MMW- Compilations - Compilation Project finals PDF

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MATHEMATICS IN THE MODERN WORLD (COMPILATIONS)

February 17, 2021

Table of Contents Chapter 1 - Statistics.......................................................................................................................3 - 24

● Introduction to Statistics ● Introduction to Population and Samples ● Quantitative and Qualitative Data ● Sampling Methods and Techniques ● Graphical Representation ● Mean, Median and Mode of Grouped/Ungrouped Data ● Hypothesis ● One Sample and Two Sample T- Test Chapter 2 - Mathematics of Finance.............................................................................................25 - 53

● Concept of Percent ● Percent of Increase and Decrease ● Ratio and Proportions ● Trade Discount ● Pricing the Merchandise ● Compensation ● Salary ● Commission ● Personal Deductions ● Bank Services and Reconciliations ● Income References..........................................................................................................................................54

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CHAPTER 1 STATISTICS Introduction to Statistics Statistics is a mathematical science including methods of collecting, organizing and analyzing data in such a way that meaningful conclusions can be drawn from them. In general, its investigations and analyses fall into two broad categories called descriptive and inferential statistics. Descriptive statistics deals with the processing of data without attempting to draw any inferences from it. The data are presented in the form of tables and graphs. The characteristics of the data are described in simple terms. Events that are dealt with include everyday happenings such as accidents, prices of goods, business, incomes, epidemics, sports data, population data. Inferential statistics is a scientific discipline that uses mathematical tools to make forecasts and projections by analyzing the given data. This is of use to people employed in such fields as engineering, economics, biology, the social sciences, business, agriculture and communications.

Introduction to Populations and Samples A population often consists of a large group of specifically defined elements. For example, the population of a specific country means all the people living within the

boundaries of that country. Usually, it is not possible or practical to measure data for every element of the population under study. We randomly

select a small group of elements from the population and call it a sample. Inferences about the population are then made on the basis of several samples. Example: A company is thinking about buying 50,000 electric batteries from a manufacturer. It will buy the batteries if no more that 1% of the batteries are defective. It is not possible to test each battery in the population of 50,000 batteries since it takes time and costs money. Instead, it will select a few samples of 500 batteries each and test them for defects. The results of these tests will then be used to estimate the percentage of defective batteries in the population. Quantitative and Qualitative Data Data is quantitative if the observations or measurements made on a given variable of a sample or population have numerical values. Example: height, weight, number of children, blood pressure, current, voltage. Data is qualitative if words, groups and categories represents the observations or measurements. Example: colors, yes-no answers, blood group. Quantitative data is discrete if the corresponding data values take discrete values and it is continuous if the data values take continuous values. Example of discrete data: number of children, number of cars.

Example of continuous data: speed, distance, time, pressure. Sampling Methods and Techniques Sampling is a technique of selecting individual members or a subset of the population to make statistical inferences from them and estimate characteristics of the whole population. Different sampling methods are widely used by researchers in market research so that they do not need to research the entire population to collect actionable insights. It is also a time-convenient and a cost-effective method and hence forms the basis of any research design. Sampling techniques can be used in a research survey software for optimum derivation. For example, if a drug manufacturer would like to research the adverse side effects of a drug on the country’s population, it is almost impossible to conduct a research study that involves everyone. In this case, the researcher decides a sample of people from each demographic and then researches them, giving him/her indicative feedback on the drug’s behavior. Types of Sampling: Sampling Methods Sampling in market research is of two types – probability sampling and nonprobability sampling. Let’s take a closer look at these two methods of sampling. 1. Probability sampling: Probability sampling is a sampling technique where a researcher sets a selection of a few criteria and chooses

members of a population randomly. All the members have an equal opportunity to be a part of the sample with this selection parameter. 2. Non-probability sampling: In non-probability sampling, the researcher chooses members for research at random. This sampling method is not a fixed or predefined selection process. This makes it difficult for all elements of a population to have equal opportunities to be included in a sample.

Types of probability sampling with examples: Probability sampling is a sampling technique in which researchers choose samples from a larger population using a method based on the theory of probability. This sampling method considers every member of the population and forms samples based on a fixed process.

For example, in a population of 1000 members, every member will have a 1/1000 chance of being selected to be a part of a sample. Probability sampling eliminates bias in the population and gives all members a fair chance to be included in the sample.

There are four types of probability sampling techniques:

● Simple random sampling: One of the best probability sampling techniques that helps in saving time and resources, is the Simple Random Sampling method. It is a reliable method of obtaining information where every single member of a population is chosen randomly, merely by chance. Each individual has the same probability of

being chosen to be a part of a sample.

For example, in an organization of 500 employees, if the HR team decides on conducting team building activities, it is highly likely that they would prefer picking chits out of a bowl. In this case, each of the 500 employees has an equal opportunity of being selected. ● Cluster sampling: Cluster sampling is a method where the researchers divide the entire population into sections or clusters that represent a population. Clusters are identified and included in a sample based on demographic parameters like age, sex, location, etc. This makes it very simple for a survey creator to derive effective inference from the feedback. For example, if the United States government wishes to evaluate the number of immigrants living in the Mainland US, they can divide it into clusters based on states such as California, Texas, Florida, Massachusetts, Colorado, Hawaii, etc. This way of conducting a survey will be more effective as the results will be organized into states and provide insightful immigration data. ● Systematic sampling: Researchers use the systematic sampling method to choose the sample members of a population at regular intervals. It requires the selection of a starting point for the sample and sample size that can be repeated at regular intervals. This type of

sampling method has a predefined range, and hence this sampling technique is the least time-consuming. For example, a researcher intends to collect a systematic sample of 500 people in a population of 5000. He/she numbers each element of the population from 15000 and will choose every 10th individual to be a part of the sample (Total population/ Sample Size = 5000/500 = 10). ● Stratified random sampling: Stratified random sampling is a method in which the researcher divides the population into smaller groups that don’t overlap but represent the entire population. While sampling, these groups can be organized and then draw a sample from each group separately.

For example, a researcher looking to analyze the characteristics of people belonging to different annual income divisions will create strata (groups) according to the annual family income. Eg – less than $20,000, $21,000 – $30,000, $31,000 to $40,000, $41,000 to $50,000, etc. By doing this, the researcher concludes the characteristics of people belonging to different income groups. Marketers can analyze which income groups to target and which ones to eliminate to create a roadmap that would bear fruitful results. Types of non-probability sampling with examples The non-probability method is a sampling method that involves a collection of feedback based on a researcher or statistician’s sample selection capabilities

and not on a fixed selection process. In most situations, the output of a survey conducted with a non-probable sample leads to skewed results, which may not represent the desired target population. But, there are situations such as the preliminary stages of research or cost constraints for conducting research, where non-probability sampling will be much more useful than the other type. Four types of non-probability sampling explain the purpose of this sampling method in a better manner:

● Convenience sampling: This method is dependent on the ease of access to subjects such as surveying customers at a mall or passers-by on a busy street. It is usually termed as convenience sampling, because of the researcher’s ease of carrying it out and getting in touch with the subjects. Researchers have nearly no authority to select the sample elements, and it’s purely done based on proximity and not representativeness. This non-probability sampling method is used when there are time and cost limitations in collecting feedback. In situations where there are resource limitations such as the initial stages of research, convenience sampling is used. For example, startups and NGOs usually conduct convenience sampling at a mall to distribute leaflets of upcoming events or promotion of a cause – they do that by standing at the mall entrance and giving out pamphlets randomly.

● Judgmental or purposive sampling: Judgemental or purposive samples are formed by the discretion of the researcher. Researchers purely consider the purpose of the study, along with the understanding of the target audience. For instance, when researchers want to understand the thought process of people interested in studying for their master’s degree. The selection criteria will be: “Are you interested in doing your masters in …?” and those who respond with a “No” are excluded from the sample. ● Snowball sampling: Snowball sampling is a sampling method that researchers apply when the subjects are difficult to trace. For example, it will be extremely challenging to survey shelterless people or illegal immigrants. In such cases, using the snowball theory, researchers can track a few categories to interview and derive results. Researchers also implement this sampling method in situations where the topic is highly sensitive and not openly discussed—for example, surveys to gather information about HIV Aids. Not many victims will readily respond to the questions. Still, researchers can contact people they might know or volunteers associated with the cause to get in touch with the victims and collect information. ● Quota sampling: In Quota sampling , the selection of members in this sampling technique happens based on a pre-set standard. In this case, as a sample is formed based on specific attributes, the created sample will have the same qualities found in the total population. It is a rapid method of collecting samples.

Graphical Representations Graphical Representation is a way of analysing numerical data. It exhibits the relation between data, ideas, information and concepts in a diagram. It is easy to understand and it is one of the most important learning strategies. It always depends on the type of information in a particular domain. There are different types of graphical representation. Some of them are as follows: ● Line Graphs – Line graph or the linear graph is used to display the continuous data and it is useful for predicting future events over time. ● Bar Graphs – Bar Graph is used to display the category of data and it compares the data using solid bars to represent the quantities. ● Histograms – The graph that uses bars to represent the frequency of numerical data that are organised into intervals. Since all the intervals are equal and continuous, all the bars have the same width. ● Line Plot – It shows the frequency of data on a given number line. ‘ x ‘ is placed above a number line each time when that data occurs again. ● Frequency Table – The table shows the number of pieces of data that falls within the given interval. ● Circle Graph – Also known as the pie chart that shows the relationships of the parts of the whole. The circle is considered with 100% and the categories occupied are represented with that specific percentage like 15%, 56%, etc. ● Stem and Leaf Plot – In the stem and leaf plot, the data are organised from least value to the greatest value. The digits of the least placed values from the leaves and the next place value digit forms the stems.

● Box and Whisker Plot – The plot diagram summarises the data by dividing into four parts. Box and whisker show the range (spread) and the middle ( median) of the data.

Mean, Median and Mode of Ungrouped Data Measures of central tendency are a key way to discuss and communicate with graphs. The term central tendency refers to the middle, or typical, value of a set of data, which is most commonly measured by using the

three m's: mean, median, and mode. The mean, median, and mode are known as the measures of central tendency. In this lesson, you will explore these three concepts. The mean, often called the average, of a numerical set of data, is simply the sum of the data values divided by the number of values. This is also referred to as the arithmetic mean. The mean is the balance point of a distribution. Mean = Sum of the Values / Number of the Values For example, Stephen has been working on programming and updating a Web site for his company for the past 15 months. The following numbers represent the number of hours Stephen has worked on this Web site for each of the past 7 months: 24, 25, 31, 50, 53, 66, 78 What is the mean (average) number of hours that Stephen worked on this Web site each month? Step 1: Add the numbers to determine the total number of hours he worked. 24 + 25 + 33 + 50 + 53 + 66 + 78 = 329 Step 2: Divide the total by the number of months. 329 / 7 = 47 The median is the number that falls in the middle position once the data has been organized. Organized data means the numbers are arranged from

smallest to largest or from largest to smallest.The median for an odd number of data values is the value that divides the data into two halves. For example, find the median of the following data: 12, 2, 16, 8, 14, 10, 6

The mode of a set of data is simply the value that appears most frequently in the set. For example, Find the mode of the following data: 76, 81, 79, 80, 78, 83, 77, 79, 82, 75 Answer: There is no need to organize the data, unless you think that it would be easier to locate the mode if the numbers were arranged from least to greatest. In the above data set, the number 79 appears twice, but all the other

numbers appear only once. Since 79 appears with the greatest frequency, it is the mode of the data values.

Mean, Median and Mode for Grouped Data Let’s have the table below as an example.

Finding the grouped mean is easy. Simply, follow the formula below.

The table below contains the explanation of the notation. Element

Description

Group mean

The frequency of the

The midpoint of the

observation

x

The sample size

Using the example from above, we get the group mean performing the following steps. Age Groups

Frequency

0 - 10

40

10 - 20

53

14.5

768.5

20 - 30

58

24.5

1421

30 - 40

64

34.5

2208

*

40 - 50

72

44.5

3204

50 - 60

49

54.5

2670.5

60 - 70

36

64.5

2322

70 - 80

25

74.5

1862.5

Total

397

14636.5

Plugging this into the formula, we get,

We attain 36.9, meaning that the mean is somewhere between 30 and 40. Grouped Median

Similarly, finding the median for grouped data requires a different process. To find the group mean, you must follow the formula below.

The table below contains the explanation of the notation. Element

Description

Group median

The lower limit of the median group

The sample size

The cumulative frequency of all groups below the median group

The frequency of the group with the median

The width of the groups

Using the same example, we can see that the median of all the groups is roughly the middle point of the total frequency.

The 199th point occurs somewhere in the group 30 - 40 (in reality 30 to 39). We can do this estimation because the data are in order. The cumulative frequency can be found in the table below. Age Groups

Frequency

0 - 10

40

10 - 20

53

20 - 30

58

30 - 40

64

215

40 - 50

72

287

50 - 60

49

336

60 - 70

36

372

70 - 80

25

397

From the previous calculations, we get the following values. Element

Value

30

397

151

64

10

Plugging these values into the formula, we get

Our estimate of the median is about 37. Grouped Mode The formula for the mode of grouped data is as follows.

The table below contains the explanation of the notation. Element

Description

Group mode

The lower limit of the group with the mode (the group with the highest frequency)

Frequency of the group with the mode

Frequency of the group before the one with the mode

Frequency of the group after the one with the mode

The width of the groups

Using the same example, we get the following. Element

Value

40

72

64

49

10

Plugging this into the formula, we get

Which gives us a mode of about 43.

Hypotheses A hypothesis is an educated guess about something in the world around you. It should be testable, either by experiment or observation. For example: ● A new medicine you think might work. ● A way of teaching you think might be better. ● A possible location of new species. ● A fairer way to administer standardized tests.

Two types of Hypothesis Null hypothesis (H0) The null hypothesis states that a population parameter (such as the mean, the standard deviation, and so on) is equal to a hypothesized value. The null hypothesis is often an initial claim that is...