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+---
+title: "INTRODUCTION TO PROBABILITY USING R"
+output:
+  word_document: default
+  html_document:
+    df_print: paged
+---
+BEING A SHORT TRAINING PRESENTED AT OFFA R USERS GROUP MEETING ON 13TH FEBRUARY, 2024 AT STATISTICAL LABORATORY, STATISTICS DEPARTMENT, THE FEDERAL POLYTECHNIC OFFA, NIGERIA
+BY
+UDOKANG, ANIETIE EDEM (OGANIZER, ORUG)
+CHIF LECTURER, STATISTICS DEPARTMENT,THE FEDERAL POLYTECHNIC OFFA, NIGERIA
+
+#Introduction
+This topic is carefully chosen because of the important of probability in Statistics, most especially when it has to do with inferential Statistics.
+We deal with probability in many fields of Statistics such as Econometrics, Time Series, Biostatistics, Medical Statistics, Sampling Theory, Inference and Business Statistics. 
+Welcome on board as we discuss the elementary part of Statistics that has to do with the Concepts of Probability.
+Before then, let’s get ourselves familiar with R Software.
+#What is R?
+R is a free statistical software created by statisticians Ross Ihaka and Robert Gentleman and supported by R Core Team and the R Foundation for Statistical Computing.
+#Where can I Find R?
+Visit: https://cran.r-project.org/ 
+Follow the instructions for free download and install in your system.
+Display of the R Console (A Tool to type command and see the result of the command)
+ 
+#What is Probability?
+Probability can be defined as a quantity from 0 to 1used to measure the likelihood or the chance of an event occurring.
+#Two Approaches of Probability
+*Classical Approach -This probability approach assumes equally likely outcomes based on prior knowledge.
+
+*Relative Frequency – This approach is based on observed occurrences over a large number of trials.
+Formula
+P(A) = n(A)/n(S) 
+Or
+P(A) = f/n
+Where 
+P(A) = number of event A
+n(S) = number all possible events = sample space
+f = frequency of a sub group
+n = Total frequency of all the groups = sample size
+
+##llustrative Examples (R-4.4.0)
+#Example 1 (Classical Approach)
+An unbiased coin has two sides Head (H) and Tail (T). This implies that each side has an equal chance of occurrence when toss or flip. What is the probability of having a head when the coin is tossed once? 
+
+```{r}
+outcomes <- c("Head", "Tail")
+total_outcomes <- length(outcomes)
+total_outcomes
+```
+```{r}
+head.outcome <- length(outcomes[outcomes == "Head"])
+head.outcome 
+```
+```{r}
+prob.head <- head.outcome / total_outcomes
+prob.head
+```
+#Example 2 (Classical Approach)
+Consider a situation that an unbiased coin is tossed twice or two of the coins are tossed twice. The sample space is HH, HT, TH and TT. 
+What is the probability of having two heads?
+```{r}
+outcomes <- c("HH", "HT", "TH", "TT")
+total_outcomes <- length(outcomes)
+total_outcomes
+```
+```{r}
+head.outcomes <- length(outcomes[outcomes == "HH"])
+head.outcomes
+```
+```{r}
+prob.head <- head.outcomes / total_outcomes
+prob.head
+```
+#Example 3 (Relative Frequency Approach)
+The scores and grades students in an examination are 75 (A), 75 (A), 70 (AB), 70 (AB), 70 (AB), 70 (AB), 70(AB), 66(B), 66 (B) and 66(B). 
+What is the probability of a student having AB in the examination?
+There are ten students in the sample space, out which three have AB.
+```{r}
+grades <- c("A", "A", "AB", "AB","AB","AB","AB","B","B","B")
+total.grades <- length(grades)
+total.grades
+```
+```{r}
+AB.grades <- length(grades[grades == "AB"])
+AB.grades
+```
+```{r}
+prob.AB <- AB.grades / total.grades
+prob.AB
+```
+We can as well put up a frequency distribution table and get the probabilities of 75 (A), AB (70) and B (66).
+```{r}
+Score<-c(75,75,70,70,70,70,70,66,66,66)
+factor(Score)
+table(Score)
+prop.table(table(Score))
+```
+#Probability Properties with Illustrations and Examples 
+1. 0<P(A)<1, where A is an event.
+2. P(A) = 1 – P(AI), where AI is the complement of A.
+
+3. P(Φ) = 0, where Φ is the null or empty set.
+
+4. P(S) = 1, where S is the sample space.
+##Illustrative Examples of the properties
+Let a set S be a sample space with all integers from 1 to 10 inclusive. 
+Then S = {1,2,3,4,5,6,7,8,9,10}
+Let A be all the elements in S less than 5.
+A = {1,2,3,4,}
+Let B be all the elements in S more than 4.
+B = {5,6,7,8,9,10}
+This example will be use to explain properties 1 to 4.
+#Property 1       0<P(A)<1
+Property one will be seen while illustrating propertie 2to 3.
+#Property 2       P(A) = 1 – P(AI)
+R Code (the step by step codes are written to aid in the illustration)
+```{r}
+S <- c(1,2,3,4,5,6,7,8,9,10)
+A<- c(1,2,3,4)
+B<- c(5,6,7,8,9,10)
+A.Complement=res <- S[is.na(pmatch(S,A))]
+A.Complement
+```
+```{r}
+length(S)
+```
+```{r}
+length(A)
+```
+```{r}
+length(A.Complement)
+```
+```{r}
+prob.A <- sum(length(A)) / length(S)
+prob.A
+```
+```{r}
+prob.A.Complement <- sum(length(A.Complement)) / length(S)
+prob.A.Complement
+```
+```{r}
+prob.A=1- prob.A.Complement 
+prob.A
+```
+##Property 3        P(Φ) = 0
+In the example let’s find the intersection between A and B with its probability.
+```{r}
+S <- c(1,2,3,4,5,6,7,8,9,10)
+length(S)
+```
+
+
+```{r}
+A<-c(1,2,3,4)
+B<-c(5,6,7,8,9,10)
+phi=intersect(A, B)
+length(phi)
+```
+```{r}
+prob_to_find <- c(phi)
+prob.phi<- sum(length(phi)) / length(S)
+prob.phi
+```
+#Property 4                 P(S) = 1
+```{r}
+S <- c(1,2,3,4,5,6,7,8,9,10)
+length(S)
+```
+```{r}
+prob.S <- sum(length(S)) / length(S)
+prob.S
+```
+#Rules or Laws of Probability
+An extension to the properties above are addition, multiplication and conditional laws of probability.
+##Addition Law of Probability
+If A, B and C are subsets of S, then
+*1.P(A U B) = P(A) + P(B) if A and B are independent events also P(A U C) = P(A) + P(C) if A and C are independent events.
+*2.P(A U B) = P(A) + P(B) – P(A ∩ B) if A and B are dependent events.
+*3.P(A U B U C) = P(A) + P(B) + P(C) - P(A∩B) – P(A∩C) – P(B∩C) + P(A∩B∩C) if A, B and C are dependent events.
+#Illustrative Example
+Set theory will be used to explain the laws of probability laws.
+Let’s consider a sample space S with only three sets A and B.
+Let S = {21,43, 22,35,50,60,20,45, 48, 57,64,67,82,33,44,80,90} 
+A = {21,43,22,50,48,,57,80}
+B = {22,35,60,20,45,64,67,82,33,44}
+C ={22,43,50,45,82,33,82,60,90}
+What is the probability of AUB?
+P(AUB) = n(AUB)/n(S)
+P(A) = n(A)/n(S)
+P(B) = n(B)/n(S)
+P(A ∩ B) = n(A ∩ B)/n(S)
+```{r}
+S <- c(21,43, 22,35,50,60,20,45, 48, 57,64,67,82,33,44,80,90)
+length(S)
+```
+
+
+```{r}
+A<-c(21,43,22,50,48,57,80)
+length(A)
+```
+
+
+```{r}
+prob.A<- sum(length(A)) / length(S)
+	prob.A
+```
+
+
+```{r}
+B<-c(22,35,60,20,45,64,67,82,33,44)
+length(B)
+```
+
+
+```{r}
+prob.B<- sum(length(B)) / length(S)
+	prob.B
+AUB=union(A,B)
+AUB
+```
+
+
+```{r}
+length(AUB)
+```
+
+
+```{r}
+prob.AUB<- sum(length(AUB)) / length(S)
+prob.AUB
+```
+
+
+```{r}
+AB=intersect(A, B)
+AB
+```
+
+
+```{r}
+length(AB)
+```
+
+
+```{r}
+prob_to_find <- c(AB)
+prob.AB<- sum(length(AB)) / length(S)
+prob.AB
+```
+
+# Lets cross check the result of P(AUB)
+```{r}
+prob.AUB = 0.4117647+ 0.5882353- 0.05882353
+prob.AUB
+```
+Venn Diagram
+```{r}
+library(VennDiagram) 
+grid.newpage() 
+draw.pairwise.venn(area1=7, area2=10,cross.area=1, 
+category=c("A","B"),fill=c("Green","Blue"))
+```
+###Try P(AUBUC)
+#Multiplication Law of Probability
+If A and B are subsets of S, then
+*1.P(A ∩ B) = P(A). P(B) if A and B are independent events
+*2.P (A ∩ B) = P(A).P(B/A) if A and B are dependent events
+
+Illustrative Example
+Let’s consider in a football field there are 100 footballs of two colours of 30 ash and 70 blue.
+Let A represents ash colour footballs
+      B represents blue colour footballs
+      S represents ash and blue colour footballs
+If two footballs are selected for training, one after the other with replacement, what is the probability that the first is ash and the second blue?
+P(ash colour football) = P(A) = n(A)/n(S)
+P(blue colour football) = P(B) = n(B)/n(S)
+P(first ash and second blue) = P(A and B) = P(AB) 
+= P(A ∩ B) = P(A). P(B) [since the process of selection is with replacement]
+#Let S represent all the football
+#Let A represent ash colour football
+#Let B represent blue colour football
+```{r}
+n.S=100
+n.A=30
+n.B=70
+n.B=70
+	prob.A=n.A/n.S
+	prob.A
+```
+```{r}
+prob.B=n.B/n.S
+	prob.B
+```
+```{r}
+prb.AB=prob.A*prob.B
+	prb.AB
+```
+#What is the probability that the first is ash and the second blue if the process of selection is without replacement?
+P(blue given that it was ash) = P(B/A) = n(B)/(n(S)-1)
+P(first ash and second blue) = P(A and B) 
+= P(AB) = P(A ∩ B) = P(A). P(B/A) [since the process of selection is without replacement]
+```{r}
+prob.A=n.A/n.S
+prob.A
+```
+```{r}
+prob.BgivenA=n.B/(n.S-1)
+prob.BgivenA
+```
+```{r}
+prb.AB=prob.A*prob.BgivenA
+prb.AB
+
+```
+#Conditional Law of Probability
+This probability explains how to get the probability of having an event base on the fact that another event occurred.
+
+##Probability of event A given that B has occurred.
+      
+      P(A/B)=P(AB)/P(B)
+	 
+#Illustrative Example
+Let’s consider the illustration in addition law of probability using sets.
+Let’s consider a sample space S with only three sets A and B.
+Let S = {21,43, 22,35,50,60,20,45, 48, 57,64,67,82,33,44,80,90}
+    A = {21,43,22,50,48,,57,80}
+    B = {22,35,60,20,45,64,67,82,33,44}
+    C ={22,43,50,45,82,33,82,60,90}
+What is the probability of P(A/B)?
+```{r}
+S <- c(21,43, 22,35,50,60,20,45, 48, 57,64,67,82,33,44,80,90)
+length(S)
+```
+```{r}
+A<-c(21,43,22,50,48,57,80)
+length(A)
+```
+
+
+```{r}
+prob.A<- sum(length(A)) / length(S)
+	prob.A
+```
+```{r}
+B<-c(22,35,60,20,45,64,67,82,33,44)
+length(B)
+```
+```{r}
+prob.B<- sum(length(B)) / length(S)
+	prob.B
+```
+```{r}
+AB=intersect(A,B)
+length(AB)
+```
+```{r}
+prob.AB<- sum(length(AB)) / length(S)
+prob.AB
+```
+```{r}
+prob.AgivenB= prob.AB/prob.B
+prob.AgivenB
+```
+#We have come to the end of the training today. The Offa-R-Users-Group (ORUG) is a place to learn and share knowledge in the use of R. I wish to see you next time. If you are a guest, find time to register as a member to actualize your goal in using R. 
+The ORUG (https://www.meetup.com/fedpofa-r-users-group/ ) is sponsored by R Consortium and AniKem_Consult. For any Enquiry Contact the Organizer (WhatsApp: +2349030912602, email: anietieeu@yahoo.com)
+
+#THANK YOU FOR PARTICIPATING IN THIS EVENT
diff --git a/Introduction-to-r-for-data-science-part-i.Rmd b/Introduction-to-r-for-data-science-part-i.Rmd
new file mode 100644
index 0000000..298c38a
--- /dev/null
+++ b/Introduction-to-r-for-data-science-part-i.Rmd
@@ -0,0 +1,184 @@
+---
+title: "INTRODUCTION TO R FOR DATA SCIENCE PART I"
+output:
+  word_document: default
+  html_document:
+    df_print: paged
+---
+##BEING A SHORT TRAINING MATERIAL USED AT OFFA R USERS GROUP MEETING ON 31ST OCTOBER, 2023 AT NO. 15 OLD IRRA GARAGE IRRA ROAD, OFFA, KWARA STATE, NIGERIA
+
+PREPARED BY
+UDOKANG, ANIETIE EDEM (OGANIZER, ORUG)
+CHIF LECTURER, STATISTICS DEPARTMENT, THE FEDERAL POLYTECHNIC OFFA, NIGERIA
+
+
+#Introduction
+Today we want to look at the world of data science, the different between data scientist and statistician and the role/use of R software in data science. Data science is an exciting discipline that allows you to turn raw data into understanding, insight, and knowledge. The world today needs data science more than ever before because people prefer visualization than descriptions. The training will focus also on some aspects of data cleaning which mostly spent in data analysis.
+#What is R?
+R is a language and environment for statistical computing and graphics. It is a GNU project which is similar to the S language and environment which was developed at Bell Laboratories (formerly AT&T, now Lucent Technologies) by John Chambers and colleagues. R can be considered as a different implementation of S. There are some important differences, but much code written for S runs unaltered under R (https://www.r-project.org/about.html). GNU (pronounced guh-noo) is a Unix-like computer operating system developed by the GNU project, Free Software Foundation.
+R has so many packages that can be installed in R. These packages can be found in Comprehensive R Archive Network (CRAN) repository. It's a huge repository of R packages that users can easily contribute to. 
+
+#Why R?
+The following points is why R is attractive.
+*i.	R is mainly used when the data analysis tasks require standalone computing or analysis on individual servers.
+*ii.	R focuses on better, user-friendly data analysis, statistics and graphical models.
+*iii.	R has been used primarily in academics and research. However, it's rapidly expanding into the enterprise market.
+*iv.	Statistical models can be written with only a few lines in R and the same piece of functionality can be written in several ways in R.
+*v.	Once you know the basics, you can easily learn advanced techniques and R is not hard for experienced programmers. 
+
+#How to download R and its Packages
+Visit https://cran.r-project.org/ to download and install R version of your choice free of charge. 
+Open the installed R and go to Packages - Install Package(s) – Select CRAN Mirror – Select Package to install and upload.
+#The R Environment
+R software environment is used to run code written in R language. This made up of R console and R editor.
+Go to file – New script to get R-Editor
+ 
+#R Code
+The code can be written on the R console or run from R editor or copy and paste.
+#What is Data Science?
+This is a process of collecting/recording, storing and analyzing of data, mostly big data, from sources like social media, internet, satellite images,  e-commerce sites, healthcare surveys and develop methods to effectively extract useful information for decision making on real life situation.
+Therefore, data science can also be described as “The application of data-centric computational and inferential thinking to understand the world and solve problems” according to Professor Joseph Gonzalez of University of California, Berkeley.
+
+#What are the Difference between Statisticians and Data Scientists?
+The audience today are mostly statisticians or familiar with statistical analysis. Let me briefly discuss the difference between statisticians and data scientists.
+
+#Statisticians	
+*Deal with small-scale data collection using methods such as surveys, polls, and experiments. 
+*Work on improving one simple model to best fit the data.
+*Only analysis data.	 
+#Data Scientist
+*Work on massive data (big data) which involves a lot of time in tasks like large-scale data collection and cleaning.
+*Try out different methods to create machine learning models, and then they choose the method that results in the best model.
+*Go beyond data analysis to implement algorithms that process data automatically. This enables data scientists to provide automated predictions and actions. An example is automation that helps with weather forecasts
+The critical stage of data science is data cleaning.
+#What is Data Cleaning?
+When data is collected mostly in raw form it has to be arranged/transformed to have a reasonable structure that can be meaningful and useful. Data cleaning usually takes care of some of the issues as the missing values, the formatting of values, the structure of the data, extracting information from complex values and unit conversion. One of the R packages that handles this aspect is tidyr.
+To use the package in R, use the library function.
+
+#Creating data frame (df) in R
+```{r}
+X<-c(1,3,4,5,6,8,10,12,5,8,9,11)
+Y<-c(2,3,6,8,9,12,23,21,54,30,44,13)
+df<-data.frame(X,Y)
+df
+```
+
+#OR
+```{r}
+X=c(1,3,4,5,6,8,10,12,5,8,9,11)
+Y=c(2,3,6,8,9,12,23,21,54,30,44,13)
+df<-data.frame(X,Y)
+df
+```
+
+#To be sure df is a data frame
+```{r}
+print(class(df))
+print(df)
+```
+#More examples of data frame
+```{r}
+x<-c("a","b","c","d","e","f","g","h","I","j","k")
+f<-c(2,3,5,2,1,4,3,2,5,7,8)
+df<-data.frame(x,f)
+df
+```
+```{r}
+x<-c("a","b","c","d","e","f","g","h","I","j","k")
+f.<-c(2,3,5,2,1,4,3,2,5,7,8)
+y<-c("Ben","Philip","Segun","Ani","Tom","Okon","Jide","Chidi","Betty","Bros","Ken")
+f..<-c(21,23,45,26,19,46,43,62,52,71,88)
+df<-data.frame(x,f.,y,f..)
+df
+```
+
+```{r}
+print(df)
+```
+
+```{r}
+df<-data.frame(rating=1:11,x,f.,y,f..)
+df
+```
+
+#Structuring of the data into character/variable and corresponding values
+```{r}
+library(tidyr)
+xf<- tibble(x = c("x1", "x2","x3","x4"),f1 = c(1, 4, 2, 5), f2 = c(3, 4, 6, 2),)
+xf
+```
+
+xf is the data frame (df)
+
+#Removing missing data
+```{r}
+xf<- tibble(x = c("x1", NA,"x3","x4"),f = c(1, NA, 2, 5),)
+xf
+drop_na(xf)
+```
+
+#Replacing missing value with specified value
+```{r}
+xf<- tibble(x = c("x1", NA,"x3","x4"),f = c(1, NA, 2, 5),)
+xf %>% replace_na(list(x = "x2", f = 4))
+```
+
+#Filling missing values up with the preceding values
+```{r}
+xf<-tibble(x=c("x1","x2","x3","x4","x5","x6","x7","x8","x9","x10"),f=c(1,4,2, NA,NA,6,7,9,NA,NA),)
+xf
+xf1 <- xf %>% fill(f, .direction = 'up')
+xf1
+```
+
+#Filling missing values down with the preceding values
+```{r}
+xf2 <- xf1 %>% fill(f, .direction = 'down')
+xf2
+```
+#Splitting columns using “separate” function
+```{r}
+GP<- tibble(id = 1:2, x = c("m-360", "f-580"))
+GP
+```
+
+#GP is the data frame (df) representing population by gender
+```{r}
+df %>% separate(x, c("gender", "unit"))
+```
+#Splitting columns using “strsplit” function
+```{r}
+Students<- data.frame(Programme = c("ND Statistics", "HND Computer", "NID Welding", "BTECH Civil"), Pop = c(120, 200, 180, 78) )
+Students
+```
+
+```{r}
+Students[c("Programme","Department")]<- do.call(rbind, strsplit(Students$Programme, " "))
+print(Students)
+```
+
+#Uniting columns using “unite” function
+```{r}
+x<-c("B","P","S","A","T","O","J","C","BT","BR","K")
+y<-c("Ben","Philip","Segun","Ani","Tom","Okon","Jide","Chidi","Betty","Bros","Ken")
+df<-data.frame(x,y)
+df
+```
+
+```{r}
+ z<-data.frame(x=c("B","P","S","A","T","O","J","C","BT","BR","K"),y=c("Ben","Philip","Segun","Ani","Tom","Okon","Jide","Chidi","Betty","Bros","Ken"))
+df %>% unite("z", x:y, remove = FALSE)
+```
+```{r}
+df %>% unite("z", x:y, remove = TRUE)
+```
+
+#We appreciate your patient and interest in participating in the discussion today. The Offa-R-Users-Group (ORUG) is a place to learn and share knowledge in the use of R. I wish to see you next time. If you are a guest, find time to register as a member to actualize your goal in using R. 
+The ORUG (https://www.meetup.com/fedpofa-r-users-group/ ) is sponsored by R Consortium and AniKem_Consult. For any Enquiry Contact the Organizer (WhatsApp: +2349030912602, email: anietieeu@yahoo.com)
+
+#THANK YOU FOR PARTICIPATING IN THIS EVENT
+
+
+
+
+
diff --git a/LICENSE b/LICENSE
index 0e259d4..354f1e0 100644
--- a/LICENSE
+++ b/LICENSE
@@ -1,121 +1,121 @@
-Creative Commons Legal Code
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-CC0 1.0 Universal
-
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+For these and/or other purposes and motivations, and without any
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+elects to apply CC0 to the Work and publicly distribute the Work under its
+terms, with knowledge of his or her Copyright and Related Rights in the
+Work and the meaning and intended legal effect of CC0 on those rights.
+
+1. Copyright and Related Rights. A Work made available under CC0 may be
+protected by copyright and related or neighboring rights ("Copyright and
+Related Rights"). Copyright and Related Rights include, but are not
+limited to, the following:
+
+  i. the right to reproduce, adapt, distribute, perform, display,
+     communicate, and translate a Work;
+ ii. moral rights retained by the original author(s) and/or performer(s);
+iii. publicity and privacy rights pertaining to a person's image or
+     likeness depicted in a Work;
+ iv. rights protecting against unfair competition in regards to a Work,
+     subject to the limitations in paragraph 4(a), below;
+  v. rights protecting the extraction, dissemination, use and reuse of data
+     in a Work;
+ vi. database rights (such as those arising under Directive 96/9/EC of the
+     European Parliament and of the Council of 11 March 1996 on the legal
+     protection of databases, and under any national implementation
+     thereof, including any amended or successor version of such
+     directive); and
+vii. other similar, equivalent or corresponding rights throughout the
+     world based on applicable law or treaty, and any national
+     implementations thereof.
+
+2. Waiver. To the greatest extent permitted by, but not in contravention
+of, applicable law, Affirmer hereby overtly, fully, permanently,
+irrevocably and unconditionally waives, abandons, and surrenders all of
+Affirmer's Copyright and Related Rights and associated claims and causes
+of action, whether now known or unknown (including existing as well as
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+3. Public License Fallback. Should any part of the Waiver for any reason
+be judged legally invalid or ineffective under applicable law, then the
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+extent the Waiver is so judged Affirmer hereby grants to each affected
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+irrevocable and unconditional license to exercise Affirmer's Copyright and
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+will not (i) exercise any of his or her remaining Copyright and Related
+Rights in the Work or (ii) assert any associated claims and causes of
+action with respect to the Work, in either case contrary to Affirmer's
+express Statement of Purpose.
+
+4. Limitations and Disclaimers.
+
+ a. No trademark or patent rights held by Affirmer are waived, abandoned,
+    surrendered, licensed or otherwise affected by this document.
+ b. Affirmer offers the Work as-is and makes no representations or
+    warranties of any kind concerning the Work, express, implied,
+    statutory or otherwise, including without limitation warranties of
+    title, merchantability, fitness for a particular purpose, non
+    infringement, or the absence of latent or other defects, accuracy, or
+    the present or absence of errors, whether or not discoverable, all to
+    the greatest extent permissible under applicable law.
+ c. Affirmer disclaims responsibility for clearing rights of other persons
+    that may apply to the Work or any use thereof, including without
+    limitation any person's Copyright and Related Rights in the Work.
+    Further, Affirmer disclaims responsibility for obtaining any necessary
+    consents, permissions or other rights required for any use of the
+    Work.
+ d. Affirmer understands and acknowledges that Creative Commons is not a
+    party to this document and has no duty or obligation with respect to
+    this CC0 or use of the Work.
diff --git a/Parameter-Estimation-in-Non-Linear-Regression-Model-Using-r.Rmd b/Parameter-Estimation-in-Non-Linear-Regression-Model-Using-r.Rmd
new file mode 100644
index 0000000..a257fb3
--- /dev/null
+++ b/Parameter-Estimation-in-Non-Linear-Regression-Model-Using-r.Rmd
@@ -0,0 +1,245 @@
+---
+title: "Parameter Estimation in Non-Linear Regression Model Using R"
+output: html_notebook
+---
+
+Being a Short Training Material Presented at the Offa-R-Users-Group (ORUG) on 30th May, 2024.
+By 
+Udokang, Anietie Edem
+Organizer
+
+#Introduction#
+The previous discussion on regression analysis in this group was based on linear relationship, where the parameters and explanatory variable(s) were both linear. Today we want to consider other situations whereby the parameters or explanatory or both, are not linear referred to as non-linear relationship which can as well be explained by a non-linear model.
+It should be noted that for the parameters of a non-linear regression model to be estimated, the said model can to be transformed to a linear model using various transformation methods suitable to the model. Therefore, non-linear models will be presented in terms of linearity in parameters and explanatory variables and use it to compute the parameters of the transformed model which will be used to get the estimate of the initial parameters. 
+#1. Polynomial Non-Linear Regression Model#
+This type of regression model can be described as non-linear regression model because its non-linearity in variables, though its parameters may be linear. The polynomial non-linear regression model may not require the model to be transformed to linear model but action should be taken on variables with the power greater than 1.There are two types of polynomial regression models which are polynomial in one variable and polynomial in k variables.
+##i. Polynomial in One Variable##
+Polynomial regression model with only one independent variables with different powers is polynomial regression model in one variable. 
+#The model:#
+#The 2nd polynomial (quadratic) regression model in one variable
+$y=a_{0}+a_{1}x+a_{2}x^{2}+e$
+
+#The 3rd polynomial (cubic) regression model in one variable
+$y=a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+e$
+
+#The 4th polynomial (quatic) regression model in one variable
+$y=a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+a_{4}x^{4}+e$
+
+#The nth polynomial regression model in one variable
+$y=a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+a_{4}x^{4}+\dots+a_{n}x^{n}+e$
+
+The estimation of the parameters of the nth polynomial regression model takes the process of least square estimation of the linear regression.
+The illustration will be done on quadratic and cubic polynomial regression which the same procedure will be followed to kth polynomial.
+
+
+```{r}
+#Illustrative Example
+#Quadratic (2nd) Polynomial Regression  	
+x<-c(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40)
+y<-c(65,63,62,61,60,57,56,55,56,58,60,62,63,64,65,66)
+plot(x, y)
+```
+
+
+```{r}
+df <- data.frame(x, y)
+m1= lm(y ~ x+I(x^2))
+m1
+```
+
+
+```{r}
+summary(m1)
+```
+
+
+```{r}
+predict(m1)
+```
+
+#Plot of the actual values and the regression line
+```{r}
+library(ggplot2)
+ggplot(df, aes(x, y)) +geom_point() +geom_line(aes(x, predict(m1)),col="red") +
+ggtitle("Quadratic Polynomial Regression")
+```
+
+#Illustrative Example
+#Cubic (3rd) Polynomial Regression  	
+```{r}
+x<-c(6,9,12,16,22,28,33,40,47,51,55,60)
+y<-c(14,28,50,64,67,57,55,57,68,74,88,110)
+plot(x, y)
+```
+
+```{r}
+df <- data.frame(x, y)
+m2= lm(y ~ x+I(x^2)+ I(x^3))
+m2
+```
+
+```{r}
+summary(m2)
+```
+
+```{r}
+predict(m2)
+```
+
+```{r}
+library(ggplot2)
+ggplot(df, aes(x, y))+geom_point() +geom_line(aes(x, predict(m2)),col="blue") +ggtitle("Cubic Polynomial Regression")
+```
+
+##ii. Polynomial Regression Model in K Variables
+The polynomial regression with number of explanatory variables more than 0ne ( ) is said to Polynomial Regression Model in K Variables.
+#The polynomial regression model is
+
+#The 2nd polynomial (quadratic) regression model in two variables
+$y=a_{0}+a_{1}x+a_{2}z+a_{3}x^{2}+a_{4}z^{2}+e$
+
+#The 3rd polynomial (cubic) regression model in one variable
+$y=a_{0}+a_{1}x+a_{2}z+a_{3}x^{2}+a_{4}z^{2}+a_{5}x^{3}+a_{6}z^{3}+e$
+
+#The 4th polynomial (quatic) regression model in one variable
+$y=a_{0}+a_{1}x+a_{2}z+a_{3}x^{2}+a_{4}z^{2}+a_{5}x^{3}+a_{6}z^{3}+a_{7}x^{4}+a_{8}z^{4}+e$
+
+#The nth polynomial regression model in one variable
+$y=a_{0}+a_{1}x+a_{2}z+a_{3}x^{2}+a_{4}z^{2}+a_{5}x^{3}+a_{6}z^{3}+\dots+a_{n-1}x^{n-1}+a_{n}z^{n}+e$
+
+The estimation of the model parameters is done just like the linear in variable case, where least square method of estimation is used.
+We will consider the quadratic case and the idea can be applied to cubic.
+#Illustrative Example
+#Quadratic (2nd) Polynomial Regression  	
+
+
+```{r}
+x<-c(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40)
+y<-c(65,63,62,61,60,57,56,55,56,58,60,62,63,64,65,66)
+z<-c(23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38)
+plot(x, y)
+```
+
+```{r}
+plot(z, y)
+```
+
+```{r}
+df <- data.frame(x, y,z)
+m3= lm(y ~ x+I(x^2)+ z+I(z^2))
+m3
+```
+
+```{r}
+summary(m3)
+```
+
+```{r}
+predict(m3)
+```
+
+#Plot of regression line with x 
+```{r}
+library(ggplot2)
+ggplot(df, aes(x, y)) +geom_point() +geom_line(aes(x, predict(m3)),col="red") +
+ggtitle("Quadratic Polynomial Regression")
+```
+
+#Plot regression line with z 
+```{r}
+#Plot regression line with z 
+ggplot(df, aes(z, y)) +geom_point() +geom_line(aes(z, predict(m3)),col="red") +
+ggtitle("Quadratic Polynomial Regression")
+```
+
+##2. Exponential Non-Linear Regression Model with Natural Logarithm Transformation
+This type of non-linear model can be transformed with with logarithm to base e to make it a linear model. 
+An example of such model is  $y=ae^{bx}$ which can be transformed to linear model taking the natural log of both sides of the equation.
+$log_{e}y=log_{e}a+bx$
+#let
+$log_{e}y=y^*$ and $log_{e}a=b_{0}$
+#Therefore, the estimate of the response variable from the new model gives the regression line equation as
+$y^*=b_{0}+bx$
+from $log_{e}a=b_{0}$
+Therefore,
+$a=e^{b_{0}}$
+Then we can now estimate $y=ae^{bx}$
+```{r}
+#Illustrative Example
+x <- c(2,4,6,8,10,12,14,16,18)
+y <- c(5,7,8,9,10,12,14,16,20) 
+df <- data.frame(x, y)
+plot(x, y)
+```
+
+```{r}
+m4= lm(log(y) ~ x)
+m4
+```
+
+```{r}
+b0= 1.38460 #The constant term of the transformed regression model (b0 = loga )
+a=exp(b0) #The constant term of exponential regression model
+a
+```
+
+```{r}
+b= 0.09282 #The coefficient of the transformed model = the coefficient of the exponential model 
+predicty=a*exp(b*x) #To predict the exponential regression model
+predicty
+```
+
+#Plot of regression line with actual values
+```{r}
+ggplot(df, aes(x, y)) +geom_point() +geom_line(aes(x, predicty),col="red") +
+ggtitle("Exponential Regression")
+```
+#This an example of exponential growth where b>1 (if 0<b<1 it is exponential decay)
+#Note that b must not be –ve. Meaning that it must not be less than zero.
+#You can also use y = abx
+$y=ab^{x}$
+
+##3. Power Non-Linear Regression Model with Logarithm to Base 10 Transformation
+The non-linear relationship of the type given below is an example of a non-linear regression model that requires logarithm to base 10 transformation to be a linear model.
+
+$y=ax^{b}$
+$logy=loga+blogx$
+#Let
+$y*=logy$
+$x*=logx$
+$b_{0}=loga$
+Therefore, the estimate of the response variable from the new model gives the regression line equation as
+$y^*=b_{0}+bx^*$
+$a=antilog{b_{0}}$
+#Illustrative Example
+
+
+```{r}
+x=1:20
+y=c(2, 9, 6, 8, 7, 21, 16, 20, 24, 38, 34, 39, 50, 51, 57, 53, 71, 90, 98, 116) 
+df <- data.frame(x, y)
+plot(x, y)
+```
+
+```{r}
+m5 <- lm(log(y)~ log(x))
+m5
+```
+
+```{r}
+b0= 0.562 #The constant term of the transformed regression model (b0 = loga )
+a=exp(b0) #The constant term of power regression model
+a
+```
+```{r}
+b= 1.281 #The coefficient of the transformed model = the coefficient of the power regression model 
+predicty=a*x^b #To predict the exponential regression model
+predicty
+```
+
+#Plot of regression line with actual values
+```{r}
+library(ggplot2)
+ggplot(df, aes(x, y)) +geom_point() +geom_line(aes(x, predicty),col="red") +
+ggtitle("Power Regression")
+```
diff --git a/README.md b/README.md
index 788d9ee..e69de29 100644
--- a/README.md
+++ b/README.md
@@ -1,26 +0,0 @@
-# rugs-materials
-This repository is for any presentations and material you might want to share from your R User Groups.
-
-This is a community resources so add resources as you like via a PR.
-
-For any problems or information, please email [operations@r-consortium.org](operations@r-consortium.org)
-
-
-## Apply for funding
-
-You can apply for funding on our [website here](https://www.r-consortium.org/all-projects/r-user-group-support-program).
-
-
-## Tips for success
-
-There are a lot of resources and ideas for doing well, [here](https://blog.revolutionanalytics.com/tips-on-starting-an-r-user-group.html) is one blog post that we think gets them all right.
-
-
-## Submitting Material
-
-All documents should be shown as either a quarto file or R notebook file.
-
-Presentations can be in quarto, linked google slides or other accptable data formats. 
-
-All code should be in an executable code block so users know it is code that should be copied. 
-
diff --git a/SELECTED-DESCRIPTIVE-AND-INFERENTIAL-STATISTICS-IN-R-FOR-UNDERGRADUATE-PROJECT.docx b/SELECTED-DESCRIPTIVE-AND-INFERENTIAL-STATISTICS-IN-R-FOR-UNDERGRADUATE-PROJECT.docx
new file mode 100644
index 0000000..8f06497
Binary files /dev/null and b/SELECTED-DESCRIPTIVE-AND-INFERENTIAL-STATISTICS-IN-R-FOR-UNDERGRADUATE-PROJECT.docx differ
diff --git a/Selected-descriptive-and-inferential-statistics-in-r-for-undergraduate-project.Rmd b/Selected-descriptive-and-inferential-statistics-in-r-for-undergraduate-project.Rmd
new file mode 100644
index 0000000..cc2db55
--- /dev/null
+++ b/Selected-descriptive-and-inferential-statistics-in-r-for-undergraduate-project.Rmd
@@ -0,0 +1,247 @@
+---
+title: "SELECTED DESCRIPTIVE AND INFERENTIAL STATISTICS IN R FOR UNDERGRADUATE PROJECT"
+output:
+  word_document: default
+  html_document:
+    df_print: paged
+---
+BEING A PRACTICAL AND DEMONSTRATION SESSSION HELD ON THURSDAY, 18TH MAY, 2023 AT STATISTICS DEPARTMENT, THE FEDERAL POLYTECHNIC OFFA
+
+BY
+DR. ABDULKABIR MURITALA
+
+
+# Descriptives Statistics simulation
+```{r}
+x=rnorm(100)
+mean(x)
+```
+
+
+```{r}
+var(x)
+```
+
+
+```{r}
+sd(x)
+```
+
+
+```{r}
+median(x)
+```
+
+
+```{r}
+summary(x)
+```
+
+
+```{r}
+barplot(x)
+```
+
+
+```{r}
+hist(x)
+```
+
+
+```{r}
+qqnorm(x)
+```
+
+
+```{r}
+boxplot(x)
+```
+
+
+```{r}
+dotchart(x)
+```
+
+#Real life data
+```{r}
+x=c(1,3,5,6,7,8,9,3,2,4)
+y=c(2,3,5,6,7,8,4,3,2,1)
+z=c(2,3,4,1,2,5,6,7,8,9)
+boxplot(x,y,z)
+```
+
+
+```{r}
+boxplot(x,y,z,horizontal=T)
+```
+
+
+```{r}
+boxplot(x,y,z,horizontal=T,names=c("A","B","C"))
+```
+
+
+```{r}
+boxplot(x,y,z,horizontal=T,names=c("A","B","C"),col=c("red","black","blue"))
+```
+
+
+```{r}
+par(mfrow=c(2,2))
+barplot(x,y,z)
+hist(x)
+```
+
+#Inferential Statistics
+##chi sqaure
+```{r}
+data <- matrix(c(100, 70, 20, 90, 75, 25), ncol=3, byrow=TRUE)
+colnames(data) <- c("Rep","Dem","Ind")
+rownames(data) <- c("Male","Female")
+data <- as.table(data)
+data
+```
+
+
+```{r}
+chisq.test(data)
+```
+
+#one sample t.test
+```{r}
+daily.sales=c(5260,5470,5640,6180,6390,6515,6805,7515,7515,8230,8770)
+```
+
+
+```{r}
+mean(daily.sales)
+```
+
+
+```{r}
+sd(daily.sales)
+```
+
+
+```{r}
+quantile(daily.sales)
+```
+
+
+```{r}
+t.test(daily.sales)
+```
+
+
+```{r}
+t.test(daily.sales,mu=7000)
+```
+
+#Two sample
+```{r}
+dat=read.csv("hnd.csv")
+attach(dat)
+t.test(exp~sex,var.equal=T)
+```
+
+#Paired sample t-test
+```{r}
+dat=read.csv("hnd2.csv")
+t.test(dat$pre,dat$post,paired=T)
+```
+
+#Regression
+```{r}
+x1=runif(50,2,3)
+x2=rnorm(50,4,1)
+y=2*x1+x2
+dat=data.frame(y,x1,x2)
+m1=lm(y~x1+x2,dat)
+m1
+```
+
+```{r}
+summary(m1)
+```
+
+
+```{r}
+predict(m1)
+```
+
+```{r}
+plot(y,x1)
+```
+#Correlation
+```{r}
+x1=rnorm(100,2,1)
+x2=rnorm(100,2,4)
+cor(x1,x2)
+```
+
+
+```{r}
+cor.test(x1,x2)
+```
+
+
+```{r}
+cor.test(x1,x2,method="spearman")
+```
+
+
+```{r}
+cor.test(x1,x2,method="kendall")
+```
+
+#One way ANOVA
+```{r}
+x1=runif(50,2,3)
+y=2*x1+x1
+anova(lm(y~x1))
+```
+#Two way Anova
+```{r}
+x1=runif(50,2,3)
+x2=rnorm(50,4,1)
+y=2*x1+x2
+dat=data.frame(y,x1,x2)
+anova(lm(y~x1+x2,dat))
+```
+
+#Two way Anova with replication
+```{r}
+x1=runif(50,2,3)
+x2=rnorm(50,4,1)
+y=2*x1+x2
+dat=data.frame(y,x1,x2)
+anova(lm(y~x1*x2,dat))
+```
+
+
+#Time series
+```{r}
+Infla=ts(c(13.8,15.7,3.2,5.4,13.2,34.4,23.7,15.6,16.6),start= c(1970,1))
+plot(Infla)
+```
+
+
+```{r}
+acf(Infla)
+```
+
+
+```{r}
+pacf(Infla)
+```
+
+
+```{r}
+library(tseries)
+adf.test(Infla)
+```
+
+
+```{r}
+adf.test(diff(Infla))
+```
diff --git a/Test-for-the-Assumptions-of-Linear-Regression-Using R.Rmd b/Test-for-the-Assumptions-of-Linear-Regression-Using R.Rmd
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+---
+title: "Test for the Assumptions of Linear Regression Using R"
+output:
+  word_document: default
+  html_document:
+    df_print: paged
+---
+Test for the Assumptions of Linear Regression Using R
+
+BEING A SHORT TRAINING PRESENTED AT OFFA R USERS GROUP MEETING ON 26TH MARCH, 2024 AT STATISTICAL LABORATORY, STATISTICS DEPARTMENT, THE FEDERAL POLYTECHNIC OFFA, NIGERIA
+BY
+UDOKANG, ANIETIE EDEM (OGANIZER, ORUG)
+CHIF LECTURER, STATISTICS DEPARTMENT, THE FEDERAL POLYTECHNIC OFFA, NIGERIA
+
+#Simple Linear Regression
+ $y=a_{0}+a_{1}x+e$
+Where
+ $y$- the dependent variable
+ $x$ - independent variable
+ $a_{0}$- constant or the intercept on  - axis
+ $a_{1}$- coefficient or slope
+ $e$- random error
+
+#Multiple Linear Regression
+ $y=a_{0}+a_{1}x_{1}+a_{2}x_{2}+\dots+a_{k}x_{k}+e$
+Where
+ $y$- the dependent variable
+ $x$ - independent variable
+ $a_{0}$- constant or the intercept on  - axis
+ $a_{1},a_{2},a_{3},\dots,a_{k}$- coefficients
+ $e$- random error
+ 
+#Assumptions
+*i.	There must be a linear relationship between the variables
+*ii.	The error term must be homoscedastic (Equal Variance)
+*iii.	There must not be autocorrelation in the error term
+*iv.	There is no existence of multicollinearity in the data
+*v.	The residuals must be normally distributed
+
+In the multiple Linear Regression of  
+$y=a_{0}+a_{1}x_{1}+a_{2}x_{2}+\dots+a_{k}x_{k}+e$
+
+Let the $y$ be Gross Domestic Product represented by GDP and $x_{1}$ be Import represented by IMP and $x_{2}$ be Export represented by EXP.
+
+Therefore,
+ $yGDP=a_{0}+a_{1}IMP+a_{2}EXP+e$
+ 
+This model will be used to illustrate the different tests that will be carried out.
+
+#1. Test for Linearity
+##Scatter Diagram
+The plot of GDP against IMP, EXP and GVTREV will be done using Scatter Diagram to determine linearity.
+```{r}
+EXIMGDP=read.csv('EXIMGDP.csv',head=T)
+plot(EXIMGDP$EXP, EXIMGDP$GDP, xlab="EXP,IMP and GVTREV", ylab="GDP",main= "Plot of GDP Against EXP,IMP and GVTREV",col="red",pch=16,xlim=c(20,50),ylim=c(60,110)) 
+points(EXIMGDP$IMP, EXIMGDP$GDP,col="blue",pch=16)
+points(EXIMGDP$GVTREV, EXIMGDP$GDP,col="green",pch=16) 
+legend(20,110,legend=c('EXP', 'IMP', 'GVTREV'), pch=c(16,16,16),col=c('red', 'blue', 'green'))
+```
+
+# There is linear relationship as indicated by the scatter diagram between the response variable and the explanatory variables.
+*Action:* No action required. 
+If any of explanatory variables did not show linearity, then it should be transformed using any of the appropriate method such as logarithm and differencing.
+#Estimation of the Tentative Model for other Tests
+
+```{r}
+head(EXIMGDP,20) 	
+lm1=lm(GDP~IMP+EXP+GVTREV,data= EXIMGDP)
+lm1
+```
+
+
+```{r}
+summary(lm1)
+```
+
+
+#The p-value of 9.278e-14 indicates the model is suitable to the data (the model has passed the goodness of fit test). 
+
+This notwithstanding, the test for some important assumptions of regression model can further improve the model.
+
+#2. Homoscedasticity (Constant Variance)
+##Goldfield-Quandit Test
+$H_{0}$: There is homoscedasticity (Constant variance) 
+Vs
+$H_{1}$: There is heteroscedasticity (Variance are not constant)
+```{r}
+library(lmtest)
+gqtest(lm1)
+```
+#Since p-value = 0.9895 is not less than 0.05 level of significance, there is homoscedasticity.
+*Action:*Nil.
+If there is heteroscedasticity, the data needs to be transformed using an appropriate transformation technique such as logarithm and reciprocal.
+
+#3. Autocorrelation (The Error Terms are Independent)
+
+##Durbin-Watson Test
+$H_{0}$: There is no autocorrelation 
+Vs 
+$H_{1}$: There is autocorrelation
+```{r}
+library(lmtest)
+dwtest(lm1) 
+```
+
+#The DW = 1.316 and p-value = 0.01142<0.05, meaning that there is autocorrelation.
+*Action:* The original data should be transformed using autocorrelation of the residuals (random term-U) between $U_{t}$ and $U_{t-1}$.
+
+#4. No Multicollinearity (No Existence of High Correlation between the Explanatory Variables)
+##Variance Inflation Factor (VIF)
+$H_{0}$: There is no multicollinearity 
+Vs 
+$H_{1}$: There is multicollinearity
+#Decision Rule: 
+VIF = 1, there is no multicollinearity.
+1<VIF<=5, there is moderate multicollinearity
+VIF>5, there is high correlation between a given explanatory variable and other explanatory variables, hence existence of multicollinearity
+```{r}
+library(car)
+vif(lm1)
+```
+
+#There is existence of multicollinearity in IMP and EXP at severe level. But GVTREV has a moderate existence of autocorrelation which may require any action.
+*Action:* Remove EXP with the highest VIF or find an appropriate way of combining the two of EXP and IMP.
+
+#5. Residuals must be Normally Distributed
+##Q-Q Plot
+```{r}
+library(forecast)	
+resid<- resid(lm1)
+qqnorm(resid)
+```
+
+
+#This is near normality even though the points are not in a straight line but are close it, except one point which normality can be considered by these observation.
+
+#Let this be sunjected to a test of hypothesis using Shapiro-Wilk Test.
+ 
+ ##Shapiro-Wilk Test.
+ 
+$H_{0}$: The residuals are normally distributed 
+Vs 
+$H_{1}$: The residuals are not normally distributed 
+```{r}
+shapiro.test(residuals(lm1))
+```
+
+# The Shapiro test statistic W = 0.95848, p-value = 0.5139>0.05, hence the residuals have a normal distribution.
+*Action:* No action. 
+If the residuals are not normally distributed the original can be transformed or another model of non-linear form can be used.
+
+#This is end of today’s short training hosted by the Offa-R-Users-Group (ORUG) a place for learning and using R. I wish to you to be part of the training session either online or physical. If you are a guest, find time to register as a member to actualize your goal in using R. 
+The ORUG (https://www.meetup.com/fedpofa-r-users-group/ ) is sponsored by R Consortium and AniKem_Consult. For any Enquiry Contact the Organizer (WhatsApp: +2349030912602, email: anietieeu@yahoo.com)
+
+#BY NEXT QUARTER WE SHALL CONSIDER ACTIONS TO BE TAKEN WHEN THE ASSUMPTIONS ARE VIOLATED.
+
+THANKS FOR BEING PART OF THIS SHORT TRAINING.
+
+
+
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