The document provides an outline and explanation of key concepts related to the normal distribution. It begins with an introduction to probability distributions for continuous random variables and the definition of a density curve. It then defines terms and symbols used in the normal distribution, including mean, standard deviation, and z-scores. The document explains the characteristics of the normal distribution graphically and provides examples of finding areas under the normal curve using z-tables. It concludes with examples of finding unknown z-values and calculating probabilities for specific scenarios involving the normal distribution.
The Normal Distribution is a symmetrical probability distribution where most results are located in the middle and few are spread on both sides. It has the shape of a bell and can entirely be described by its mean and standard deviation.
The Normal Distribution is a symmetrical probability distribution where most results are located in the middle and few are spread on both sides. It has the shape of a bell and can entirely be described by its mean and standard deviation.
It includes various cases and practice problems related to Binomial, Poisson & Normal Distributions. Detailed information on where tp use which probability.
Normal Distribution – Introduction and PropertiesSundar B N
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It includes various cases and practice problems related to Binomial, Poisson & Normal Distributions. Detailed information on where tp use which probability.
Normal Distribution – Introduction and PropertiesSundar B N
In this video you can see Normal Distribution – Introduction and Properties.
Watch the video on above ppt
https://www.youtube.com/watch?v=ocTXHLWsec8&list=PLBWPV_4DjPFO6RjpbyYXSaZHiakMaeM9D&index=4
Subscribe to Vision Academy
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The Normal Distribution:
There are different distributions namely Normal, Skewed, and Binomial etc.
Objectives:
Normal distribution its properties its use in biostatistics
Transformation to standard normal distribution
Calculation of probabilities from standard normal distribution using Z table.
Normal distribution:
- Certain data, when graphed as a histogram (data on the horizontal axis, frequency on the vertical axis), creates a bell-shaped curve known as a normal curve, or normal distribution.
- Two parameters define the normal distribution, the mean (µ) and the standard deviation (σ).
Properties of the Normal Distribution:
Normal distributions are symmetrical with a single central peak at the mean (average) of the data.
The shape of the curve is described as bell-shaped with the graph falling off evenly on either side of the mean.
Fifty percent of the distribution lies to the left of the mean and fifty percent lies to the right of the mean.
-The mean, the median, and the mode fall in the same place. In a normal distribution the mean = the median = the mode.
- The spread of a normal distribution is controlled by the
standard deviation.
In all normal distributions the range ±3σ includes nearly
all cases (99%).
Uni modal:
One mode
Symmetrical:
Left and right halves are mirror images
Bell-shaped:
With maximum height at the mean, median, mode
Continuous:
There is a value of Y for every value of X
Asymptotic:
The farther the curve goes from the mean, the closer it gets to the X axis but it never touches it (or goes to 0).
The total area under a normal distribution curve is equal to 1.00, or 100%.
Using Normal distribution for finding probability:
While finding out the probability of any particular observation we find out the area under the curve which is covered by that particular observation. Which is always 0-1.
Transforming normal distribution to standard normal distribution:
Given the mean and standard deviation of a normal distribution the probability of occurrence can be worked out for any value.
But these would differ from one distribution to another because of differences in the numerical value of the means and standard deviations.
To get out of this problem it is necessary to find a common unit of measurement into which any score could be converted so that one table will do for all normal distributions.
This common unit is the standard normal distribution or Z
score and the table used for this is called Z table.
- A z score always reflects the number of standard deviations above or below the mean a particular score or value is.
where
X is a score from the original normal distribution,
μ is the mean of the original normal distribution, and
σ is the standard deviation of original normal distribution.
Steps for calculating probability using the Z-
score:
-Sketch a bell-shaped curve,
- Shade the area (which represents the probability)
-Use the Z-score formula to calculate Z-value(s)
-Look up Z-values in table
Probability
Random variables and Probability Distributions
The Normal Probability Distributions and Related Distributions
Sampling Distributions for Samples from a Normal Population
Classical Statistical Inferences
Properties of Estimators
Testing of Hypotheses
Relationship between Confidence Interval Procedures and Tests of Hypotheses.
Python is an interpreted, object-oriented, high-level programming language with dynamic semantics. Its high-level built in data structures, combined with dynamic typing and dynamic binding, make it very attractive for Rapid Application Development, as well as for use as a scripting or glue language to connect existing components together
Discusses on how to determine the area under the normal curve using the z-table. Discusses the skewness and kurtosis of the curve. It includes examples how to determine the area below, above, between, or the end-tails.
2. TOPIC OUTLINE
The Normal Distribution
1) Introduction
2) Definition of Terms and Statistical Symbols Used
3) How To Find Areas Under the Normal Curve
4) Finding the Unknown Z represented by Zo
5) Examples
Hypothesis Testing
3. The Normal Distribution
Introduction
Before exploring the complicated Standard
Normal Distribution, we must examine how the
concept of Probability Distribution changes when the
Random Variable is Continuous.
4. The Normal Distribution
Introduction
A Probability Distribution will give us a Value of P(x) = P(X=x)
to each possible outcome of x. For the values to make a Probability
Distribution, we needed two things to happen:
1. P (x) = P (X = x)
2. 0 ≤ P (x) ≤ 1
For a Continuous Random Variable, a Probability Distribution
must be what is called a Density Curve. This means:
1. The Area under the Curve is 1.
2. 0 ≤ P (x) for all outcomes x.
5. The Normal Distribution
Introduction: Example:
Suppose the temperature of a piece of metal is always between 0°F
and 10°F. Furthermore, suppose that it is equally likely to be any
temperature in that range. Then the graph of the probability distribution
for the value of the temperature would look like the one below:
0.10
Uniform Distribution
P (X < 5) ValuesAREA
are spread
uniformly across
Probability 0.05 0.8
the range 0 to 10
0.5 4
P (3 < X ≤ 7) Probability:
Area: 80% P (X > 2)
(4)(0.1) =
0.00 Finding the Area Under the Curve
0.4
7 - 3 = 4
2 3 5
5 7
10
X
Illustration of the fundamental fact about DENSITY CURVE
6. The Normal Distribution: Definition
of Terms and Symbols Used
Normal Distribution Definition:
1) A continuous variable X having the symmetrical, bell shaped distribution
is called a Normal Random Variable.
2) The normal probability distribution (Gaussian distribution) is a
continuous distribution which is regarded by many as the most significant
probability distribution in statistics particularly in the field of statistical
inference.
Symbols Used:
“z” – z-scores or the standard scores. The table that transforms every normal
distribution to a distribution with mean 0 and standard deviation 1. This
distribution is called the standard normal distribution or simply
standard distribution and the individual values are called standard
scores or the z-scores.
“µ” – the Greek letter “mu,” which is the Mean, and
“σ” – the Greek letter “sigma,” which is the Standard Deviation
7. The Normal Distribution: Definition
of Terms and Symbols Used
Characteristics of Normal Distribution:
1) It is “Bell-Shaped” and has a single peak at the center of the
distribution,
2) The arithmetic Mean, Median and Mode are equal.
3) The total area under the curve is 1.00; half the area under the
normal curve is to the right of this center point and the other
half to the left of it,
4) It is Symmetrical about the mean,
5) It is Asymptotic: The curve gets closer and closer to the X –
axis but never actually touches it. To put it another way, the
tails of the curve extend indefinitely in both directions.
6) The location of a normal distribution is determined by the
Mean, µ, the Dispersion or spread of the distribution is
determined by the Standard Deviation, σ.
8. The Normal Distribution: Graphically
Normal Curve is Symmetrical
Two halves identical
Theoretically, curve Theoretically, curve
extends to - ∞ extends to + ∞
Mean, Median
and Mode are
equal.
9. AREA UNDER THE NORMAL CURVE
P(x1<X<x2)
P(X<x1) P(X>x2)
x1 x2
10. AREA UNDER THE NORMAL CURVE
Let us consider a variable X which is normally distributed with a mean of 100 and
a standard deviation of 10. We assume that among the values of this variable are
x1= 110 and x2 = 85.
110 100 85 100
z1 1 . 00 z2 1 . 50
10 10
0.7745
0.4332
0.3413
-1.5 1
11. The Standard Normal Probability
Distribution
The Standard Normal Distribution is a Normal
Distribution with a Mean of 0 and a Standard Deviation
of 1.
It is also called the z distribution
A z –value is the distance between a selected value
, designated X, and the population Mean µ, divided by
the Population Standard Deviation, σ.
The formula is :
13. How to Find Areas Under the
Normal Curve
Let Z be a standardized random variable
P stands for Probability
Φ(z) indicates the area covered
under the Normal Curve.
a.) P (0 ≤ Z≤≤ZZ≤≤2.45) Φ(-0.35) – Φ (0.91)
b.) (-0.35 1.53) = Φ Φ (2.45)
c.) (0.91 0) = (1.53)
= 0.4929 - 0.3186
= 0.1368
0.4370
= 0.1743
This shaded area, from
This shaded area, from
This shaded area, from
Z Z = -0.35Z = 1.53, = 2.45,
= 0 and until Z = 0,
Z – 0.91 until Z
represents the probability
represents the probability
represents the probability
value ofof0.1368
value of0.4370.
value 0.1743
14. How to Find Areas Under the
Normal Curve
Let Z be a standardized random variable
P stands for Probability
Φ(z) indicates the area covered
under the Normal Curve.
d.) P (-2.0 ≤ Z ≤ 0.95) = Φ (-2.0) + Φ (0.95)
=0.4772 + 0.3289
= 0.8061
This shaded area, from
Z = -2.0 until Z = 0.95
represents the probability
value of 0.8061.
15. How to Find Areas Under the
Normal Curve
Let Z be a standardized random variable
P stands for Probability
Φ(z) indicates the area covered
under the Normal Curve.
e.) P (-1.5 ≤ Z ≤ -0.5) = Φ (-1.5) – Φ (-0.5)
=0.4332 - 0.1915
= 0.2417
This shaded area, from
Z = -1.5 until Z = -0.5
represents the probability
value of 0.2417.
16. How to Find Areas Under the
Normal Curve
Let Z be a standardized random variable
P stands for Probability
Φ(z) indicates the area covered
under the Normal Curve.
f.) P(Z ≥ 2.0) = 0.5 – Φ (2.0) This shaded area, from
= 0.5 – 0.4772 Z = 2.0 until beyond Z = 3
represents the probability
= 0.0228
value of 0.0228.
17. How to Find Areas Under the
Normal Curve
Let Z be a standardized random variable
P stands for Probability
Φ(z) indicates the area covered
under the Normal Curve.
g.) P (Z ≤ 1.5) = 0.5 + Φ (1.5)
= 0.5 + 0.4332
= 0.9332
This shaded area, from
Z = 1.5 until beyond Z = -3
represents the probability
value of 0.9332
19. Finding the unknown Z represented
by Zo
P(Z ≤ Z0) = 0.8461
0.5 + X = 0.8461
X = 0.8461 – 0.5
X = 0.3461
Z0 = 1.02 ans.
• P(-1.72 ≤ Z ≤ Z0) = 0.9345
• Φ (-1.72) + X = 0.9345
• X = 0.9345 – 0.4573
• X = 0.4772
• Z0 = 2.0
20. Finding the unknown Z represented
by Zo
CASE MNEMONICS
0 ≤ Z ≤ Zo Φ(Zo )
(-Zo ≤ Z ≤ 0) Φ(Zo )
Z1 ≤ Z ≤ Z2 Φ(Z2 ) – Φ(Z1)
(-Z1 ≤ Z ≤ Z2) Φ(Z1) + Φ(Z2 )
(-Z1 ≤ Z ≤ - Z2) Φ(Z1) - Φ(Z2 )
Z ≥ Zo 0.5 – Φ(Zo )
Z ≤ Zo 0.5 + Φ(Zo )
Z ≤ -Zo 0.5 – Φ(Zo )
Z ≥ -Zo 0.5 + Φ(Zo )
21. The event X has a normal distribution with mean µ = 10 and
Variance = 9. Find the probability that it will fall:
a.) between 10 and 11
b.) between 12 and 19
c.) above 13
d.) at x = 11
e.) between 8 amd 12
22. x 10 10
a .) Z 0
3
x 11 10 1
Z 0 . 33
3 3
P ( 10 X 11 ) P(0 Z 0 . 33 )
( 0 . 33 ) 0 . 1293
23. x 12 10 2
b .) Z 0 . 67
3 3
x 19 10 9
Z 3
3 3
P ( 12 X 19 ) P ( 0 . 67 Z 3)
( 3) ( 0 . 67 ) 0 . 4987 0 . 2486 0 . 251
24. x 13 10 3
c .) Z 1
3 3
P( X 13 ) P( Z 1) 0 .5 ( 1)
0 .5 0 . 3413 0 . 1587
25. d .) P ( X 11 ) 0
x 8 10 2
e .) Z 0 . 67
3 3
P( 8 X 12 ) P ( 0 . 67 Z 0 . 67 )
( 0 . 33 ) 0 . 1293
26. 2. A random variable X has a normal distribution
with mean 5 and variance 16.
a.) Find an interval (b,c) so that the probability of X
lying in the interval is 0.95.
b.) Find d so that the probability that X ≥ d is 0.05.
1
Solution: A.
P (b ≤ X ≤ c) = P (Z b ≤ Z ≤ Zc) 2
= P (-1.96 (4) ≤X -5 ≤ 1.96 (4)
= P (-7.84 + 5 ≤ X ≤ 7.84 + 5)
P (b ≤ X ≤ c) = P (-2.84 ≤ X ≤ 12.84 )
thus: b = -2.84 and c = 12.84
27. 2. A random variable X has a normal distribution
with mean 5 and variance 16.
a.) Find an interval (b,c) so that the probability of X
lying in the interval is 0.95.
b.) Find d so that the probability that X ≥ d is 0.05.
1 0.5 – 0.05 = 0.45
Solution B:
P ( X ≥ d ) = P (Z ≥ Zd) = 0.05≥ from the table
0.45 Z = 1.64
= P (X -5 ≥ 6.56)
= P (X ≥ 6.56 + 5)
P ( X ≥ d ) = P (X ≥ 11.56)
thus: d = 11.56
28. 3. A certain type of storage battery last on the
average 3.0 years, with a standard deviation σ of
0.5 year. Assuming that the battery are normally
distributed, find the probability that a given battery
will last less than 2.3 years.
Solution: X 2 .3 3 0 .7
z 1.4
0 .5 0 .5
P (X < 2.3) = P (Z < -1.4)
= 0.5 – Φ (-1.4)
= 0.5 – 0.4192
P (X < 2.3) = 0.0808