Question

Tree ring dating is often used to estimate the age of an archeological site.? However different trees give slightly different age estimates. A random sample of trees from the Sitka Terrace Archeological Site (STAS) gave the following age estimates:

1275, 1317, 1260, 1314, 1268, 1272, 1267, 1271, 1179, 1200

It is reasonable to treat these measurements as coming from a normal distribution with unknown mean μ and unknown variance σ2

a)Use the data to calculate an unbiased point estimate of the true mean, μ, of tree ring age estimates from STAS

b)Use the data to find an unbiased point estimate of the population variance, σ2 of tree ring age estimates from STAS.

c) Find the sample standard deviation of the above data

d) Find the maximum likelihood estimate for σ2 based upon this data.

e)Create a 99% confidence interval for μ. ( , )

f) What critical value did you use to calculate the 99% confidence interval in d?

g)Create a 99% prediction interval for μ. ( , )

h) Copy your R script for the above into the text box here.

Answer 1

##........reading the data
tree<-c(1275, 1317, 1260, 1314, 1268, 1272, 1267, 1271, 1179, 1200)
##-------------- a) Arithmatic mean
mean(tree)

##.........b) population variance estimate....
bvar<-var(tree) ##------it will give biased estimate
ubvar<-length(tree)/(length(tree)-1)*bvar ##-----unbiased estimate of variance

##.......c) Sample Standard Deviation -----
sd(tree)

##.......d)Since the data is normal then MLE of variance is simply the var function in r
var(tree)

##........e) confidence interval for μ
Average<-mean(tree) ##------- Simply Arithmatic mean
Error<-qnorm(0.995)*sd(tree)/sqrt(length(tree))#...we take 0.995 because 0.005 from both side of normal curve
Upper_interval<- Average+Error #....upper confidence interval
lower_interval<-Average-Error #....Lower confidence interval

##------f) 2.58 will be the critical value

##-------g) predict interval
data<- as.data.frame(tree)
PI<-predict(lm(data$tree~ 1), interval="predict",level = 0.99)
PI[1,]