Chapter 1 Mathematical Statistics PDF

Faculty of Applied Health & Science Technology Dr: Maha Hamed 1 Mathematical statistic  Faculty of Applied Health & Science Technology 1.1 Statistic science: DEFINITION: Statistic science is a...

Faculty of Applied Health & Science Technology Dr: Maha Hamed 1 Mathematical statistic  Faculty of Applied Health & Science Technology 1.1 Statistic science: DEFINITION: Statistic science is a branch of applied mathematics that involves the collection, description, analysis, and inference of conclusions from quantitative data. Then we can conclude that the statistical science in four steps: ✓ Data collection. ✓ Data summarization. ✓ Data analysis. ✓ Making decisions to solve the problem. 1.2 Types of statistic science: o Descriptive Statistics Is a branch of statistics focused on summarizing, organizing, and presenting data using tables, graphs, and summary measures such as mean, median, and variance. o Inferential Statistics Is a branch of statistics that makes the use of various analytical tools to draw inferences about the population data from sample data. Statistics in healthcare can include: ▪ Tracking disease prevalence ▪ Assessing treatment safety and effectiveness ▪ Monitoring birth and death rates ▪ Evaluating access to and use of healthcare ▪ Analyzing the quality and efficiency of the healthcare system 2 Mathematical statistic  Faculty of Applied Health & Science Technology ▪ Calculating individual health metrics such as resting heart rate, blood pressure, and weight fluctuations. 1.3Data(observations): Data is a collection of facts, such as numbers, words, measurements, observations etc. There are two types of data, Data collection methods There are two types of data collection: o Comprehensive survey method o Sample method -The data are collected for each and - is a process for choosing sample every element. members from a population. -It is useful when case intensive study - Conclusions derived from the is required or the area is limited. small sample are generalized for the whole population. Population: It is the collection of all individuals under consideration in a statistical study. Sample: It is the part of population from which formation is collected.: Data Types There are two types of data 1. Qualitative data. Data 2. Quantitative data Qualitative Quantitative ordinal data Non-ordinal Discrete continous 3 Mathematical statistic  Faculty of Applied Health & Science Technology Qualitative Data: It is descriptive data and non numerical. It is ordinal data (for example, stages of breast cancer (I, II, III, or IV), degree of pain (minimal, moderate, severe or unbearable) or non-ordinal data (for example, place of birth, blood type, type of drug). Quanitatve Data: It is numerical data which represents the numerical value. It is discrete (for example, age of patient, the number of people suffering from diabetes) or continous (for example, weight, temperature). Variables are characteristics or qualities of a person, animal, or object that you can count or measure. 1.4 Representation of Data The data in the statistics can be summarized by three ways: a) tabular method. b) graphical method. c) Calculating statistical measures. a) Tabular method Representation of Representation of Qualitative variables Quantitative variables 4 Mathematical statistic  Faculty of Applied Health & Science Technology Representation of Qualitative variables Example 1: A school conducted a blood donation camp. The blood groups of 30 students were recorded as follows. Solution: The type of data (variable) in this example is qualitative variable, so we can summarize these data by counting all the values of variable and how often each of the values is repeated We can represent this data in a tabular form as ▪ This table is known as a frequency distribution table. ▪ You can observe that all the collected data is organized under two columns. ▪ This makes it easy for us to understand the given information. ▪ Thus, summary statistics condenses the data to a simpler form so that it is easy for us to observe its features at a glance. 5 Mathematical statistic  Faculty of Applied Health & Science Technology Representation of Quantitative Variable Discrete variable Continuous variable Example 2: In a sample of 30 hospitals, the number of cases that were admitted to the hospital and were suffering from pneumonia was as follows 2 5 6 1 3 8 0 9 6 3 2 5 2 6 8 5 0 2 4 7 7 3 3 2 5 2 7 8 2 5 Represent this data in the form of a frequency distribution table. Solution: The variable of these data is discrete variable take the values 0,1,2,3,4,5,6,7,8,9 Number of patients frequence 0 2 1 1 2 7 3 4 4 1 5 5 6 3 7 3 8 3 9 1 Sum 30 Example 3: Given below are the weekly pocket expenses (in $) of a group of 25 worker selected at random. 37, 41, 39, 34, 41, 26, 46, 31, 48, 32, 44, 39, 35, 39, 37, 49, 27, 37, 33, 38, 49, 45, 44, 37, 36 Construct a grouped frequency distribution table with class intervals of equal widths, starting from 25 - 30, 30 - 35, and so on. Also, find the range of weekly pocket expenses. Solution: The following table represents the given data: 6 Mathematical statistic  Faculty of Applied Health & Science Technology Weekly expenses (in $) Number of students 25-30 2 30-35 4 35-40 10 40-45 4 45-50 5 Total 25 Weekly expenses (in $) Number of students Example 4: 100 hospitals counted the number of patients in a certain day Represent the given data in the form of frequency distribution 95, 67, 28, 32, 65, 65, 69, 33, 98, 96, 76, 42, 32, 38, 42, 40, 40, 69, 95, 92, 75, 83, 76, 83, 85, 62, 37, 65, 63, 42, 89, 65, 73, 81, 49, 52, 64, 76, 83, 92, 93, 68, 52, 79, 81, 83, 59, 82, 75, 82, 86, 90, 44, 62, 31, 36, 38, 42, 39, 83, 87, 56, 58, 23, 35, 76, 83, 85, 30, 68, 69, 83, 86, 43, 45, 39, 83, 75, 66, 83, 92, 75, 89, 66, 91, 27, 88, 89, 93, 42, 53, 69, 90, 55, 66, 49, 52, 83, 34, 36. Solution: to arrange a large amount of discrete variable, we need the following steps 1. Determine the highest and lowest value of the set. 2. Define the interval for the data values, all interval must be the same, by the following steps: The numbers of interval K=1+3.3log(n), n is the number of values. Calculating the range R= highest value-lowest value. 1. The lowest value is 23, and the highest𝑅value is 98. The length of interval 𝑤 =. 𝐾 2. The number of interval K=1+3.3log (100) = 7.6 ≅8. 7 Mathematical statistic  Faculty of Applied Health & Science Technology The range R=98-23=75. 75 The length of interval 𝑤 = = 9.3 ≅ 9. 8 𝑢𝑝𝑝𝑒𝑟 𝑙𝑖𝑚𝑖𝑡 𝑜𝑓 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙+𝑙𝑜𝑤𝑒𝑟 𝑙𝑖𝑚𝑖𝑡 𝑜𝑓 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙 Center of interval=. 2 So, we draw a table of 8 intervals every interval with length 9. Number of Patients Center of interval frequency 20-29 24.5 3 30-39 34.5 14 40-49 44.5 12 50-59 45.5 8 60-69 46.5 18 70-79 47.5 10 80-89 48.5 23 90-99 49.5 12 total ----------- 100 b) Graphical Method Graphical Representation of Data Examples is a way of analyzing numerical data. 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. The general representation of statistical data are: Pictograph Bar Graph Histogram Pie Chart Frequency Distribution Line Graph 8 Mathematical statistic  Faculty of Applied Health & Science Technology 1.5 Cumulative Frequency Distributions ▪ In statistics, the frequency of the first-class interval is added to the frequency of the second class, and this sum is added to the third class and so on then, frequencies that are obtained this way are known as cumulative frequency (c.f.). ▪ A table that displays the cumulative frequencies that are distributed over various classes is called a cumulative frequency distribution or cumulative frequency table. Type of cumulative frequency ▪ There are two types of cumulative frequency 1. lesser than type 2. greater than type. Cumulative frequency is used to know the number of observations that lie above (or below) a particular frequency in a given data set. Example: Write down less than type cumulative frequency and greater than type cumulative frequency for the following data. 9 Mathematical statistic  Faculty of Applied Health & Science Technology Height (in cm) Frequency (students) 140 – 145 2 145 – 150 5 150 – 155 3 155 – 160 4 160 – 165 1 Solution: We would have less than type and more than type frequencies as: 2+5= 3+5+2= 155 5 10 Mathematical statistic  Faculty of Applied Health & Science Technology Exercise: 1. The following data are the heights (in inches to the nearest half inch) of 100 male semiprofessional soccer players. The heights are continuous data since height is measured. 60, 60.5, 61, 61, 61.5,63.5, 63.5, 63.5,64, 64, 64, 64, 64, 64, 64, 64.5, 64.5, 64.5, 64.5, 64.5, 64.5, 64.5, 64.5,66, 66, 66, 66, 66, 66, 66, 66, 66, 66, 66.5, 66.5, 66.5, 66.5, 66.5, 66.5, 66.5, 66.5, 66.5, 66.5, 66.5, 67, 67, 67, 67, 67, 67, 67, 67, 67, 67, 67, 67, 67.5, 67.5, 67.5, 67.5, 67.5, 67.5, 67.5,68, 68, 69, 69, 69, 69, 69, 69, 69, 69, 69, 69, 69.5, 69.5, 69.5, 69.5, 69.5,70, 70, 70, 70, 70, 70, 70.5, 70.5, 70.5, 71, 71, 71,72, 72, 72, 72.5, 72.5, 73, 73.5,74 Construct a grouped frequency distribution table with class intervals of equal widths, then construct the upper and lower cumulative frequency table. 2. The following represents scores of a class of 20 students received on their most recent Biology test. Construct the frequence distribution table. 58, 79, 81, 99, 68, 92, 76, 84, 53, 57, 81, 91, 77, 50, 65, 57, 51, 72, 84, 89. 3. By following up the number of days spent by 25 patients who were admitted to a hospital in May, the results were as follows: Construct the frequence distribution table, then find the number of patients who spent days between 6-8. Find the number of patients who spent days less than 8 days 11 Mathematical statistic