531 HW 1 solution

STA 431/531 Homework 1 Due: Thursday 9/7 Name: Directions –You may work with others on the homework, but you must write and turn in your own copy. This does not mean that you can simply copy someone else ' s work!! Come and see me if you have any questions. Show your work and carry your answer to 3 decimal points. Write your answer on separate sheets of paper and clearly label your problems. Submit your R code along with R output. 1. Given the following data set: X Y 44.0 13.2 41.8 17.7 42.6 16.4 33.2 9.7 56.2 13.8 53.7 18.8 34.0 19.6 34.3 11.3 39.0 21.8 58.8 15.7 47.3 11.1 30.0 12.6 47.8 12.0 52.6 14.1 29.1 10.7 Use R to calculate the following things WITHOUT using R ' s built-in functions to calculate them directly. Again, you may not use the R functions, such as “mean”, “sd”, “cor” or “lm” which would calculate these directly (except to check your answers). You may use other R functions such as “sum” or “length” in your program. (a) Calculate the (sample) mean of Y. > x<-c (44.0 ,41.8 ,42.6 ,33.2 ,56.2 ,53.7 ,34.0 ,34.3 ,39.0 ,58.8 ,47.3 ,30.0 ,47.8 ,52.6 ,29.1 > y<-c (13.2 ,17.7 ,16.4 ,9.7 ,13.8 ,18.8 ,19.6 ,11.3 ,21.8 ,15.7 ,11.1 ,12.6 ,12.0 ,14.1 ,10.7) > mean.y<-sum(y)/length(y) > mean.y Page 1 of 6

[1] 14.56667 (b) Calculate the (sample) standard deviation of Y. > var.y<-sum ((y-mean(y))^2) /( length(y) -1) > sd.y<-sqrt(var.y) > sd.y [1] 3.630165 (c) Calculate the correlation coefficient between X and Y. > mean.x<-sum(x)/length(x) > var.x<-sum ((y-mean.x)^2) /( length(y) -1) > sd.x<-sqrt(var.x) > rho <-sum ((x-mean.x)*(y-mean.y))/(sd.x*sd.y) > rho [1] 0.8996201 Page 2 of 6

> index <-which(tennis.dfr $ major.titles >=6) > greatest.dfr <- tennis.dfr[index ,] (f) In the most efficient way possible, have R calculate the sample means for each of the numeric variables in the tennis.dfr data set. (Hint: Extract the appropriate subset of the data frame first.) > tennis.dfr2 <-tennis.dfr [,c(" major.match.wins"," major.match.losses "," overall.match.wins"," overall.match.losses", "major.titles "," overall.titles "," major.winning.pct"," overall.winning.pct ")] > apply(tennis.dfr2 , 2, mean ) major.match.wins major.match.losses overall.match.wins overall .match.losses major.titles overall.titles 159.9666667 41.7000000 700.9000000 225.8333333 6.3666667 50.7000000 major.winning.pct overall.winning.pct 0.7791792 0.7511338 3. Use the write.table() function to write the data set tennis.dfr to an external file. Make sure the external file includes the column names. Also, make sure the players ' names are NOT surrounded by quotes in the external file. write.table (tennis.dfr , "tennis2.txt",quote =FALSE ,col.names =TRUE ,row. names =F) Page 4 of 6

4. The Fibonacci numbers are the sequence of numbers defined by the linear recurrence equa- tion F n = F n − 1 + F n − 2 , where F 1 = F 2 = 1 and by convention F 0 = 0. For example, the first 8 Fibonacci numbers are 1, 1, 2, 3, 5, 8, 13, 21. (a) For a given n , compute the n th Fibonnaci number using a for loop. > Fibonacci1 <-function(n){ + r<-rep(NA ,n) + for(i in 1:n){ + if(i==1||i==2){ + r[i]<-1 + } + else{ + r[i]<-r[i -1]+r[i-2] + } + } + return(r[n]) + } > Fibonacci1 (15) [1] 610 (b) For a given n , compute the n th Fibonnaci number using a while loop. > Fibonacci2 <-function(n){ + r<-rep(NA ,n) + counter <-1 + while(counter <=n){ + if(counter ==1|| counter ==2){r[counter ]<-1} + else{r[counter]<-r[counter -1]+r[counter -2]} + counter <-counter +1 + } + return(r[n]) + } > Fibonacci2 (15) [1] 610 You can also calculate Fibonacci numbers recursively. > Fibonacci3 <-function(n){ + r<-NULL + if(n==1||n==2){r<-1} + else{r<-Fibonacci3(n-1)+Fibonacci3(n-2)} + r + } > Fibonacci3 (15) Page 5 of 6