Data Management mathematics in the modern world PDF

Preview

Summary

This is a book for academic purpose it has to be precise and many students should learn from it always remember always choose to learn bow...

Description

Republic of the Philippines

CAMARINES NORTE STATE COLLEGE F. Pimentel Avenue, Brgy. 2, Daet, Camarines Norte – 4600, Philippines

NAME OF DELIVERY UNIT

GEC 3 – Mathematics in the Modern World

Page 1 of 14

Republic of the Philippines

CAMARINES NORTE STATE COLLEGE F. Pimentel Avenue, Brgy. 2, Daet, Camarines Norte – 4600, Philippines

NAME OF DELIVERY UNIT

GEC 3 – MATHEMATICS IN THE MODERN WORLD ENGR. FRANCES ANGELIQUE T. UBANA Instructor I Contact Details Contact Number: 09489322205 E-mail Address: [email protected] Web Address: n/a Consultation Schedule T-Th 8-10am

OUTLINE OF LEARNING TOPICS

TIME ALLOCATION

B. Mathematics as a Tool Week 7 to Week 12 [Midterm] B.1 Data Management -Data: Gathering and Organizing Data; Representing using Graphs and Charts; Interpreting Organized Data -Measures of Central Tendency: Mean, Median, Mode, AWM -Measures of Dispersion: Range, Standard Deviation and Variance -Measures of Relative Position: z-scores, Percentiles, Quartiles -Basic/Elementary Probability -Inferential Statistics: t-test, ANOVA & Pearson r Coefficient

INTENDED LEARNING OUTCOMES (ILOs) At the end of the topic, students should be able to • Use a variety of statistical tools to process and manage numerical data. • Use the methods of linear regression and correlations to predict the value of a variable given certain conditions • Advocate the use of statistical data in making important decisions INSTRUCTIONS TO THE LEARNERS This learning material serves as a reflection among one of the flexible learning strategies that complement the outcomes-based education approach. This material contains the essential discussions for the specified topic together with a learning activity in order to achieve the indicated intended learning outcomes. In addition, students will undergo pre-test and post-test short-examination. The pre-test questionnaires will be given at the start of each rating period (Prelims, Midterms, Finals) while the post-test questionnaires will be given at the end of each rating period. The results of the assessment will serve as one of the key indicators that determine the effectiveness of this learning material. Thus, exemplifying honesty and rectitude in this particular undertaking are highly appreciated and commendable. Always keep connected and updated with announcements and relevant information concerning this course. Lastly, do not hesitate to ask for assistance and raise your concerns to your instructor / professor. GEC 3 – Mathematics in the Modern World

Page 2 of 14

Republic of the Philippines

CAMARINES NORTE STATE COLLEGE F. Pimentel Avenue, Brgy. 2, Daet, Camarines Norte – 4600, Philippines

NAME OF DELIVERY UNIT

B. Mathematics as a Tool Introduction Mathematics is a powerful tool for global understanding and communication. Using it, students can make sense of the world and solve complex and real problems. Rethinking math in a global context offers students a twist on the typical content that makes the math itself more applicable and meaningful for students. For students to function in a global context, math content needs to help them get to global competence, which is understanding different perspectives and world conditions, recognizing that issues are interconnected across the globe, as well as communicating and acting in appropriate ways. In math, this means reconsidering the typical content in a typical ways and showing students how the world consists of situations, events and phenomena that can be sorted out using the right math tools. In this learning material, you will find out how mathematics is applied as a powerful tool in our nature.

B.1 Data Management B.1.1. Data Data is everywhere. It is observable or measurable. With the advancement of technology every day, data can be accessed anywhere and by anyone. When data is correct, valid analysis and interpretation can be generated to produce valuable information.

Gathering and Organizing Data Data are the quantities Data are the quantities (numbers) or qualities (attributes) measured or observed that are to be collected and analyzed (Asaad, 2004). There are two types of data: the qualitative and quantitative data. Qualitative data deals with categories or attributes. Examples are color of eyes, ethnicity and brand of ice cream. Quantitative data are numerical data. Quantitative data can be discrete or continuous. Discrete data is obtained through counting. Continuous data is obtained by measuring. The number of households in a particular community is an example of discrete data while family income and weight of an individual are some of the examples of continuous data. Another way is to classify data into four levels of measurement such as nominal, ordinal, interval and ratio. The nominal level of measurement is the lowest of the four ways to characterize data. Nominal data deals with names, categories, or labels. Data at the nominal level is qualitative. Colors of eyes, yes or no responses to a survey, and favorite breakfast cereal all deal with the nominal level of measurement. Ordinal level of measurement ranks qualitative data. Winners in a pageant and the academic rank of teachers are examples of ordinal data. Interval level of measurement deals with data that can be ordered, and in which differences between the data does make sense. Data at this level does not have a starting point. The Fahrenheit and Celsius scales of temperatures are both examples of data at the interval level of measurement. The fourth and highest level of measurement is the ratio level. Data at the ratio level possess all of the features of the interval level, in addition to a zero value. Examples are weight, the time to answer a quiz and the number of absences of students in a class.

GEC 3 – Mathematics in the Modern World

Page 3 of 14

Republic of the Philippines

CAMARINES NORTE STATE COLLEGE F. Pimentel Avenue, Brgy. 2, Daet, Camarines Norte – 4600, Philippines

NAME OF DELIVERY UNIT Learning Activity No. 1 Part 1. Abstraction (Classification) Instructions: For each of these variables, identify whether the variable is qualitative or quantitative, and if quantitative, state whether it is discrete or continuous. 1) Number of family members in a particular household Answer: _________________________ 2) Ownership of a cell phone among family members

Answer: _________________________

3) Length (in minutes) of a longest call made per month

Answer: _________________________

4) Amount spent on food in a day

Answer: _________________________

5) Occupation of household head

Answer: _________________________

Part 2. Abstraction (Classification) Instructions: Identify the level of measurement for each of the following variable 1) Highest education attainment Answer: _________________________ 2) Hair color

Answer: _________________________

3) Body Temperature

Answer: _________________________

4) Civil Status

Answer: _________________________

5) Total household expenditures in Pesos

Answer: _________________________

Representing using Graphs and Charts; and Interpreting Organized Data After the data have been collected and processed, data need to be organized to produce meaningful information. There are three methods in presenting information from the data set.

 Textual or paragraph or narrative form This describes the data by enumerating some of the important feature of the data set like giving the highest, lowest or the average values. In case there are only few observations, say less than ten observations, the values could be enumerated if there is a need to do so. Data could also be presented using tables.

 The tabular method of presentation This is applicable for large data sets. A frequency is the number of times a value of the data occurs. A frequency distribution is the organization of raw data in table form, using classes and frequencies. A relative frequency is the ratio (fraction or proportion) of the number of times a value of the data occurs in the set of all outcomes to the total number of outcomes. To find the relative frequencies, divide each frequency by the total number of students in the sample . Relative frequencies can be written as fractions, percent, or decimals. Cumulative relative frequency is the accumulation of the previous relative frequencies. To find the cumulative relative frequencies, add all the previous relative frequencies to the relative frequency for the current row. GEC 3 – Mathematics in the Modern World

Page 4 of 14

Republic of the Philippines

CAMARINES NORTE STATE COLLEGE F. Pimentel Avenue, Brgy. 2, Daet, Camarines Norte – 4600, Philippines

NAME OF DELIVERY UNIT Let us refer to the following set of examples Example 1 Twenty-five employees were given a blood test to determine their blood type. Construct a frequency distribution for the data. Raw Data: A, B, B, AB, O

O, O, B, AB, B

B, B, O, A, O

A, O, O, O, AB

AB, A, O, B, A

Table 1. Frequency Table of Employees Blood Type with Relative Frequencies

Class A B O AB

Tally IIII IIII II IIII IIII IIII

Frequency

Relative Frequency (%)

5 7 9 4 n=25

20 28 36 16 100

When the range of the data is large, grouped frequency distributions are used. The smallest and largest possible data values in a class are the lower- and upper-class limits. Class boundaries separate the classes. To find a class boundary, average the upper-class limit of one class and the lower-class limit of the next class. The class width can be calculated by subtracting successive lower-class limits (or boundaries) or successive upper-class limits (or boundaries). The class midpoint X m can be calculated by averaging upper and lower class limits (or boundaries). Rules for Classes in Grouped Frequency Distributions There should be 5-20 classes, the class width must be an odd number, mutually exclusive, continuous, exhaustive and must be equal in width (except in open-ended distributions). Example 2 The following data represent the record high temperatures for each of the 50 states. Construct a grouped frequency distribution for the data. 112 100 127 120 134 118 105 110 109 112 110 118 117 116 118 122 114 114 105 109 107 112 114 115 118 117 118 122 106 110 116 108 110 121 113 120 119 111 104 111 120 113 120 117 105 110 118 112 114 114 Steps in Constructing a Grouped Frequency Distribution Step 1. Determine the range. Range = Highest score – Lowest score = 134 – 100 = 34 Step 2. Determine the no. of classes k, k = 1 + 3.32 log n, where n is the no. of population. k = 1 + 3.32 log n; 1+ 3.32 log 50 = 6.64 𝑅

Step 3. Obtain the class size or class width, i, i = 𝑘 =

34 = 6.64

5.12 or 5

Step 4. Make the classes. For convenience sake, we will choose the lowest data value, 100 for the first lower class limit. The subsequent lower-class limits are found by adding the width to the previous lower-class limits. GEC 3 – Mathematics in the Modern World

Page 5 of 14

Republic of the Philippines

CAMARINES NORTE STATE COLLEGE F. Pimentel Avenue, Brgy. 2, Daet, Camarines Norte – 4600, Philippines

NAME OF DELIVERY UNIT Step 5. Tally the data. Table 2. Grouped Frequency Distribution of Recorded High Temperature of 50 States

Class Limits

Class Boundaries

Frequency

Cumulative Frequency

130-134 125-129 120-124 115-119 110-114 105-109 100-104

129.5 - 134.5 124.5-129.5 119.5-124.5 114.5-119.5 109.5-114.5 104.5-109.5 99.5-104.5

1 1 7 13 18 8 2 n=50

50 49 48 41 28 10 2

Based from the constructed table, answer the following questions: a. How many states experienced the temperature 115-119? b. What temperature was experienced by 18 states? c. What is the cumulative for the temperature 120-124?

_________________ _________________ _________________

Learning Activity No. 2 – Abstraction (Critical Thinking) Instructions: Follow what each item requires with the given set of data. 1. Twenty farmers were asked how many hours they worked per day. Their responses, in hours, are as follows: 5; 6; 3; 3; 2; 4; 7; 5; 2; 3; 5; 6; 5; 4; 4; 3; 5; 2; 5; 3. Construct a frequency distribution and interpret the data. 2. Construct a frequency distribution for the following data and interpret. 11 16 21 11 19

19 16 11 17 13

12 15 13 24 18

15 16 21 12 20

10 27 29 23 11

18 16 15 26 11

15 23 24 15 12

10 11 12 11 18

25 17 21 14 12

13 12 12 13 16

 Graphical Presentation The graphical presentation on the other hand, is a visual presentation of the data. A graph is a tool that helps you learn about the shape or distribution of a sample. The graph can be a more effective way of presenting data than a mass of numbers because we can see where data clusters and where there are only a few data values. Graphs are commonly used in oral presentation. There are several forms of graphs to use like the pie chart, pictograph, bar graph, line graph, histogram, frequency polygon and box-plot. 1. Histogram A histogram is a graph that displays the data by using vertical bars of various heights to represent the frequencies of the classes. The horizontal axis is labeled with what the data represents. The vertical axis is labeled either frequency or relative frequency. Histograms use class boundaries and frequencies of the classes.

GEC 3 – Mathematics in the Modern World

Page 6 of 14

Republic of the Philippines

CAMARINES NORTE STATE COLLEGE F. Pimentel Avenue, Brgy. 2, Daet, Camarines Norte – 4600, Philippines

NAME OF DELIVERY UNIT Example Construct a histogram to represent the data for the record high temperatures for each of the 50 states. Class Limits

Class Boundaries

Frequency

130-134 125-129 120-124 115-119 110-114 105-109 100-104

129.5 - 134.5 124.5-129.5 119.5-124.5 114.5-119.5 109.5-114.5 104.5-109.5 99.5-104.5

1 1 7 13 18 8 2 n=50

2. Frequency Polygon Frequency polygon is a graph that displays the data by using lines that connect points plotted for the frequencies at the class midpoints. The frequencies are represented by the heights of the points and the class midpoints are represented on the horizontal axis. Frequency polygons use class midpoints and frequencies of the classes. Example Using the example above, construct a frequency polygon. Class Limits

Class Midpoints

Frequency

130-134 125-129 120-124 115-119 110-114 105-109 100-104

132 127 122 117 112 107 102

1 1 7 13 18 8 2 n=50

GEC 3 – Mathematics in the Modern World

Page 7 of 14

Republic of the Philippines

CAMARINES NORTE STATE COLLEGE F. Pimentel Avenue, Brgy. 2, Daet, Camarines Norte – 4600, Philippines

NAME OF DELIVERY UNIT A frequency polygon is anchored on the x-axis before the first class and after the last class.

3. Box Plot A box plot is also known as box-and-whisker plots or box-whisker plots. It shows how far the extreme values are from most of the data. A box plot is constructed from five values: the minimum value, the 1st quartile, the median, the 3rd quartile, and the maximum value. We use these values to compare how close other data values are to them. To construct a box plot, use a horizontal or vertical number line and a rectangular box. The smallest and largest data values label the endpoints of the axis. The 1st quartile marks one end of the box and the 3rd quartile marks the other end of the box. The "whiskers" extend from the ends of the box to the smallest and largest data values. The median or 2nd quartile can be between the 1st & 3rd quartiles, or it can be one, or the other, or both. The box plot gives a good, quick picture of the data. Example Consider the dataset. 1; 1; 2; 2; 4; 6; 6.8; 7.2; 8; 8.3; 9; 10; 10; 11.5 The 1st quartile is 2, the median is 7, and the 3rd quartile is 9. The smallest value is 1, and the largest value is 11.5. The following image shows the constructed box plot.

4. Pareto Chart A Pareto chart is a bar graph. The lengths of the bars represent frequency or cost (time or money), and are arranged with longest bars on the left and the shortest to the right.

GEC 3 – Mathematics in the Modern World

Page 8 of 14

Republic of the Philippines

CAMARINES NORTE STATE COLLEGE F. Pimentel Avenue, Brgy. 2, Daet, Camarines Norte – 4600, Philippines

NAME OF DELIVERY UNIT Other Types of Graphs 1. Bar Graph A bar graph or a bar chart is used to represent data visually using bars of different heights or lengths. Data is graphed either horizontally or vertically, allowing viewers to compare different values and draw conclusions quickly and easily.

2. Time Series Graph A time series chart, also called a times series graph or time series plot, is a data visualization tool that illustrates data points at successive intervals of time. Each point on the chart corresponds to both a time and a quantity that is being measured.

3. Pie Chart A Pie Chart (or Pie Graph) is a special chart that uses "pie slices" to show relative sizes of data. The chart is divided into sectors, where each sector shows the relative size of each value.

GEC 3 – Mathematics in the Modern World

Page 9 of 14

Republic of the Philippines

CAMARINES NORTE STATE COLLEGE F. Pimentel Avenue, Brgy. 2, Daet, Camarines Norte – 4600, Philippines

NAME OF DELIVERY UNIT Learning Activity No. 3 – Abstraction (Critical Thinking) Instructions: Follow what each item requires with the given set of data. 1. Make a box plot using the data below. 12, 5, 22, 30, 7, 36, 14, 42, 15, 53, 25 2. Make a histogram and frequency polygon.

3. Make a pie chart using the data below Science Club = 30% Math Club = 40%

English Club = 20%

History Club = 10%

B.1.2. Measures of Central Tendency Any measure indicating the center of a set of data arranged in an array is known as Measure of Central Tendency. Measure of Central Tendency provides a very convenient method of describing a set of scores with a single value that is used to describe the “center” of the data. The most commonly used measures of Central tendency are the mean, median and mode. Mean (𝒙ഥ) is the set of scores or observations being added and divided by the number of scores. It is also called as average or arithmetic mean. 𝑥 =

∑𝑥

where: 𝑥 = sample mean ∑ 𝑥 = sum of the scores or observation n = number of scores or observation

𝑛

Let us refer to the following set of examples Example 1 The data below are the ages of samples of 10 pupils in a certain school. Find the mean. 9, 8, 10, 7, 7, 8, 12, 9, 10,11 Solution:

𝑥 =

∑𝑥 𝑛

=

9+8+10+7+7+8+12+9+!0+11 10

=

91 10

= 9.1

The average age of 10 pupils in a certain school is 9.1. GEC 3 – Mathematics in the Modern World

Page 10 of 14

Republic of the Philippines

CAMARINES NORTE STATE COLLEGE F. Pimentel Avenue, Brgy. 2, Daet, Camarines Norte – 4600, Philippines

NAME OF DELIVERY UNIT Example 2 The data below are the number of employees in a certain store. Find the mean. 3, 6, 5, 4, 7, 8,10 Solution: 𝑥 =

∑𝑥<...