Example R programs and commands
		 6. Use, power and sample size in Student t tests

# All lines preceded by the "#" character are my comments.
# All other left-justified lines are my input.
# All other indented lines are the R program output.

# Perform a one-sample, two-sided t-test of H_0: mu=1.7 on data vector x:
x<-c(1,4,2,2,3,1,5,3,4,2,1,3); t.test(x,mu=1.7)

	   One Sample t-test

   data:  x
   t = 2.3334, df = 11, p-value = 0.03963
   alternative hypothesis: true mean is not equal to 1.7
   95 percent confidence interval:
    1.750127 3.416539
   sample estimates:
   mean of x
    2.583333


# Perform a one-sample, one-sided t-test of H_A: mu>1.7 on data vector x:
x<-c(1,4,2,2,3,1,5,3,4,2,1,3); t.test(x,mu=1.7, alternative="greater")

	   One Sample t-test

   data:  x
   t = 2.3334, df = 11, p-value = 0.01982
   alternative hypothesis: true mean is greater than 1.7
   95 percent confidence interval:
    1.903482      Inf
   sample estimates:
   mean of x
    2.583333


# Perform a two-sample, two-sided t-test of H_0:mu_x=mu_y on data vectors x,y
# of unequal length, assuming both populations have the same variance:
x<-c(1,4,2,2,3,1,5,3,4,2,1,3); y<-c(3,6,4,5,7,5,6); t.test(x,y, var.equal=TRUE)

	   Two Sample t-test

   data:  x and y
   t = -4.0666, df = 17, p-value = 0.0008027
   alternative hypothesis: true difference in means is not equal to 0
   95 percent confidence interval:
    -3.887451 -1.231596
   sample estimates:
   mean of x mean of y
    2.583333  5.142857

# ...and using Welch's correction for not assuming equal variance:
x<-c(1,4,2,2,3,1,5,3,4,2,1,3); y<-c(3,6,4,5,7,5,6); t.test(x,y, var.equal=FALSE)

	   Welch Two Sample t-test

   data:  x and y
   t = -4.0378, df = 12.415, p-value = 0.001540
   alternative hypothesis: true difference in means is not equal to 0
   95 percent confidence interval:
    -3.935537 -1.183511
   sample estimates:
   mean of x mean of y
    2.583333  5.142857



# Perform a two-sample, one-sided t-test of H_A:mu_x-mu_y < -1.5 on data vectors x,y
# of unequal length, assuming both populations have the same variance:
x<-c(1,4,2,2,3,1,5,3,4,2,1,3); y<-c(3,6,4,5,7,5,6); 
t.test(x,y, mu= -1.5, alternative="less", var.equal=TRUE)

	   Two Sample t-test

   data:  x and y
   t = -1.6834, df = 17, p-value = 0.05529
   alternative hypothesis: true difference in means is less than -1.5
   95 percent confidence interval:
	 -Inf -1.464607
   sample estimates:
   mean of x mean of y
    2.583333  5.142857

# ...and using Welch's correction for not assuming equal variance:
t.test(x,y, mu= -1.5,alternative="less", var.equal=FALSE)

	   Welch Two Sample t-test

   data:  x and y
   t = -1.6715, df = 12.415, p-value = 0.05981
   alternative hypothesis: true difference in means is less than -1.5
   95 percent confidence interval:
	 -Inf -1.432897
   sample estimates:
   mean of x mean of y
    2.583333  5.142857


# Perform a paired-sample, one-sided t-test of H_A:mu_d < -1.5 on data vectors x,y
# of equal length:
x<-c(1,4,2,2,3,1,5,3,4,2,1,3); y<-c(3,6,4,2,6,5,5,7,5,6,4,6); 
t.test(x,y, mu= -1.5, paired=TRUE, alternative="less")

	   Paired t-test

   data:  x and y
   t = -2.011, df = 11, p-value = 0.03474
   alternative hypothesis: true difference in means is less than -1.5
   95 percent confidence interval:
	 -Inf -1.589141
   sample estimates:
   mean of the differences
		 -2.333333



# Perform a paired-sample, two-sided t-test of H_0:mu_d = 0 on data vectors x,y
# of equal length:
x<-c(1,4,2,2,3,1,5,3,4,2,1,3); y<-c(3,6,4,2,6,5,5,7,5,6,4,6); 
t.test(x,y, paired=TRUE)  # mu=0 and alternative="two.sided"  are defaults

	   Paired t-test

   data:  x and y
   t = -5.6308, df = 11, p-value = 0.0001531
   alternative hypothesis: true difference in means is not equal to 0
   95 percent confidence interval:
    -3.245395 -1.421272
   sample estimates:
   mean of the differences
		 -2.333333





# Find the power in a two-sided two-sample Student t test with n1=n2=n=33
# samples per population, a true difference |mu1-mu2|=1.7, equal standard
# deviations s1=s2=2.1, using a Type I error probability of alpha=0.05:
power.t.test(n=33,delta=1.7,sd=2.1, sig.level=0.05, power=NULL)

     Two-sample t test power calculation

              n = 33
          delta = 1.7
             sd = 2.1
      sig.level = 0.05
          power = 0.8994822
    alternative = two.sided

 NOTE: n is number in *each* group

# Find the minimum sample size, if the desired power level is 95%:
power.t.test(n=NULL,delta=1.7,sd=2.1, sig.level=0.05, power=0.95)

     Two-sample t test power calculation

              n = 40.64486
          delta = 1.7
             sd = 2.1
      sig.level = 0.05
          power = 0.95
    alternative = two.sided

 NOTE: n is number in *each* group


# Find the minimum detectable difference with 50 samples, same distribution
# parameters, alpha=0.05 and power=0.95:
power.t.test(n=50,delta=NULL,sd=2.1, sig.level=0.05, power=0.95)

     Two-sample t test power calculation

              n = 50
          delta = 1.529104
             sd = 2.1
      sig.level = 0.05
          power = 0.95
    alternative = two.sided

 NOTE: n is number in *each* group


# Given d, find the the number n of samples needed from an N(mu,sigma)
# probability space so that the average m of the n samples makes a 95%
# confidence interval [m-d,m+d] for mu:
variance <- 12.8;  sigma <- sqrt(variance)   # example variance and its s.d.
conf <- 0.95; sig <- 1-conf     # 95% confidence is significance level 5%  
err <- 2.0;                # allowed error; we want to estimate mu within err
power.t.test(n=NULL,       #  this is computed and returned
              sig.level=sig,     # 1 - confidence 
              delta=err,         # half-width of conf. interval 
              sd=sigma,          # estimated standard deviation
              power=0.50,        # fixed value!
              type="one.sample")  # required flag for non-default

# This "magic" value power=0.50 is what it takes to get the right value of n