Week 5

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Week 5
The Normal Probability Distribution

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OUTLINE

• Importance of the Normal distribution
• Characteristics of the Normal distribution.
• 68 - 95 - 99.7 rule.
• Standard Normal distribution.
• How to answer probability questions with the z
  table.

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OMNIPRESENCE

• Pharmacy
• Weights of tablets taken from a single batch
• Chemistry
• The concentration of a solution, the errors
  associated with any calibration process
• The random deviation from a baseline in a
  chromatogram
• The hydrogen ion concentration in rain at a
  given location
• The mass of a beaker if weighted many times
• Many other distributions that are not themselves
  normal can be made normal by transforming the
  data onto a different scale

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IMPORTANCE OF THE NORMAL DISTRIBUTION

• Physicians often rely on a knowledge of normal
  limits to classify patients as healthy or
  otherwise.
• For example, a serum cholesterol level above 250
  mg/dl is widely regarded as indicating a
  significantly increased risk of coronary heart
  disease. An accurate determination of such a
  value is of critical importance - may be a matter
  of life or death.
• Serum albumin is the chief protein of blood
  plasma and tend to follow a normal distribution.
  (- 2 s.d. from the mean from a group of
  presumably healthy persons)
• However, not all variables follow N. D. (well
  known counterexamples urea and alkaline
  phosphatase) clinical limits are the lower and
  upper 2.5 percentage points for any distribution,
  of healthy persons. They are obtained empirically!

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Entire Population
Sample
Inference
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Entire Population
Sample
Classical (Parametric) Inference is based on the
assumption that the population has a NORMAL
distribution
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CHARACTERISTICS OF A NORMAL PROBABILITY
DISTRIBUTION

• The normal curve is bell-shaped and has a single
  peak at the exact center of the distribution.
• The arithmetic mean, median, and mode of the
  distribution are equal and located at the peak.
• Half the area under the curve is above this
  center point (peak), and the other half is below
  it.

• The normal probability distribution is symmetric
  about its mean.
• It is asymptotic - the curve gets closer and
  closer to the x-axis but never actually touches
  it.

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CHARACTERISTICS OF A NORMAL DISTRIBUTION
Normal curve is symmetrical - two halves
identical -
Tail
Tail
Theoretically, curve extends to infinity
Theoretically, curve extends to - infinity
Mean, median, and mode are equal
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Normal Distributions with Equal Means but
Different Standard Deviations
s 3.1 s 3.9 s 5.0
m 20
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NOTE

• You can also have normal distributions with the
  same standard deviation but with different means
• different means and standard deviations

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Normal Probability Distributions with Different
Means and Standard Deviations.
m 5, s 3 m 9, s 6 m 14, s 10
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The equation

• The normal curve is not a single curve, rather it
  is an infinite number of possible curves, all
  described by the same algebraic expression
• s standard deviation of the normal curve
• m mean of the normal curve
• X value of the observation
• e base of natural logarithms, 2.718
• Y ordinate of normal curve, a function of X

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AREAS UNDER THE NORMAL CURVE

• About 68 percent of the area under the normal
  curve is within plus one and minus one standard
  deviation of the mean. This can be written as
  m 1s.
• About 95 percent of the area under the normal
  curve is within plus and minus two standard
  deviations of the mean, written m 2s.
• Practically all (99.7 percent) of the area under
  the normal curve is within three standard
  deviations of the mean, written m 3s.

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68 - 95 -99.7 RULE
X N(?, ?)
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EXERCISE 1
Women participating in a three-day experimental
diet regime have been demonstrated to have
normally distributed weight loss with mean 600 g
and a standard deviation 200 g. a) What
percentage of these women will have a weight loss
between 400 and 800 g? b) What percentage of
women will lose weight too quickly on the diet
(where too much weight is defined as gt1000g)?
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X (600,200)
a)
600
800
1000
400
200
0
1200
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X (600,200)
b)
600
800
1000
400
200
0
1200
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How to calculate probabilities?

• We see that normal curve is specified by two
  quantities only mean and the standard deviation.
  Hence X N (????) tells it all as soon as the
  values of the mean and standard deviation is
  known.
• Question How do we calculate the probabilities?
  Do we have to integrate?
• Fortunately, NO. We can transform any normal
  distribution into so-called standard normal
  distribution for which the probabilities have
  been already calculated.

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THE STANDARD NORMAL PROBABILITY DISTRIBUTION

• A normal distribution with a mean of 0 and a
  standard deviation of 1 is called the standard
  normal distribution.
• z value The distance between a selected value,
  designated X, and the population mean m, divided
  by the population standard deviation, s.
• A standardized score is simply the number of
  standard deviations an individual falls above or
  below the mean for the whole group. (Values
  above the mean have positive standardized scores,
  while those below the mean have negative ones).

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Table of the Standard Normal Distribution

• Cumulative probability for the Standard Normal
  Distribution
• Second digit of Z
• z 0.00 0.01 0.02 0.03 0.04
  0.05 0.06 0.07 0.08 0.09
• -3.5 0.0002 0.0002 0.0002 0.0002 0.0002
  0.0002 0.0002 0.0002 0.0002 0.0002
• -3.4 0.0003 0.0003 0.0003 0.0003 0.0003
  0.0003 0.0003 0.0003 0.0003 0.0002
• -3.3 0.0005 0.0005 0.0005 0.0004 0.0004
  0.0004 0.0004 0.0004 0.0004 0.0003
• -3.2 0.0007 0.0007 0.0006 0.0006 0.0006
  0.0006 0.0006 0.0005 0.0005 0.0005
• -3.1 0.0010 0.0009 0.0009 0.0009 0.0008
  0.0008 0.0008 0.0008 0.0007 0.0007
• -3.0 0.0013 0.0013 0.0013 0.0012 0.0012
  0.0011 0.0011 0.0011 0.0010 0.0010
• -2.9 0.0019 0.0018 0.0018 0.0017 0.0016
  0.0016 0.0015 0.0015 0.0014 0.0014
• -1.0 0.1587 0.1562 0.1539 0.1515 0.1492
  0.1469 0.1446 0.1423 0.1401 0.1379
• -0.9 0.1841 0.1814 0.1788 0.1762 0.1736
  0.1711 0.1685 0.1660 0.1635 0.1611
• -0.8 0.2119 0.2090 0.2061 0.2033 0.2005
  0.1977 0.1949 0.1922 0.1894 0.1867
• -0.7 0.2420 0.2389 0.2358 0.2327 0.2296
  0.2266 0.2236 0.2206 0.2177 0.2148
• -0.6 0.2743 0.2709 0.2676 0.2643 0.2611
  0.2578 0.2546 0.2514 0.2483 0.2451
• -0.5 0.3085 0.3050 0.3015 0.2981 0.2946
  0.2912 0.2877 0.2843 0.2810 0.2776

What is the probability that Z is less than
-2.92?
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THE STANDARD NORMAL CURVE
Z N(0,1)
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Standardisation
Normal with mean???and standard deviation?
?
???
???
????
????
X
Standard normal with mean???and standard
deviation?
Z
-2
0
1
2
-1
-1
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Computing Normal Probabilities

• 1. State the problem.
• 2. What is the appropriate probability statement?
• 3. Draw a picture and shade required area
• 4. Convert to a standard normal distribution
• 5. Find the probability in the standard normal
  table

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EXAMPLE 1
Classification of arterial diastolic blood
pressure (mm Hg) in adults, 18 years and older
(According to the 1998 report of the Joint
National Committee on the Detection, Evaluation,
and Treatment of High Blood Pressure, Arch Intern
Med 1988, 148, 1023)
Consider that blood pressure readings are
obtained from nearly 200,000 participants in a a
large-scale community blood pressure screening
program, and that these measurements follow a
normal distribution. The mean is 85 mm Hg, with a
standard deviation of 13 mm Hg.
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EXAMPLE 1

• a) What proportion of our sample will NOT be
  categorised as severely hypertensive?
• b) Suppose that we recommend that a physician be
  consulted if an individual has an arterial
  diastolic blood pressure equal or greater than 90
  mm Hg. What proportion of individuals in our
  screening program will be asked to consult a
  physician?
• c) What proportion of individuals have diastolic
  blood pressure in the mildly hypertensive range?
• d) What diastolic blood pressure will 75 of the
  population be above?

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a) What proportion of our sample will NOT be
categorized as severely hypertensive?
X (85, 13)
a)
115
85
98
111
72
59
46
124
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After the standardization
Z (0, 1)
a)
2.31
0
1
2
-1
-2
-3
3
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b) What proportion will be asked to consult a
physician?
X (85, 13)
b)
90
85
98
111
72
59
46
124
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After standardization
Z (0, 1)
b)
0
1
2
-1
-2
-3
3
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c) What proportion have blood pressure in the
mildly hypertensive range?
X (85, 13)
c)
85
98
111
72
59
46
124
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After we have standardised BOTH values
Z (0, 1)
c)
0
1
2
-1
-2
-3
3
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Finding a Value (X) given a Probability

• 1. State the problem
• 2. Draw a picture
• 3. Use table to find the probability closest to
  the one you need
• 4. Read off the z-value
• 5. Unstandardise i.e. x m zs

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d) What diastolic blood pressure will 25 of the
population be below?
X (85, 13)
d)
X ?
85
98
111
72
59
46
124
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d) Draw the picture of the Z curve
Z (0, 1)
d)
Z ?
0
1
2
-1
-2
-3
3
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EXERCISE 2
A measurement is taken of a solution with a known
concentration of 0.25Mg. It can be assumed that
the measurement process can be approximated by a
normal distribution with mean ? 0.25 and
standard deviation ? 0.001. Would you be
surprised by a measurement of 0.255?
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EXERCISE 3
Exam Question 4 1998 (8/100 marks) (a) The
distribution of serum levels of alpha tocopherol
(serum vitamin E) is approximately normal with
mean 860 mg/dL and standard deviation
340mg/dL. Show all working. What serum level
will 85 of the population be below? (b) Suppose
a person is identified as having toxic levels of
alpha tocopherol if his or her serum level is
greater than 2000mg/dL. What percentage of
people will be so identified?