# Lower and Upper confidence limit on estimated arithmetic mean using Land's exact procedure

I have to compute the LCL95% and UCL95% using Land's "exact" method. I computed the LCL and UCL for this lognormal distribution using another technique and I can't find anything for Land's exact procedure.

My data set x = {0.043, 0.236, 0.057, 0.016}

Here is what I tried

$y =$ mean of $\ln{x}$

$s^2 =$ standard deviation of $\ln x$.

Confidence limits $= \exp\left(y + s^2/2 \pm z\sqrt{s^2/n + s^4/2(n-1)}\right)$

and I got UCL: 2.98 and LCL: 0.139 but the answer using Land's exact is UCL95%:11.6 and LCL95%: 0.039

Here is what I have calculated already:

• Mean: 0.088
• Standard deviation: 0.1
• Geometric mean: 0.0552
• Geometric standard deviation: 3.04
• Estimated arithmetic mean using MVUE: 0.085
• 95th percentile: 0.343
• Upper limit of tolerance: 16.8
• mean of $\ln{x} = -2.898$
• standard deviation of $\ln{x} = 1.112$

Can anyone please help me sketch out an algorithm for the formula when using Land's exact method?

• (i) when you say "I couldn't find anything" what did you try? (ii) When you say "I have to" ... from where does this compulsion arise? Is this for a class, say? – Glen_b -Reinstate Monica Jul 23 '14 at 0:00
• If you're converting from an excel spreadsheet, you have a spreadsheet formula, do you not? The procedure is found in Land, C. E. (1988), Hypothesis tests and interval estimates, In Lognormal Distribution, (Crow and Shimizu eds), New York: Marcel Dekker, pp 87-112 ... but I am not sure you'll find it enlightening, since it takes about three pages to do it (p103-106) and it seems to assume you have tables that are in other documents. There's also R code for it here (see p 40) – Glen_b -Reinstate Monica Jul 23 '14 at 2:19
• The table is on my Web site (as an Excel workbook) linkable through quantdec.com/envstats/software. – whuber Jul 23 '14 at 15:47
• Averaging is not a good idea. Gilbert quotes Land in recommending that cubic interpolation in these tables should be adequate. A link on the web page I referenced ("Perform linear, quadratic, cubic, etc. interpolation (used to interpolate Land's H factor tables)") will send you to a spreadsheet showing how to do that. With a sample count of $4$ the results are heavily dependent on the assumption of lognormality, so you have much more serious issues to deal with in evaluating the sensitivity to that assumption. – whuber Jul 23 '14 at 18:00
• Perhaps it is interpolating differently--but that's a pretty big difference. (The implied value of $H$ they are using is more than $1$ less than you are computing.) Because that workbook does not allow you to view the formulas or the macros, it cannot be considered authoritative or trustworthy. – whuber Jul 23 '14 at 20:27

If you are looking for a statistical software implementation of Land's Exact method then in R this is provided by the EnvStats package available on CRAN.
The elnormAlt() function from this package can be used to calculate Land's confidence interval as well as other (Zou et al and Parkin et al and Cox) methods of confidence intervals for the lognormal distribution.