# Topic 13 a  Confidence intervals when sigma is known
     # find y such that P( -y < X < y ) = 0.95
qnorm( (1-0.95)/2, lower.tail=FALSE)
     # bring this back to a N( mu, 17.6/sqrt(32))
     # first recompute but save the value
y <- qnorm( (1-0.95)/2, lower.tail=FALSE)
y
-y*17.6/sqrt( 32 )
y*17.6/sqrt( 32 )
     # we can do a few problems
     # find the 95% confidence interval for the mean of a 
     # population with known standard deviation 3.67 based
     # on a sample of size 28 with sample mean = 76.9
     #
sigma <- 3.67  #population standard deviation  
n <- 28        # sample size        
x_bar <- 76.9  # sample mean
alpha <- 0.95  # desired confidence level
z <- qnorm( (1-alpha)/2, lower.tail = FALSE ) # z value for N(0,1)
     # get low and high sides of the confidence interval
x_bar - z * sigma / sqrt( n )  # low side
x_bar + z * sigma / sqrt( n )  # high side
     # find the 93% confidence interval for the mean of a 
     # population with known standard deviation 8.06 based
     # on a sample of size 35 with sample mean = 1.27
     #
sigma <- 8.06  #population standard deviation  
n <- 35        # sample size        
x_bar <- 1.27  # sample mean
alpha <- 0.93  # desired confidence level
z <- qnorm( (1-alpha)/2, lower.tail = FALSE ) # z value for N(0,1)
# get low and high sides of the confidence interval
x_bar - z * sigma / sqrt( n )  # low side
x_bar + z * sigma / sqrt( n )  # high side
# find the 98.5% confidence interval for the mean of a 
# population with known standard deviation 15.4 based
# on a sample of size 11 with sample mean = 26.7
#
sigma <- 15.4  #population standard deviation  
n <- 11        # sample size        
x_bar <- 26.7  # sample mean
alpha <- 0.985 # desired confidence level
z <- qnorm( (1-alpha)/2, lower.tail = FALSE ) # z value for N(0,1)
# get low and high sides of the confidence interval
x_bar - z * sigma / sqrt( n )  # low side
x_bar + z * sigma / sqrt( n )  # high side
     #
     # It is clear that we could do sample after sample in 
     # this way.  There are a few points to observe here, however,
     # Each time we do the pair of lines, 45 and 46, we 
     # have R compute  z * sigma / sqrt( n ) two times.
     # We will call this value the Margin of Error, or MOE.
     # Then we can compute it once.
#
     # Find the 85% confidence interval for the mean of a 
     # population with known standard deviation 6.97 based on
     # a sample of size 26 with sample mean = 437.2.
z <- qnorm( 0.15/2, lower.tail=FALSE)
MOE <- z*6.97/sqrt( 26 )
437.2 - MOE  # the low value
437.2 + MOE  # the high value
     # do this again with our function
source("../ci_known.R")
ci_known( 6.97, 26, 437.2, 0.85 )
     # for a population with known sigma=13.4, taking a sample
     # of size 15, getting a sample of sample mean=8.74
     # look at changes for different levels of confidence
ci_known( 13.4, 15, 8.74, 0.80 )   # 80%
ci_known( 13.4, 15, 8.74, 0.90 )   # 90%
ci_known( 13.4, 15, 8.74, 0.98 )   # 98%
ci_known( 13.4, 15, 8.74, 0.999 )  # 99.9%
     # do the same thing but keep level at 0.90 and change 
     # the sample size
ci_known( 13.4, 25, 8.74, 0.90 )   # n=25
ci_known( 13.4, 45, 8.74, 0.90 )   # n=45
ci_known( 13.4, 85, 8.74, 0.90 )   # n=85