Transcript CHAPTER 6: DISCRETE PROBABILITY DISTRIBUTIONS

A
Characteristic is a measurable description
of an individual such as height, weight or a
count meeting a certain requirement.
 A Parameter is a numerical summary of a
characteristic from an entire population such
as mean, proportion or standard deviation.
 A Statistic is a numerical summary of a
characteristic from a sample taken from an
entire population such as mean, proportion
or standard deviation.
A
Point Estimate is the Statistic from
experimental data that estimates the
Parameter.
 The confidence Interval is an interval that
contains the Parameter (with some level of
confidence) and is based on the Point
Estimate.
 Confidence Level is the expected proportion of
intervals that will contain the Parameter.
 Margin of Error is distance from the Point
Estimate to the ends of the Confidence Interval
(the possible error of the Point Estimate.
 To
Find the Confidence Interval of a
Proportion, the Z-distribution can be used to
find the Critical Value IF the requirements for
using the Normal Approximation of a Binomial
are met (Section 8.2).
 For a Proportion the experiment consists of a
sample size (n) and the count (x) from the
sample that meets the criteria producing the
point estimate of p ( p ) and a confidence
Level and p  x . If given p , then x  p * n .
n
 The
Margin of Error is the
(Standard Deviation) *(Critical Value).
Z
 Find the Critical Value    using the Z1
2 ).
Distribution as in Chapter 7: InvNorm(
 From 8.2, recall that the standard deviation
p*q
of p is given by n . So the Margin of Error
is calculated by Z * pn* q .
p*q
pZ *
 So the Confidence Interval is
n .
 Given x, n, and a Confidence Level, find p
Confidence Intervals. Also do for n and .
 Use 1-PropZInt.
2

2

2
 If
a certain Margin of Error (E) is required, a
sample size must be calculated.
 If
a p is available, then the sample size
needed to get a value of E or less is
Z


2

p * q 
E 

 If
p is not known, then the sample size
needed to get a value of E or less is
Z


2

0.25 
 E 



If the Confidence Interval is given, the Margin of
Error and the Point Estimate can be found from
the interval maximum and minimum values:
Maximum  Minimum

Margin of Error =
2
between the two values

Point Estimate =
interval.
Maximum  Minimum
2
Half the distance
The middle of the
 To
find the confidence Interval for a Mean,
find the Margin of Error (E) and add it to and
subtract it from the point estimate x :
 x  E, x  E 
 The
Margin of Error (E) is a factor (Critical
Value) multiplied by the Standard Deviation:
E  CV
. .* or E  CV
. .* s

 The Critical Value depends on the
Confidence Level desired.
 Find
the Critical Value Z 2 from the
Confidence Level Required using the
InvNorm( 1   2 ).
 Create the table of Confidence Levels and
Critical Values for various Levels.
Z *
 Then the Margin of Error is  2
.
 And the Confidence Interval is x  Z 2 *  .
 Given an x , a  and a required Confidence
Level find the Critical Value, the margin of
Error and the Confidence Interval.
 If
the population standard deviation is unknown,
the sample standard deviation must be used.
 But for the sample standard deviation we can not
use the Z-table to find the Critical Value. The
sample standard deviation is probably not exact,
so a wider different distribution is needed.
 The larger the sample size the better estimate
the sample standard deviation is of the
population standard deviation.
 Therefore the Critical Value will depend on the
Confidence Level AND THE SAMPLE SIZE.
 The
distribution used for the Mean with
an unknown Standard Deviation is the
Student or t Distribution and will be found
in the t-table.
 The columns of the table are the  2 ' s from
the Confidence Level. The rows are the
Degrees of Freedom and are one less than
the sample size  n  1.
t
 Find  2 for several sample sizes and
Confidence Levels.

 Also use InvT( 1  2 , n-1).
 Note
that the t-distribution is symmetric like
the Z distribution but wider.
 Note that the t-distribution is not one
distribution but many – one for each sample
size. The smaller the sample size the wider the
distribution.
 As the sample size approaches infinity the t
distribution approaches the Z distribution.
 After
finding the Critical Value, the margin of
Error and Confidence Interval can now be
calculated as was done with the situation with
a known standard deviation.
 Then the Margin of Error is t * s .
2
 And the Confidence Interval is x  t * s .
 Given an x , an s and a required Confidence
Level find the Critical Value, the margin of
Error and the Confidence Interval.
 Also use Tinterval.
2
 Finding
the Confidence Interval for a
Standard Deviation (s) or Variance ( s 2) is
different than the others have been.
 There is no Error to be calculated.
 The Critical Values comes from a different
distribution that is not normal.
2
 The  (called Chi-squared) Distribution is
skewed right, starts at 0 on the left and goes
to
on the right.
 There are two Critical Values that are both
positive.

 The
columns are split into two halves. The
half on the right are for R2 - Chi-squared Right

and use the value of 2 . The half on the left
are for L2 - Chi-squared Left and use the
value of 1   2 .

 The
rows are again the Degrees of Freedom
(n - 1).
 To
find the Confidence Interval of a Variance
 n 1 s 2 n 1 s 2 
use:
 ,  


 R2
 L2 



 To find the Confidence Interval of a Standard
Deviation use: 
2
2 
n

1
s
n

1
s






,
2
2

R
L 


 There are no calculator functions
 Must be given s, n and a Confidence Level.