# Exercise 1:  one-sample t-test and confidence interval

 

# You may copy the commands below from the web-browser into the command-window of R (or into a R-script)

# A line that starts with # is a comment, and R will disregard such lines.

 

 

#At the lectures we looked an example with the age of mineral samples (cf. slide 3 from the lectures)

#We will in this exercise see how the computations for this examples may be done in R.

 

 

#Start by reading the data into R. This may be done by the command:

age=c(249, 254, 243, 268, 253, 269, 287, 241, 273, 306, 303, 280, 260, 256, 278, 344, 304, 283, 310)

 

 

#Compute mean, median and standard deviation:

mean(age)

median(age)

sd(age)

 

# Check that you get the same result as in the lectures (cf slide 3)

 

 

# Make a histogram (cf. slide 4)

hist(age)

 

 

# Plot the empirical distribution function (cf. slide 4)

plot(ecdf(age))                                         # Basic plot

plot(ecdf(age),verticals=T, do.points=F)          # Nicer looking plot

 

 

# Compute min, first quartile, median, third quartile, and max (cf. slide 5)

quantile(age)

 

 

# Make a boxplot (cf. slide 5)

boxplot(age)

 

 

#We will then consider confidence intervals and hypothesis testing.

# We will illustrate direct calculations of the quantities involved as well as the use of a special R-command.

 

 

# Compute the 97.5% percentile of the t-distribution with 18 degrees of freedom:

qt(0.975,18)

 

# Compute lower and upper limit of the 95% confidence interval:

mean(age) - qt(0.975,18)*(sd(age)/sqrt(19))      # lower limit

mean(age) + qt(0.975,18)*(sd(age)/sqrt(19))     # upper limit

 

# Check that you get the same result as in the lectures (cf slide 18)

 

 

# Compute t-statistic:

tstat=(mean(age)-265)/(sd(age)/sqrt(19))       #t-statistic

tstat

 

# Compute P-value:

1-pt(tstat,18)

 

# Check that you get the same result as in the lectures (cf slide 22)

 

 

# R has readymade commands for t-tests with corresponding confidence intervals.

# Use the command "t.test" to compute the confidence interval (this gives a two-sided test):

t.test(age,mu=265)

 

# Use the command "t.test" to compute a one-sided test (this gives a one-sided confidence interval):

t.test(age,alternative="greater",mu=265)

 

# Check that you get the same results as above.