Download The normal distribution

A normal distribution of data means
that most of the observations in a
set of data are close to the
"average," while relatively few
observations tend to one extreme
or the other.
Let's say you are interested in nutrition
You need to look at people's typical daily
kilojoule consumption. Like most data, the
numbers for people's typical consumption
probably will turn out to be normally
distributed. That is, for most people, their
consumption will be close to the mean, while
fewer people eat a lot more or a lot less than the
mean.
When you think about it, that's just common
sense.
Not that many people are getting by on a single
serving of kelp and rice. Or on eight meals of
steak and milkshakes. Most people lie
somewhere in between.
Measurements made of the following
quantities are considered to exhibit the
characteristics of a normal distribution:










heights of adults
cholesterol levels metabolic rates
crop yields per area
lengths of sardines
oxygen consumption
radiation exposures per area
IQ’s
head circumference
blood pressures
exam scores
There are many more, but if a particular
quantity does follow this distribution, many
useful statements may be made regarding
the proportions of the population that lie in
appropriate intervals.
Hypothetical data with a bell-shaped
frequency polygon would look like this:
If the frequency polygon with a normal
distribution is made into a smooth curve,
the result is a normal curve:
The x-axis (the horizontal one) is the value in
question... kilojoules consumed, dollars
earned or crimes committed, for example.
And the y-axis (the vertical one) is the
number (or frequency) of data points for each
value on the x-axis.
In other words, the number of people who eat
x kilojoules, the number of households that
earn x dollars, or the number of cities with x
crimes committed.
Not all sets of data will have graphs that look
this perfect.
Some will have relatively flat curves, others
will be pretty steep. Sometimes the mean will
lean a little bit to one side or the other.
But all normally distributed data will have
something like this same "bell curve" shape.
The standard deviation is the statistic that
tells you how tightly all the various
observations are clustered around the mean
in a set of data.
When the observations are pretty tightly
bunched together and the bell-shaped curve
is steep, the standard deviation is small.
When the examples are spread apart and the
bell curve is relatively flat, that tells you that
you have a relatively large standard
deviation.
One standard deviation away from the mean
in either direction on the horizontal axis (the
red area on the above graph) accounts for
around 68 percent of the observations in this
group.
Two standard deviations away from the mean
(the red and green areas) account for roughly
95 percent of the observations.
And three standard deviations (the red, green
and blue areas) account for about 99 percent
of the observations.
If this curve were flatter and more spread out,
the standard deviation would have to be
larger in order to account for those 68
percent or so of the people.
So that's why the standard deviation can tell
you how spread out the examples in a set
are from the mean.
Why is this useful?
Looking at the standard deviation can
help point you in the right direction
when asking why information is the way
it is.
http://www.youtube.com/watch?v=Wqw9cLRMPL0&feature=related
The normal distribution
The ends of the normal curve extend
toward infinity—they approach the
horizontal axis (x axis) but never quite
touch it.
This property represents the possibility
that, although the normal curve contains
virtually all values of a variable, a very
small number of values may exist that
reflect extremely large or extremely small
measurements (or values) of the variable
(outliers).
The horizontal axis of a normal curve can be
divided into 6 equal units
3 units between the mean and the place where
the curve approaches the axis on the left side,
and 3 units between the mean and the place
where it approaches the axis on the right side.
These six units collectively
reflect the amount of variation
that exists within virtually all
values of a normally distributed variable.
In a normal distribution of any variable,
virtually all values (except for 0.26%) fall
within these six units of variation.
Each unit corresponds to exactly one
standard deviation.
Look up a Z value of 1 on Page
442 of the textbook Area under
the Standard Normal Curve …
How does this table work?
The normal distribution showing standard
deviations
Normal distributions derived from different
data sets tend to have different means and
different standard deviations.
The figures below demonstrate how this
occurs by comparing three pairs of normal
curves.
The figure also demonstrates the fact that
normal curves can be drawn in a way that
they reflect the distribution of data.
That is, curves may be high and narrow, low
and wide, or anything in between.
Curves can be drawn to suggest the degree of
variation present in the distribution of a
variable—that is, the size of its standard
deviation.
Flatter curves suggest relatively large standard
deviations and higher ones reflect relatively
small standard deviations; however, they still
retain what is essentially a “bell shape.”
As the distribution is symmetrical, 50 percent of
the total area of the frequency polygon in a
normal distribution falls below the mean and 50
percent falls above it.
Other segments of the frequency polygon
similarly reflect standard percentages of its total
area.
The next chart displays the percentage of
the normal curve that falls between the
mean and the point (or variable) referred to
as standard deviation and between
standard deviations.
Almost all the area of the frequency polygon
(99.74%) lies between the points
-3SD and +3SD
Understanding that the percentage of the
area under a normal curve is also the
percentage of values that fall within a certain
area of a normally distributed variable is
critical to our understanding.
To summarise the characteristics of the standard normal
curve:

It is bell shaped and symmetrical.

Its’ mean, median, and mode all fall at the same point.

It is asymptotic which means the tails come close to but do
not touch the x axis.

Nearly all its area (99.74%) falls within three standard
deviations of the mean.
The centre and variation will depend on the values of mu and
sigma


The total area under the curve is 1 such that:
 the area within 1SD of mean
= 68.26%
 the area within 2 SD of mean
= 95.44%
 the area within 3SD of mean
= 99.74%
Example using area under the curve
The time taken to repair defective components
produced on an assembly line was monitored by
the manager each day over six months.
It was found that the daily repair time followed an
approximate normal distribution with a mean of 10
hours and a standard deviation of 2 hours.
Sketch the normal curve, and shade in the region
that represents the proportion of days on which
the repair time for defective components was:
Between 8 and 12 hours
more than 11 hours
Less than 9 hours
Between 11 and 14 hours
In this example, one curve is used to
represent a single distribution, and various
proportions of the observations are indicated
by shading under the curve.
We could calculate these areas and obtain
values for the proportions. After detailed
calculations, we could construct a table to
show values for all the different areas under
this curve.
However, in practice, we often need values
for areas under many normal curves (with
different means and different standard
deviations), so this particular table would
have very limited use.
It is impractical to create a table for
every normal curve and z-scores
overcome this problem by converting
all our measurements into standard
score or z-score form.
To convert a raw score of x to a standard or z-score
Standard score = z =
observed value – mean
Standard deviation
=
To convert a z-score to a raw score
If we were to repeatedly take samples from a
population and graph all the means we'd have
a normal distribution of sample means with the
standard deviation of these means called the
standard error.
This is such an important concept in statistics,
almost everything else you learn after this
depends on the fundamental concept.
It's called the central limit theorem. The
central limit theorem in it's shortest form states
that the sampling distribution of the sampling
means approaches a normal distribution as the
sample size gets larger, regardless of the shape
of the population distribution
So the sample means will be normally
distributed (especially when the
sample is above 30) even if the
population is positively skewed,
negatively skewed or even binomial
(having only 2 outcomes).
Here are two key points from the central
limit theorem:


The average of our sample means will itself be
the population mean.
The standard deviation of the sample means
equals the standard error of the population
mean.
The important message the central
limit theorem provides is
the sampling distribution of the
means is normally distributed even
if the population is not
http://www.khanacademy.org/math/statistics/v/centrallimit-theorem
Even though the individuals in a
population may have considerable
variable among themselves,
there is much less variation among the
averages of samples of many individuals.
What that allows us to do is solve
problems about means, even when the
data is not normally distributed.
Therefore the standard deviation of the
distribution of means is calculated by:
Regardless of the distribution of the parent population:

The mean of the population of means is always equal to
the mean of the parent population from which the
samples were drawn.

The standard deviation of the population of means is
always equal to the standard deviation of the parent
population divided by the square root of the sample size
(N).

The distribution of means will increasingly approximate a
normal distribution as the size N of samples increases.
A consequence of Central Limit Theorem
is that if we average measurements of a
particular quantity, the distribution of
our average tends toward a normal one.
The Central Limit Theorem explains why
we frequently we see the famous
bell-shaped "Normal distribution"
(or "Gaussian distribution")
in the measurements domain.

The notation ‘Pr’ or ‘P’ followed by ‘( z-score
takes on particular values )’ will mean:
The proportion of time that a z-score takes on
those particular values.
For example, Pr(z-score is less than 0.57) =
the proportion of z-scores that are less than
0.57.
Stat Mode 1,0, 0, xy, 0.1, ENT, 1, xy, 0.3, ENT…… RCL, 4 – mean is 1.7
100,000, ENT, 125000, ENT …. RCL, 6 x RCL 6 = 251,440
RCL 6 = 15856.86
Not to be confused with the Correlation Coefficient
The Coefficient of Variation is just showing what percentage of the mean is the
Standard Deviation
The larger the Standard Deviation is the larger the percentage is and the greater
is the dispersion of data from the mean - for example if the STD Dev was
30,000 we would have a coefficient of variation of 30,000 / 101,600 x 100 =
29.53%
4 (a)
Mean = 33.6 Standard deviation = 11.66
(i) Probability profit > $46,000
= 46 – 33.6
11.66
= 1.06
Z = 0.5 – 0.3554
= 14.46%
(ii)
Range = 33.6 + or – 1.96 (11.66)
$10,746 to $56,454
What is Hypothesis Testing?
A statistical hypothesis is an assumption about a
population parameter, often the mean.
This assumption may or may not be true.
In hypothesis testing we analyse differences between
the results we observe and the results we would expect
to obtain if the hypothesis were true.
Hypothesis testing refers to the formal procedures
used by statisticians to accept or reject statistical
hypotheses.

State the null and alternate hypotheses to be tested

Determine the level of significance

Select the test statistic

Determine the critical value

Calculate the value of the test statistic

Draw a conclusion
Statistical Hypotheses
The best way to determine whether a statistical
hypothesis is true would be to examine the entire
population.
Since that is often impractical, researchers typically
examine a random sample from the population.
If sample data are not consistent with the statistical
hypothesis, the hypothesis is rejected.

Are smoking and lung cancer related?

Are males better than females at maths?

Does fluoridation of the water supply help reduce
tooth decay?

Does a particular drug assist in curing a disease?

Has random breath testing reduced the road toll?

Is performance at high school related to
performance at TAFE?

Is profit related to the amount spent on
advertising?
Two types of statistical hypotheses
Null hypothesis
the null hypothesis, denoted by H0, is usually the
hypothesis that nothing unusual has occurred.
Alternative hypothesis
the alternative hypothesis, denoted by H1 or Ha, is
the hypothesis that something unusual has
occurred.
The hypotheses can be written together in the form
H0: (statement) v. H1: (alternate statement)
The alternative hypothesis may be classified as
two-tailed or one-tailed.
In the case of the two-tailed test (two-sided
alternative) we do the test with no pre-conceived
notion that the true vale of µ is either above or
below the hypothesised value of µ0.
The test is said to be two-sided because (or twotailed) and the alternate hypothesis is written as
H1: µ≠ µ0
A paint manufacturer claims that its new plastic all –
weather outdoor brand dries in an average of 20 minutes
when used on timber houses.
To test this claim, a scientist uses the paint on 30 different
pieces of timber that might be used in the construction of
houses.
What is the null and alternate hypotheses for this test and
what type of test is it?
In this case, there is no pre-conceived notion by the
scientist about whether the claimed 20 minutes might be
too long or too short.
Therefore, a two sided alternative is appropriate. The
hypotheses may be written
H0: µ=20 v. H1:µ≠ 20
For the one-tailed test (one-sided alternative), we do
the test with a strong conviction that if H0 is not true,
it is clear that µ is either greater than µ0 or less than
µ0
If we feel the only possible alternative is that µ is
greater than µ0 we write
H0: µ > µ0
However, if we feel the only possible alternative is
that µ is less than µ0 we write
H0: µ < µ0
The weights of full term female babies born to mothers at
a community hospital have been recorded over the years
and found to have a mean of 3200 g.
A particular group of 50 mothers were identified as being
heavy smokers throughout their pregnancy. Some medical
researchers felt that heavy smoking would usually result in
babies having lower birth weights than other babies.
What test would you use to test this and what is the
alternate hypothesis?
In this case, there is a pre-conceived notion that, if these
babies do have an unusual birth weight, it will only be in
the direction of lower.
That is, it is felt there is no chance that they will have a
higher birth weight. When setting up the alternate
hypothesis, it may well be appropriate to use a one-sided
alternative. The hypothesis may be written
H0: µ = 3200 v. H1: µ < 3200
A point of interest
can we accept the Null Hypothesis?
Some researchers say that a hypothesis test can have
one of two outcomes: you accept the null hypothesis or
you reject the null hypothesis.
Many statisticians, however, take issue with the notion of
"accepting the null hypothesis.“
Instead, they say: you reject the null hypothesis or you
fail to reject the null hypothesis.
Why the distinction between "acceptance" and "failure
to reject?"
Acceptance implies that the null hypothesis is true.
Failure to reject implies that the data are not sufficiently
persuasive for us to prefer the alternative hypothesis
over the null hypothesis.
After the appropriate hypothesis have been formulated,
the next step is to determine the
significance level (or -level) of the test.
This level represents the borderline probability between
whether an event (or sample) has occurred
by chance (ie.within acceptable boundaries of variation)
or
 an unusual event has taken place

The significance level used in testing a given hypothesis,
represents the maximum probability with which the tester
would be willing to risk a Type I error.
That is the risk the tester is prepared to take of rejecting
the null hypothesis when it is actually true.
Any significance level may be used
The most common significance level used is 0.05
Called the 5% significance level with notation  = 0.05
Another widely used level is  = 0.01or 1% significance
level
If we use a 5% significance level, what we are saying is
that an event (or sample) that occurs less than 5% of the
time is considered unusual.
In this case, we reject H0 if the probability of obtaining a
sample like ours is less than 0.05. That is, we conclude H0
may well be false.
We cannot reject H0 if this probability is more than 0.05.
That is, we conclude H0 is true.
If we use a 1% significance level, what we are saying is
that an event (or sample) that occurs less than 1% of the
time is considered unusual.
In this case, we reject H0 if the probability of obtaining a
sample like ours is less than 0.01. That is, we conclude H0
may well be false.
We cannot reject H0 if this probability is more than 0.01.
That is, we conclude H0 is true.
Decision Errors - two types of errors can
result from a hypothesis test
Type I errors occur when the researcher rejects a null
hypothesis (concludes it is false) when, in fact it is true.
The probability of committing a Type I error is equal to ,
the significance level of the test.
Type II errors occur when the researcher fails to reject a
null hypothesis (concludes it is true) when it is really false.
The probability of committing a Type II error is called Beta,
and is often denoted by β.
The probability of not committing a Type II error (1 – β)
is called the Power of the test.
Of course, we would like to avoid both errors as much as
possible.
Unfortunately, in trying to avoid one of them, we increase
the chances of making the other.
A company has developed a drug it feels may cure certain
types of cancer.
It has collected vast amounts of data as a result of clinical
trials and has asked you to examine the data and
determine whether the drug actually works.
Outline the null and alternate hypotheses, the
consequences of type I and type II errors and the
relationship between these errors in this scenario.
The null and alternate hypotheses in words H0 : the drug does not work v. H1 the drug does work
Making a type I error:
concluding the drug works when it really doesn’t

Patients are given false hope

Patients are taken off possibly more useful medication

The company will invest additional funds in the drug only
to find down the track the medication is not effective

The company may be subject to litigation
Making a type II error:
concluding the drug doesn’t work when it really does

The world and cancer patients are deprived of a cure

Many patients die who may have been cured

The company will have lost potential profits

The company may subject the statistician to litigation
In reality
Decision
H0 is true
H0 is false
Reject H0
Type I error
No error made
Not reject H0
No error made
Type II error
RCL, 6 – Std Dev of the Population, RCL 5 – Std Dev of a Sample

a Z score is used for normal distributions or
when the sample is greater or equal to 30
 otherwise we use a t test

State the null and alternate hypotheses to be
tested

Determine the level of significance

Select the test statistic

Determine the critical value

Calculate the value of the test statistic

Draw a conclusion
Check this using Z tables in Textbook
The Standard Deviation is adjusted by the size of our Sample -50 /√30
.5 - .025 =
.475
What if we had taken a sample of 100 cakes and found the mean to still be
505 grams then (505-500) / (15/10) = 3.33 - is greater than 1.96 - we
could not be 95% certain that a cake sampled at random would be close to
the expected mean of 500 grams - that is the more cakes we sample the
closer we would expect the average of the sample to be to the expected
mean of the entire production run
Check out the Excel Spread sheet – Exploring Self Test 5 Page 128 further
Equal or not equal to –
we have to test if it is outside the range - too small or too great therefore it is a two sided test.
Less than or greater than - it is a one sided test.
Note a Z score is used for normal distributions, when the sample
is greater or equal to 30, otherwise we use a t test
Question 18.
x bar = mean of Sample
u = mean of population
s = sample std dev
n = sample size
(a) (80-85) / (4/(√40)) - make sure you use memory
do √40 first - store in memory STO
then 4 / RCL = store in memory STO
then 5 / RCL = 7.9057
(a) (128-120) / 12 / (√(25-1)) = 3.27
(b) (340-330) / 23 / (√50) = 3.074
Normal populations or samples >30 - use Z and divided by √N
Otherwise use t test - it becomes √n-1
Normal populations or samples >30 - use Z and divided by √N
Otherwise use t test - it becomes √n-1
Question 20.
(41.3-42)/(.5/(sqr root 40 )) = 14
Question 22.
Critical Value 5% - .5 - .05 = .045 – Value of Z is 1.645
Value of Z test = 360 – 354 / (14 / (√30)) = 2.35
Reject the hypothesis since 2.35 > 1.645 – because it is further
from the mean – outside the acceptable range
Question 23