ciNormHalfWidth.RdCompute the half-width of a confidence interval for the mean of a normal distribution or the difference between two means, given the sample size(s), estimated standard deviation, and confidence level.
numeric vector of sample sizes. When sample.type="one.sample",
this argument denotes \(n\), the number of observations in the single sample.
When sample.type="two.sample", this argument denotes \(n_1\),
the number of observations from group 1.
Missing (NA), undefined (NaN), and infinite (Inf, -Inf) values are not allowed.
numeric vector of sample sizes for group 2. The default value is the value of n.or.n1.
This argument is ignored when sample.type="one.sample".
Missing (NA), undefined (NaN), and infinite (Inf, -Inf) values are not allowed.
numeric vector specifying the value(s) of the estimated standard deviation(s).
numeric vector of numbers between 0 and 1 indicating the confidence level
associated with the confidence interval(s). The default value is conf.level=0.95.
character string indicating whether this is a one-sample
(sample.type="one.sample") or two-sample
(sample.type="two.sample") confidence interval.
When sample.type="one.sample", the computed half-width is based on
a confidence interval for a single mean.
When sample.type="two.sample", the computed half-width is based on
a confidence interval for the difference between two means.
The default value is sample.type="one.sample" unless the argument
n2 is supplied.
If the arguments n.or.n1, n2, sigma.hat, and
conf.level are not all the same length, they are replicated to be the same length
as the length of the longest argument.
One-Sample Case (sample.type="one.sample")
Let \(\underline{x} = x_1, x_2, \ldots, x_n\) denote a vector of \(n\)
observations from a normal distribution with mean \(\mu\) and standard deviation
\(\sigma\). A two-sided \((1-\alpha)100\%\) confidence interval for \(\mu\)
is given by:
$$[\hat{\mu} - t(n-1, 1-\alpha/2) \frac{\hat{\sigma}}{\sqrt{n}}, \, \hat{\mu} + t(n-1, 1-\alpha/2) \frac{\hat{\sigma}}{\sqrt{n}}] \;\;\;\;\;\; (1)$$
where
$$\hat{\mu} = \bar{x} = \frac{1}{n} \sum_{i=1}^n x_i \;\;\;\;\;\; (2)$$
$$\hat{\sigma}^2 = s^2 = \frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})^2 \;\;\;\;\;\; (3)$$
and \(t(\nu, p)\) is the \(p\)'th quantile of
Student's t-distribution with \(\nu\) degrees of freedom
(Zar, 2010; Gilbert, 1987; Ott, 1995; Helsel and Hirsch, 1992). Thus, the
half-width of this confidence interval is given by:
$$HW = t(n-1, 1-\alpha/2) \frac{\hat{\sigma}}{\sqrt{n}} \;\;\;\;\;\; (4)$$
Two-Sample Case (sample.type="two.sample")
Let \(\underline{x}_1 = x_{11}, x_{12}, \ldots, x_{1n_1}\) denote a vector of
\(n_1\) observations from a normal distribution with mean \(\mu_1\) and
standard deviation \(\sigma\), and let
\(\underline{x}_2 = x_{21}, x_{22}, \ldots, x_{2n_2}\) denote a vector of
\(n_2\) observations from a normal distribution with mean \(\mu_2\) and
standard deviation \(\sigma\). A two-sided \((1-\alpha)100\%\) confidence
interval for \(\mu_1 - \mu_2\) is given by:
$$[(\hat{\mu}_1 - \hat{\mu}_2) - t(n_1 + n_2 - 2, 1-\alpha/2) \hat{\sigma} \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}, \, (\hat{\mu}_1 - \hat{\mu}_2) + t(n_1 + n_2 - 2, 1-\alpha/2) \hat{\sigma} \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}] \;\;\;\;\;\; (5)$$
where
$$\hat{\mu}_1 = \bar{x}_1 = \frac{1}{n_1} \sum_{i=1}^{n_1} x_{1i} \;\;\;\;\;\; (6)$$
$$\hat{\mu}_2 = \bar{x}_2 = \frac{1}{n_2} \sum_{i=1}^{n_2} x_{2i} \;\;\;\;\;\; (7)$$
$$\hat{\sigma}^2 = s_p^2 = \frac{(n_1 - 1) s_1^2 + (n_2 - 1) s_2^2}{n_1 + n_2 - 2} \;\;\;\;\;\; (8)$$
$$s_1^2 = \frac{1}{n_1 - 1} \sum_{i=1}^{n_1} (x_{1i} - \bar{x}_1)^2 \;\;\;\;\;\; (9)$$
$$s_2^2 = \frac{1}{n_2 - 1} \sum_{i=1}^{n_2} (x_{2i} - \bar{x}_2)^2 \;\;\;\;\;\; (10)$$
(Zar, 2010, p.142; Helsel and Hirsch, 1992, p.135,
Berthouex and Brown, 2002, pp.157–158). Thus, the half-width of this confidence
interval is given by:
$$HW = t(n_1 + n_2 - 2, 1-\alpha/2) \hat{\sigma} \sqrt{\frac{1}{n_1} + \frac{1}{n_2}} \;\;\;\;\;\; (11)$$
Note that for the two-sample case, the function ciNormHalfWidth assumes the
two populations have the same standard deviation.
a numeric vector of half-widths.
Berthouex, P.M., and L.C. Brown. (2002). Statistics for Environmental Engineers. Second Edition. Lewis Publishers, Boca Raton, FL.
Gilbert, R.O. (1987). Statistical Methods for Environmental Pollution Monitoring. Van Nostrand Reinhold, New York, NY.
Helsel, D.R., and R.M. Hirsch. (1992). Statistical Methods in Water Resources Research. Elsevier, New York, NY, Chapter 7.
Millard, S.P., and N. Neerchal. (2001). Environmental Statistics with S-PLUS. CRC Press, Boca Raton, FL.
Ott, W.R. (1995). Environmental Statistics and Data Analysis. Lewis Publishers, Boca Raton, FL.
USEPA. (2009). Statistical Analysis of Groundwater Monitoring Data at RCRA Facilities, Unified Guidance. EPA 530/R-09-007, March 2009. Office of Resource Conservation and Recovery Program Implementation and Information Division. U.S. Environmental Protection Agency, Washington, D.C. p.21-3.
Zar, J.H. (2010). Biostatistical Analysis. Fifth Edition. Prentice-Hall, Upper Saddle River, NJ, Chapters 7 and 8.
The normal distribution and lognormal distribution are probably the two most frequently used distributions to model environmental data. In order to make any kind of probability statement about a normally-distributed population (of chemical concentrations for example), you have to first estimate the mean and standard deviation (the population parameters) of the distribution. Once you estimate these parameters, it is often useful to characterize the uncertainty in the estimate of the mean. This is done with confidence intervals.
In the course of designing a sampling program, an environmental scientist may wish to determine
the relationship between sample size, confidence level, and half-width if one of the objectives
of the sampling program is to produce confidence intervals. The functions
ciNormHalfWidth, ciNormN, and plotCiNormDesign
can be used to investigate these relationships for the case of normally-distributed observations.
# Look at how the half-width of a one-sample confidence interval
# decreases with increasing sample size:
seq(5, 30, by = 5)
#> [1] 5 10 15 20 25 30
#[1] 5 10 15 20 25 30
hw <- ciNormHalfWidth(n.or.n1 = seq(5, 30, by = 5))
round(hw, 2)
#> [1] 1.24 0.72 0.55 0.47 0.41 0.37
#[1] 1.24 0.72 0.55 0.47 0.41 0.37
#----------------------------------------------------------------
# Look at how the half-width of a one-sample confidence interval
# increases with increasing estimated standard deviation:
seq(0.5, 2, by = 0.5)
#> [1] 0.5 1.0 1.5 2.0
#[1] 0.5 1.0 1.5 2.0
hw <- ciNormHalfWidth(n.or.n1 = 20, sigma.hat = seq(0.5, 2, by = 0.5))
round(hw, 2)
#> [1] 0.23 0.47 0.70 0.94
#[1] 0.23 0.47 0.70 0.94
#----------------------------------------------------------------
# Look at how the half-width of a one-sample confidence interval
# increases with increasing confidence level:
seq(0.5, 0.9, by = 0.1)
#> [1] 0.5 0.6 0.7 0.8 0.9
#[1] 0.5 0.6 0.7 0.8 0.9
hw <- ciNormHalfWidth(n.or.n1 = 20, conf.level = seq(0.5, 0.9, by = 0.1))
round(hw, 2)
#> [1] 0.15 0.19 0.24 0.30 0.39
#[1] 0.15 0.19 0.24 0.30 0.39
#==========
# Modifying the example on pages 21-4 to 21-5 of USEPA (2009),
# determine how adding another four months of observations to
# increase the sample size from 4 to 8 will affect the half-width
# of a two-sided 95% confidence interval for the Aldicarb level at
# the first compliance well.
#
# Use the estimated standard deviation from the first four months
# of data. (The data are stored in EPA.09.Ex.21.1.aldicarb.df.)
# Note that the half-width changes from 34% of the observed mean to
# 18% of the observed mean by increasing the sample size from
# 4 to 8.
EPA.09.Ex.21.1.aldicarb.df
#> Month Well Aldicarb.ppb
#> 1 1 Well.1 19.9
#> 2 2 Well.1 29.6
#> 3 3 Well.1 18.7
#> 4 4 Well.1 24.2
#> 5 1 Well.2 23.7
#> 6 2 Well.2 21.9
#> 7 3 Well.2 26.9
#> 8 4 Well.2 26.1
#> 9 1 Well.3 5.6
#> 10 2 Well.3 3.3
#> 11 3 Well.3 2.3
#> 12 4 Well.3 6.9
# Month Well Aldicarb.ppb
#1 1 Well.1 19.9
#2 2 Well.1 29.6
#3 3 Well.1 18.7
#4 4 Well.1 24.2
#...
mu.hat <- with(EPA.09.Ex.21.1.aldicarb.df,
mean(Aldicarb.ppb[Well=="Well.1"]))
mu.hat
#> [1] 23.1
#[1] 23.1
sigma.hat <- with(EPA.09.Ex.21.1.aldicarb.df,
sd(Aldicarb.ppb[Well=="Well.1"]))
sigma.hat
#> [1] 4.93491
#[1] 4.93491
hw.4 <- ciNormHalfWidth(n.or.n1 = 4, sigma.hat = sigma.hat)
hw.4
#> [1] 7.852543
#[1] 7.852543
hw.8 <- ciNormHalfWidth(n.or.n1 = 8, sigma.hat = sigma.hat)
hw.8
#> [1] 4.125688
#[1] 4.125688
100 * hw.4/mu.hat
#> [1] 33.99369
#[1] 33.99369
100 * hw.8/mu.hat
#> [1] 17.86012
#[1] 17.86012
#==========
# Clean up
#---------
rm(hw, mu.hat, sigma.hat, hw.4, hw.8)