# 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