Modeling data distributions #1 - Questions and Answers

Question 1 Presenting data in the form of table. For the data set shown by the table, Solve, a) Create a scatter plot for the data. b) Use the scatter plot to determine whether an exponential function, a logarithmic function, or a linear function is the best choice for modeling the data. (If applicable, you will use your graphing utility to obtain these functions.) Intensity (wattd per meter²) 0.1(loud thunder) 1(rock concert, 2 yd from speakers) 10(jackhammer) 100(jet take off, 40 yd away) Answer Step 1 Loudness Level (decibels) 110 120 130 140 a) We can draw scatter plot using Excel as follows, Enter the data in Excel and select it. Go to insert → scatter → select Ist scatter plot → add trendline to the plot → Tick on display equation on plot and display R square on the plot. We get result as follows, 150 145 140 135 130 125 t 120 115 110 105 100 0 20 40 60 80 100 Step 2 b) From the above scatter plot, we can observe that, the regression equation for logarithmic model is, ŷ = 4.3429ln (x) + 120 and R2 = 1 Since, the value of R2 = 1 for Logarithmic function, the logarithmic function is the best choice for modeling the data. Question 2 List 7 ways of representing data Answer 1 Step 1 Tables, charts, and graphs can all be used to represent data. Different ways of graphical representation are- Pie Chart Bar Chart Histogram Boxplot Line Chart Frequency Polygon Step 2 You can create simple tables or tables that display frequency distributions. Measures of central tendency and dispersion like mean, median, mode, range, and standard deviation can also be used to display data. Answer 2 Text - Representing data using plain text. Numerical values - Representing data using numbers. Tables - Representing data in tabular format. Graphs - Representing data using visual plots, such as bar graphs or line graphs. Charts - Representing data using visual representations like pie charts or scatter plots. Diagrams - Representing data using graphical illustrations or flowcharts. Maps - Representing data using geographical maps or spatial representations. Answer 3 Tabular format: A table is a common way to represent structured data. Tabular format Graphs: Graphs are used to visualize relationships between data points. Graphs Charts: Charts, such as bar charts or pie charts, are used to display data visually. Charts Matrices: Matrices are used to represent data in a grid-like format. Matrices Trees: Trees are hierarchical structures that represent relationships between data elements. Trees Networks: Networks or graphs are used to represent interconnected data elements. Networks Equations: Equations are used to represent mathematical relationships between variables. Equations These are just a few examples of ways to represent data. Depending on the context and type of data, there are many other methods and formats available. Question 3 Discuss the importance of data modeling. Answer Step 1 Data models can facilitate interaction among the designer, the applications programmer, and the end user. The Importance of the Data Modelling Data constitute the most basic information units employed by a system. Applications are created to manage data and to help transform data into information. But data are viewed in different ways by different people. So that there is a huge importance of data modeling in DBMS. For example, contrast the (data) view of a company manager with that of a company clerk. Although the manager and the clerk both work for the same company, the manager is more likely to have an enterprise-wide view of company data than the clerk. Applications programmers have yet another view of data, being more concerned with data location, formatting, and specific reporting requirements. Basically, applications programmers translate company policies and procedures from a variety of sources into appropriate interfaces, reports, and query screens. When a good database blueprint is available, it does not matter that an applications programmer's view of the data is different from that of the manager and/or the end user. Conversely, when a good database blueprint is not available, problems are likely to occur. For instance, an inventory management program and an order entry system may use conflicting product-numbering schemes, thereby costing the company thousands (or even millions) of dollars. The data model is an abstraction; you cannot draw the required data out of the data model. Step 2 The aim of data modeling is to establish and to keep the references of all concepts used or applied during succeeding software engineering processes. Engineers must have a common unique reference of the concepts of the domain. Varying references of concepts causes trouble and puts the whole project to risk. Example: 'c is const' was given as axiom..Later, one uses 'c + v' or 'c - v' to calculate sth that refers to objects that were given property 'speed' with constant value, namely c. Ongoing Changes of basic structures during devolopment causes the overall process to cycle and cycle and cycle ... Example: The discussion on special relativity is ongoing sind 111 years. This is because the theory is not sound (not consistent) and allows changes of their main concepts which is (as a intrinsically given schema) adopted by their followers to be allowed. (The list of effects of missing clear concepts or data structures is very long) Question 4 Look at the normal curve below, and find μ, μ + σ, and σ μ μ+σ = = σ = 16 18 20 22 Answer 1 Step 1 The value of mean (μ) is obtained below: The below graph represents the normal curve with the one standard deviation, two standard deviations and three standard deviations either side of the mean with their respective values. K 99,7% 85% 68% μ-30 μ-20 μ-σ μ μισ με 20 μ+30 16 18 20 22 From the graph it is clear that, parameter that lies in the center of the normal curve can be identified as the population mean (μ) which is 16. The value of population mean (μ) is 16. The value of mean (μ) is obtained by using the concept of the empirical rule. Step 2 The value of one standard deviation away from the mean (μ + σ) is obtained below: The below graph represents the normal curve with the one standard deviation, two standard deviations and three standard deviations either side of the mean with their respective values. 99,7% 85% 68% μ-30 μ-20 μ-σ μ μεσ με 20 μ+30 22 16 18 20 By the empirical rule, one standard deviation away from the mean can be identified as (μ + σ) which is 18. The value of one standard deviation away from the mean is, μ + σ = 18 The value of one standard deviation away from the mean (μ + σ) is obtained by using the concept of the empirical rule. Step 3 The value of population standard deviation is obtained below: The below graph represents the normal curve with the one standard deviation, two standard deviations and three standard deviations either side of the mean with their respective values. ← 99,7% 95% 68% K μ-30 μ-20 μ-σμ μεσ με20 μ+30 22 16 18 20 It is clear that, the value of one standard deviation away from the mean (μ +σ) is 18 and the value of the population mean (μ) is 16. Consider: μ + σ = 18 16 + σ = 18 σ = 18 – 16 σ = 2 The value of population standard deviation (σ) is 2. The value of population standard deviation (σ) is obtained by the difference between the values of one standard deviation away from the mean (μ + σ) to the population mean. Answer 2 Step 1 Look at the normal curve below, and find μ, μ + σ, and σ. 16 18 20 22 from above curve: μ = central value on the curve =16 μ + 0 = 1 standard deviaiton greater value from mean =18 σ = 18 – 16 = 2 Question 5 True or false to each of the statements in parts (a) and (b), and explain your reasoning. a. Two data sets that have identical frequency distributions have identical relative-frequency distributions. b. Two data sets that have identical relative-frequency distributions have identical frequency distributions. c. Use your answers to parts (a) and (b) to explain why relativefrequency distributions are better than frequency distributions for comparing two data sets. Answer 1 a) The frequency of the class is defined as the total number of observations in the class, and the relative frequency of the class is defined as the ratio of the class frequency to the total number of observations in the class. The claim that two data sets with the same frequency distribution also have the same relative frequency distribution is thus true. b) The distinction is that relative frequency of each class is multiplied by 100 to determine the proportion (percentage). If the percentage of a class is expressed as a decimal value, however, it is relative frequency of the class. According to the assertion, the ratio of the relative frequency distribution will equal the total of the two frequency distributions. Because doubling one distribution yields the entire value of the other, for instance, if the percentage of two distributions is 5, 4, 1, and 10, 8, 2, their values will differ even if they have the same relative frequency distribution. The claim that two data sets with the same relative frequency distribution also have the same frequency distribution is false. c) Based on the findings of parts a and b, it can be seen that data sets with the same number of observations use the frequency or relative frequency distribution. In contrast, the relative frequency distribution is the best option if the data set contains a range of observation counts since the total of each set's relative frequencies equals one, making it simple and clear to compare the relative frequency distribution with other distributions. Therefore, relative frequency distribution is preferable than frequency distribution when comparing two data sets. Answer 2 Step 1: a. The statement in part (a) is false. Two data sets that have identical frequency distributions may not necessarily have identical relative-frequency distributions. To illustrate this, let's consider two data sets: Data Set 1: 1, 2, 3, 4, 5 Data Set 2: 10, 20, 30, 40, 50 Both data sets have the same frequency distribution, as each value appears once. However, their relative-frequency distributions differ. The relative-frequency distribution for Data Set 1 is {0.2, 0.2, 0.2, 0.2, 0.2), while the relative-frequency distribution for Data Set 2 is {0.1, 0.1, 0.1, 0.1, 0.1}. Thus, the statement is false. Step 2: b. The statement in part (b) is true. Two data sets that have identical relative-frequency distributions will have identical frequency distributions. To demonstrate this, let's consider two data sets: Data Set 1: 1, 2, 3, 4, 5 Data Set 2: 10, 20, 30, 40, 50 If we calculate the relative frequencies for both data sets, we obtain the same values: {0.2, 0.2, 0.2, 0.2,0.2} for both sets. From these relative frequencies, we can infer the frequency distributions for both sets. In this case, both data sets have the same frequency distribution: {1, 1, 1, 1, 1). Therefore, the statement is true. Step 3: c. The relative-frequency distributions are better than frequency distributions for comparing two data sets because they allow for a more meaningful comparison that is not influenced by differences in sample sizes. Frequency distributions provide information about the counts or frequencies of individual values in a data set. While they can be useful, they do not account for differences in sample sizes between data sets. Comparing frequency distributions directly may lead to misleading conclusions, as larger data sets will naturally have higher frequencies. On the other hand, relative-frequency distributions express the frequencies as proportions or percentages relative to the total sample size. By doing so, they provide a standardized way of comparing data sets, regardless of their sample sizes. This allows for a fairer comparison that focuses on the relative occurrence of values within each data set. In conclusion, relative-frequency distributions are preferred over frequency distributions when comparing two data sets because they provide a more accurate and unbiased representation of the data, eliminating the influence of sample size disparities. Answer 3 a. False. Two data sets that have identical frequency distributions may have different sample sizes, resulting in different relative-frequency distributions. Relative frequencies are calculated by dividing the frequency of each observation by the total sample size. Since the sample sizes can differ between the two data sets, their relative-frequency distributions will not be identical. b. True. Two data sets that have identical relative-frequency distributions necessarily have the same proportions for each observation. This implies that the frequencies of the observations must also be the same in both data sets, leading to identical frequency distributions. c. Relative-frequency distributions are better than frequency distributions for comparing two data sets because they eliminate the influence of sample size. By focusing on proportions rather than raw frequencies, relative-frequency distributions allow for a direct comparison of the distributions, even when the sample sizes differ. This makes it easier to identify patterns and similarities between data sets without being affected by differences in scale.