We have quickly covered the statistical equations we will be using when we went over question set 3,
but some of you either missed class (about 1/5 per day) or need further clarification about when to use
which equation. Thus, the following set of equations.
Note: Common sense will not carry you through the rest of the semester, as it will be numbers in not so
simple business problems. Thus, please use common sense and come to class!
Sample size when taking one sample to determine the mean or proportion within a given Error (E) for a
given confidence level (90, 95, 99 are popular confidence levels). You know to use the equation with
proportions (p) in it when the E is expressed as a %. You know to use the equation with sigma (standard
deviation (SD)) when the error is expressed as the same unit of measure as sigma, e.g., inches,
temperature, feet, and so on. You can approximate the SD by finding the range of a small sample of the
population and dividing by 4. When in doubt about which proportion to use, use p = .5, as it will give you
the largest sample size of any proportion, thus, giving you some extra margin.
To use these equations, determine the Z score for a given CL by subtracting CL expressed as a proportion
(.95) from one which gives you alpha (α), divide alpha by 2 as indicated in the equation and then find the
corresponding Z score by using an ‘Areas for the standard normal distribution’ chart. For the chart in
Evans and Lindsay you would subtract α/2 from .5 which gives you the area expressed as a proportion
between zero and the Z score you are looking for, then just find that area in the body of the chart, and
read the Z score. 1-.95=.05  .05/2=.025  .5-.025=.4750  the Z score for .4750 is 1.96.
If you are taking samples from a population to create a distribution of the means of the sample, sample
sizes between 5 and 30 will suffice. Five if the population has a low variability and is normally
distributed. Sample size will need to increase as the variability increases and the further from the
normal distribution of the population distribution.
How to determine what proportion of a population is between two measures, including a measure and
beyond, or including a given measure or less.
When we know μ and the σ and we just want to know one of the above 3 conditions or the sample size
is one, use the following equation:
If you want to know Z between μ and some of a sample, then:
If the standard deviation of the population is not known, then we have to use the student t distribution
instead of the Z distribution to find the three conditions. Unfortunately, the tables in most books do not
allow us to easily find areas for a given t value except for a few standard probabilities, so in the past we
have been forced to use the Z table by approximating the population standard deviation. Now there is a
calculator at http://www.anu.edu.au/nceph/surfstat/surfstat-home/tables/t.php where we can easily us
the t score just like the Z score. Select the button that best represents the area you are looking for (one
sided, two sided, tails, body), subtract one from the sample size and enter it in the d.f. box, enter t in the
t value box, then hit the top arrow to calculate the probability.
Estimating population mean when population standard deviation is known:
Estimating population mean when population standard deviation is not known: