Normal, Binomial, Poisson Distributions - Department of Mathematics
Library, Teaching and Learning
Normal, Binomial, Poisson Distributions
QMET201
? 2014 Lincoln University
Did you know that QMET stands for Quantitative Methods? That is, methods for dealing with quantitative data, not qualitative data.
It is assumed you know about averages ? means in particular ? and are familiar with words like data, standard deviation, variance, probability, sample, population
You must know how to use your calculator to enter data, and from this, access the mean, standard deviation and variance
You need to become familiar with the various symbols used and their meanings ? be able to "speak" the language
Sample statistics are estimates of population parameters:
symbol used for the
symbol used for the
mean
population parameter
sample statistic
x
standard deviation
s
variance
2
s 2
standard error
s
n
n
You should appreciate that the analysis and interpretation of this data is the basis of decision making. For example
o Should the company put more money into advertising? o Should more fertilizer or water be applied to the crop? o Is it better to use Brand A or Brand B? etc
There are many analytical processes ? and this course deals with a few of the basic ones. Which process you use depends on
o What type of data you have ? discrete or continuous o How many variables - one, two, many o What you want to know
Tests and Examination preparation Practise on a regular basis ? set aside, say, half an hour each night or every second night, and/or 3 times during the weekend ? rather than a whole day or several hours just before a test.
Make sure your formula sheet is with you as you work, so that you become familiar with the information that is on it.
The following sections show summaries and examples of problems from the Normal distribution, the Binomial distribution and the Poisson distribution.
Best practice For each, study the overall explanation, learn the parameters and statistics used ? both the words and the symbols, be able to use the formulae and follow the process.
2
Normal Distribution
Applied to single variable continuous data e.g. heights of plants, weights of lambs, lengths of time
Used to calculate the probability of occurrences less than, more than, between given values e.g. "the probability that the plants will be less than 70mm", "the probability that the lambs will be heavier than 70kg", "the probability that the time taken will be between 10 and 12 minutes"
Standard Normal tables give probabilities - you will need to be familiar with the Normal table and know how to use it. First need to calculate how many standard deviations above (or below) the mean a particular value is, i.e., calculate the value of the "standard score" or "Z-score".
Use the following formula to convert a raw data value, X , to a standard score, Z :
Z X
eg. Suppose a particular population has m= 4 and = 2. Find the probability of a
randomly selected value being greater than 6.
The Z score corresponding to X = 6 is Z 6 4 1.
2 (Z=1 means that the value X = 6 is 1 standard deviation above the mean.)
Now use standard normal tables to find P(Z>1) = 0.6587 (more about this later).
Process:
o Draw a diagram and label with given values i.e.
, population mean , pop s.d.and X (raw score)
o Shade area required as per question
o Convert raw score X to standard score Z using formula
o Use tables to find probability: eg p0 Z z.
o Adjust this result to required probability
3
Example
Wool fibre breaking strengths are normally distributed with mean m = 23.56 Newtons
and standard deviation, = 4.55. What proportion of fibres would have a breaking strength of 14.45 or less?
Draw a diagram, label and shade area required:
X=14.45 xm=2233..5566 s=44.5.555
Convert raw score X to standard score Z : Z 14.45 23.56 2.0
4.55
That is, the raw score of 14.45 is equivalent to a standard score of -2.0. It is negative because it is on the left hand side of the curve.
Use tables to find probability and adjust this result to required probability:
p(X 14.45) pZ 2.0 0.5 p0 Z 2
0.5 0.4772 0.0228
That is the proportion of fibres with a breaking strength of 14.46 or less is 2.28%.
Note: Standard normal tables come in various forms. The ones used for these exercises show the probability of Z being between 0 and z, i.e. P(0 ................
................
In order to avoid copyright disputes, this page is only a partial summary.
To fulfill the demand for quickly locating and searching documents.
It is intelligent file search solution for home and business.
Related download
- the normal approximation to the binomial distribution
- binomial and geometric ap statistics
- normal binomial poisson distributions department of mathematics
- negative binomial distribution mean calculator
- the binomial distribution maynooth university
- beta binomial distribution demo md anderson cancer center
- binomial stati nspire rowan university
- 7 5 the binomial distribution graphing calculator version
- chapter 4 the poisson distribution university of wisconsin madison
- binomial distribution ti 83 84 everett community college
Related searches
- list of mathematics journals
- journal of mathematics research
- journal of mathematics education
- application of mathematics pdf
- application of mathematics in economics
- application of mathematics in science
- encyclopedia of mathematics pdf
- fundamentals of mathematics pdf
- levels of mathematics in order
- rudin principles of mathematics pdf
- international journal of mathematics research
- international journal of mathematics education