 13049 55736 Total 13450 13200 15389 16790 18900 19500 97229 Types of Statistical Charts a) Bar Graph (Vertical BaR/COLUMN Charts) is applicable for showing comparison of amount of a variable of interest collected over time. Simple Chart Grouped Column Chart Subdivided Column Figure 3. Number of Students Enrolled when Grouped According to Sex 7386 8056 2014 2015 Female Male Pie Chart d) Pie Chart is a circular graph partitioned into several section, depicting relative percentage with respect to the total distribution. Simple Line Graph e) Line Graph is a graph used to visualize data that changes continuously over time. Multiple Line Graph f) Statistical Map is used to show data in geographical areas. Statistical Map