Question

I am supposed to answer these questions using R software. However, I have little to no experience with R or any other type of programming. Can someone please help me with the R code for these questions?

Q1. Suppose you have a population of size 5 [ie. N= corresponding numbers are: You measure sorne quantity (X) and the 21,22,23, 24,25 a) Calculate the population mean () b) Calculate the population variance (ơ2) using the formula ơ2-41 -P Q2. Imagine you are taking samples (of size n -3) from this population with replacement. Recall "sampling WITH replacement" ensures independence. a) Write down every possible way that you could have a sample of size 3 with replacement from this population. (hint: there will 5 5 5 125 possible combinations) Help: if you are struggling with figuring out the combinations try this code in R: expand.grid( c(21:25), c(21:25), c(21:25)) b) For each of these samples of size 3, calculate the sample mean and record it (either as a new object in R or as a new column in excel). Lets call this new column "X.bar". So you should have 125 values in this column Q3. You should have noticed that the values in the "X bar" column are repetitive. For example, 21.3333333 will show up 3 times. a) Construct a frequency table based on the column "X bar". [i.e. write down which values showed up how many times]. Now using the frequencies (also known as counts) calculate proportion of each of those repeated values. [For example: proportion of 21.3333333 will be 3/125] b) Plot these proportions against the values and connect the points using a non-linear line. (it will look like a density plot). Does the shape of this plot look like any known distribution? c) Using the table of proportions or otherwise, calculate the mean of these 125 numbers and com- pare it to your answer of 1(a) d) Using the table of proportions or otherwise, calculate the variance of these 125 numbers. Use the population variance formula (i.e. divide by 125 not 124). What is the relationship of this answer to your answer of 1(b)? e) Which theorem did you demonstrate empirically in part b, c and d?

Answer 1

1)

x <- 21:25
mean(x)

var_pop <- sum((x - mean(x))^2) / length(x)
var_pop

 mean(x)
[1] 23
> var_pop <- sum((x - mean(x))^2) / length(x)
> var_pop
[1] 2

a) mean = 23

b) variance = 2

2)

a) and b)

data <- expand.grid(c( 21:25),c( 21:25) ,c( 21:25))
data$xbar <- 1/3* ( data$Var1 + data$Var2 +data$Var3)

last line creates a new column of average

Please post Q3 again