The Scientific Method

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The Scientific Method

• Lecture 6
• Probability and Inferential Statistics

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Probability and Inferential Statistics

• In this lesson, we will
• Define probability and probability distributions
• Explore the mathematical properties of the normal
  distribution and its relationship to the mean and
  standard deviation

• By the end of this exercise, you should be able
  to
• Explain the terms probability, binomial, Poisson
  and normal (Gaussian) distribution
• Describe the mathematical relationship between
  the mean and standard deviation in a standard
  normal distribution

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Probability and Inferential Statistics
Scientific investigations sample values of a
variable to make inferences (or predictions)
about all of its possible values in the population
BUT! There is always some doubt as to whether
observed values population values eg Jersey
cow serum iron concentrations

• Inferential statistics quantify the doubt
• What are the chances of conclusions based on a
  sample of the population holding true for the
  population as a whole ?
• are the conclusions safe ? will the prediction
  happen in most observed situations?

Probability is defined as a relative frequency
or proportion the chance of something happening
out of a defined number of opportunities
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For example
subjectively as a expectation of an event a
cow has a 60 chance of calving tonight (based on
experience, but subject to individual opinions)
a priori probability based on the theoretical
model defining the set of all probabilities of an
outcome. eg when a coin is tossed, the
probability of obtaining a head is ½ or
0.5 defined probability the proportion of times
an event will occur in a very large number of
trials (or experiments) performed under similar
conditions. e.g. the proportion of times a guinea
pig will have a litter of greater than three,
based upon the observed frequency of this event
All of these approaches are related
mathematically
Probabilities can be expressed as a percentage
(23), a fraction / proportion (23/100) or a
decimal (0.23) as parts of a whole ( parts of a
unitised number of opportunities)
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Two rules govern probabilities
Addition rule when two events are mutually
exclusive (they cant occur at the same time) the
probabilities of either of them occurring is the
sum of the probabilities of each event eg 1/5
1/5 2/5 or 0.4 for two particular biscuits out
of 5 types
Multiplication rule - when two events are
independent, the probability of both events
occurring the product of their individual
probabilities e.g. a Friesian cow inseminated on
a particular day has a probability of calving 278
days later (the mean gestation period) of 0.5
(she either calves or she doesnt!)
If two Friesian cows are inseminated on the same
day, then the probability of both of them calving
on the same day 278 days later is (0.5 x 0.5 )
0.25
Probability distributions can derive from
discrete or continuous data a discrete random
variable with only two possible values (e.g.
male/female)is called a binary variable
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The NORMAL ( or Gaussian ) DISTRIBUTION is a
theoretical distribution of a continuous random
variable (x) whose properties can be described
mathematically by the mean (?) and standard
deviation (s)
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the proportion of the values of x lying between
and 1x (times!), 1.96x and 2.58x the standard
deviation on either side of the mean. It means
100 of the data values are included within 3 sd
units either side of the mean
In a perfectly symmetrical normal
distribution,MEAN, MEDIAN and MODE have the same
value
Normal distributions with the same value of the
standard deviation (s) but different values of
the mean (?)
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It is possible to make predictions about the
likelihood of the mean value of a variable
differing from another mean value whether the
difference is likely or unlikely to be due to
chance alone
This is the basis of significance testing - if
the distribution of observed values approximates
to the normal distribution, it becomes possible
to compare means of variables with the
theoretical distribution and estimate whether
their observed differences are significantly
different from the expected values of each
variable if they are truly normally distributed
eg Students t Test
We carry out an experiment on guinea pigs to test
the hypothesis that dietary lipid sources rich in
?3 polyunsaturated fatty acids improve coat
condition We compare the breaking strength of
hairs from two groups of 10 guinea pigs fed a
normal mix compared with a diet supplemented with
cod liver oil, recording the max. weight their
hair will support as tensile strength in g. We
want to decide whether the mean strength of hairs
from the control and experimental groups differ
significantly at the end of the trial
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Calculating the t statistic

The steps for doing this manually are best set
out in a table
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Calculating the t statistic
na 10
nb 10
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Calculating the t statistic
We then compare our calculated value of t with
those in the table of critical values for the
value of t
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Significance and confidence
These are the significance levels for the t
statistic at 10, 5, 1 and 0.1, (from left to
right)
If our value for t (2.78) exceeds any of
the tabulated values of t for 18 df, which it
does for p 0.05, but not for p 0.01, we can
say
the means are different at the 5 level of
significance and we can reject H0 (the null
hypothesis of no difference between the two
treatments)
The confidence level is simply 100
(significance level) So, alternatively, we could
say we can be 95 confident that there is a
significant difference between the two means
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• Now attempt
• the practice Students t exercise
• in your workbook!