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CABT Statistics & Probability – Grade 11 Lecture Presentation
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The Normal Distribution
A Grade 11Statistics & Probability
Lecture
3
The Normal Distribution
The Normal DistributionThe NORMAL DISTRIBUTION is a continuous, symmetric, bell-shaped distribution of a random variable. The graph of this distribution is called a NORMAL CURVE.
a normal curve
CABT Statistics & Probability – Grade 11 Lecture Presentation
The Normal Distribution
The Normal DistributionThe equation of the theoretical normal distribution is given by the formula
2
2212
x
f x e
where is the mean of the distribution, is the standard deviation, and e and are irrational constants (e = 2.718… and = 3.1415….)
CABT Statistics & Probability – Grade 11 Lecture Presentation
The Normal DistributionProperties of the Normal Distribution1. The distribution curve
is bell-shaped.2. The curve is symmetric
about its center, the mean.
3. The mean, the median, and the mode coincide at the center.
4. The width of the curve is determined by the standard deviation of the distribution.
CABT Statistics & Probability – Grade 11 Lecture Presentation
The Normal DistributionProperties of the Normal Distribution5. The tails of the curve
flatten out indefinitely along the horizontal axis, always approaching the axis but never touching it. That is, the curve is asymptotic to the base line.
6. The area under the curve is 1. Thus, it represents the probability or proportion or the percentage associated with specific sets of measurement values.
asymptotic to the x-axis
The Normal DistributionThe Distribution of Area Under the Normal Curve- a.k.a. the empirical rule or the
“68% - 95% - 99.7%” ruleThe area under the part of a normal curve that lies:• within 1 standard deviation of the
mean is approximately 0.68, or 68%;• within 2 standard deviations, about
0.95, or 95%within 3 standard deviations, about 0.997, or 99.7%CABT Statistics & Probability – Grade 11 Lecture Presentation
The Normal DistributionThe Distribution of Area Under the Normal Curve
CABT Statistics & Probability – Grade 11 Lecture Presentation
The Normal DistributionThe Standard Normal Distribution
The STANDARD NORMAL DISTRIBUTION of a random
variable is a normal distribution with mean = 0 and standard
deviation = 1.The letter Z is used to denote the standard normal random variable. The specific value z of the r.v. Z is
calledthe z-score. CABT Statistics & Probability – Grade 11 Lecture Presentation
The Normal DistributionThe Standard Normal Distribution
The probability function of a random variable Z with a
standard normal distribution by is given by
2
212
z
y p z e
CABT Statistics & Probability – Grade 11 Lecture Presentation
The Normal DistributionThe Standard Normal Distribution
The graph of the standard normal distribution
is shown below:
The Normal DistributionAreas Under the Standard Normal CurveThe Table of Areas under the Normal Curve is also known as the z-Table. The z-score is a measure of relative standing. It is calculated by subtracting the mean from the measurement X and then dividing the result by the standard deviation. The final result, the z-score, represents the distance between a given measurement X and the mean, expressed in standard deviations.
The Normal Distribution
The standard normal distribution table to be used in this course gives areas under thestandard normal curve for the variable Z ranging from 0 to a positive number z.
Areas Under the Standard Normal Curve
The Normal Distribution
In the table, the area A gives the PROBABILITY that the value of Z lies between 0 and a constant z0; i.e.
CABT Statistics & Probability – Grade 11 Lecture Presentation
00A P Z z
Areas Under the Standard Normal Curve
The Normal Distribution
Four-Step Process in Finding the Areas Under the Normal Curve Given a z-ValueStep 1. Express the given z-value into a three-
digit form. Step 2. Using the z-Table, find the first two
digits on the left column. Step 3. Match the third digit with the
appropriate column on the right. Step 4. Read the area (or probability) at the
intersection of the row and the column. This is the required area. CABT Statistics & Probability – Grade 11 Lecture Presentation
Areas Under the Standard Normal Curve
The Normal Distribution
1Find the area that
corresponds toz = 1.
The area is A = 0.3413CABT Statistics & Probability – Grade 11 Lecture Presentation
Areas Under the Standard Normal Curve
The Normal Distribution
2Find the area that
corresponds toz = 2.58.
The area is A = 0.4951
CABT Statistics & Probability – Grade 11 Lecture Presentation
Areas Under the Standard Normal Curve
The Normal Distribution
FACT: The area between 0 and a positive value z is the same as the area between z and 0.To find the area between z and 0, use the value in the table corresponding to positive z.
Both regions have the same area.
CABT Statistics & Probability – Grade 11 Lecture Presentation
Areas Under the Standard Normal Curve
The Normal Distribution
3The area that corresponds toz = 2.58 is the same as the area that corresponds to z = 2.58, which isA = 0.4951.
The area is A = 0.4951CABT Statistics & Probability – Grade 11 Lecture Presentation
Areas Under the Standard Normal Curve
The Normal DistributionProbabilities and Areas Under the Standard Normal Curve
5Find the area that corresponds toz = 1.15.
The area is A = 0.3749CABT Statistics & Probability – Grade 11 Lecture Presentation
The Normal Distribution
6What is the probability that the value of a standard normal random variable Z lies betweena. 0 and 1.28?b. 2.07 and 0?
0 1.28A = 0.3997
02.07A = 0.4808
CABT Statistics & Probability – Grade 11 Lecture Presentation
Areas Under the Standard Normal Curve
Do you have any QUESTIONs?
The Normal Distribution
Finding areas of other regions
CABT Statistics & Probability – Grade 11 Lecture Presentation
tails - right of a positive z
or left of a negative z
between two z valueswith the same sign
between two z values
with opposite signs
cumulative - left of a positive zor right of a negative z
Areas Under the Standard Normal Curve
The Normal Distribution
Finding areas of other regionsCASE REGION INVOLVED ILLUSTRATION
1tails - right of a
positive z or left of a negative z
2 between two z values with the same sign
3 between two z values with opposite signs
4cumulative - left of a
positive z or right of a negative z
Areas Under the Standard Normal Curve
The Normal DistributionProbabilities and Areas Under the Standard Normal Curve
To find the area at any tail:• Look up the z score
to get the area.• Subtract the area
from 0.5.
CASE 1: Finding areas of region in the TAILS
CABT Statistics & Probability – Grade 11 Lecture Presentation
The Normal Distribution
CASE 1: Finding areas of region in the TAILS
Probabilities and Areas Under the Standard Normal Curve
To the RIGHT of or GREATER THAN +z: 00.5A P Z z A
To the LEFT of or LESS THAN z: 0.5 A P Z z P Z z
00.5A P Z z A
Let A0 be the area between 0 and +z (value in table)
CABT Statistics & Probability – Grade 11 Lecture Presentation
The Normal Distribution
7Find the area under the standard normal curve for z greater than 2.
2
Look for the value in the table corresponding to z = 2:
0 0.4772A Subtract the table value from 0.5 to find the area.
0.5 0.4772 0.0228A CABT Statistics & Probability – Grade 11 Lecture Presentation
Areas Under the Standard Normal Curve
The Normal Distribution
2
NOTE: The area under the standard normal curve for z greater than 2 (or to the right of 2) is the same as the area for z less than 2 (or to the left of 2). The table value at z = 2 is
0 0.4772A The area corresponding to z <
2 is0.5 0.4772 0.0228A
Areas Under the Standard Normal Curve
The Normal Distribution
8Find the area to the left of z = 1.5.
Probabilities and Areas Under the Standard Normal Curve
1.5
Look for the value in the table corresponding to z = 1.5:
0 0.4332A
Subtract the table value from 0.5 to find the area.
0.5 0.4332A 0.0668
CABT Statistics & Probability – Grade 11 Lecture Presentation
The Normal DistributionProbabilities and Areas Under the Standard Normal Curve
Between two z scores on the same side of the mean (or with the SAME SIGN):• Look up both z scores
to get the areas.• Subtract the smaller
area from the larger area.
CASE 2: Finding areas between two values of z with the SAME SIGN
CABT Statistics & Probability – Grade 11 Lecture Presentation
The Normal Distribution
CASE 2: Finding areas between two values of z with the SAME SIGN
Probabilities and Areas Under the Standard Normal Curve
Let z1 = number nearer zero z2 = number farther from
zeroIf A1 = area corresponding to z1
A2 = area corresponding to z2
2 1A A A The area of the region between z1 and z2 is
CABT Statistics & Probability – Grade 11 Lecture Presentation
The Normal Distribution
9Find the area of the region betweenz = 1.23 and z = 2.57 Let z1 = 1.23 and z2 =
2.57.
1.23 2.57
From the table:For z1 : A1 = 0.3907For z2 : A2 = 0.4911
The area of the region is
2 1A A A 0.4911 0.3907 0.1104
Areas Under the Standard Normal Curve
The Normal DistributionProbabilities and Areas Under the Standard Normal CurveNote: The area of the region betweenz = 1.23 and z = 2.57 is EQUAL to the area between z = 2.57 and z = 1.23.Let z1 = 1.23 and z2 = 2.57. From the table:
-1.23-2.57
For z1 : A1 = 0.3907For z2 : A2 = 0.4911The area of the region is
2 1A A A 0.1104CABT Statistics & Probability – Grade 11 Lecture Presentation
The Normal DistributionProbabilities and Areas Under the Standard Normal Curve
10Find the area of the region betweenz = 0.96 and z = 0.36. Let z1 = 0.36 and z2 = 0.96. From the
table:
1.23 2.57
For z1 (use z = 0.36): A1 = 0.1406For z2 (use z = 0.96): A2 = 0.3395The area of the region is
2 1A A A 0.3395 0.1406 0.1909
The Normal DistributionProbabilities and Areas Under the Standard Normal Curve
Between two z scores on DIFFERENT sideS of the mean (or with DIFFERENT SIGNS):• Look up both z scores
to get the areas.• Add the two areas.
CASE 3: Finding areas between two values of z with DIFFERENT SIGNS
CABT Statistics & Probability – Grade 11 Lecture Presentation
The Normal Distribution
CASE 3: Finding areas between two values of z with DIFFERENT SIGNS
CABT Statistics & Probability – Grade 11 Lecture Presentation
Let z1 = negative z valuez2 = positive z value
If A1 = area corresponding to z1
A2 = area corresponding to z2
1 2A A A The area of the region between z1 and z2 is
Areas Under the Standard Normal Curve
The Normal DistributionProbabilities and Areas Under the Standard Normal Curve
10Find the area of the region betweenz = 2.46 and z = 1.55. Let z1 = 2.46 and z2 = 1.55. From the
table:
-2.46 1.55
For z1 (use z = 2.46): A1 = 0.4931For z2 (use z = 1.55): A2 = 0.4394The area of the region is
1 2A A A 0.4931 0.4394 0.9325
CABT Statistics & Probability – Grade 11 Lecture Presentation
The Normal DistributionProbabilities and Areas Under the Standard Normal Curve
To find the area to the left of any positive z score or to the right of a negative z score:
CASE 4: Finding areas of regions to the LEFT of a positive z or to the RIGHT of a negative z
CABT Statistics & Probability – Grade 11 Lecture Presentation
• Look up the z score to get the area.
• Add 0.5 to the area.
The Normal DistributionProbabilities and Areas Under the Standard Normal Curve
11Find the area of the region to the left of z = 2.37.
From the table, the area corresponding to z = 2.37 isThe area of the region is
00.5A A 0.5 0.4911 0.9911
0 0.4911A
2.37
CABT Statistics & Probability – Grade 11 Lecture Presentation
The Normal Distribution
12Find the area of the region to the right of z = 2.37.
From the table, the area corresponding to z = 2.37 is the same for z = 2.37:The area of the region is
00.5A A 0.5 0.4911 0.9911
0 0.4911A
2.37
CABT Statistics & Probability – Grade 11 Lecture Presentation
Areas Under the Standard Normal Curve
Check your understanding
The Normal Distribution
CABT Statistics & Probability – Grade 11 Lecture Presentation
Determine the area of the indicated region under the standard normal curve.1. to the left of z = 1.312. to the right of z = 1.923. to the left of z = 2 4. between z = 1.23 and z = 1.95. between z = 1.98 and z =
1.466. between z = 3 and z = 1.5
0.90490.97260.02280.08060.04820.9319
The Normal DistributionProbabilities and Areas Under the Standard Normal CurveRecall that the area under the graph of a continuous probability function corresponds to the value of a probability in an interval.
CABT Statistics & Probability – Grade 11 Lecture Presentation
PROBABILITY CORRESPONDING AREAP(Z > a) to the right of aP(Z < a) to the left of a
P( a < Z < b) between a and bNOTE: The area won’t change even if “>” and “<” are replaced by “” and “”, respectively.
The Normal Distribution
14If Z is a standard normal random variable, what is the probability that
Probabilities and Areas Under the Standard Normal Curve
a. 0 < Z < 0.33?
b. Z > 2? c. Z < 1.67?
d. 1.03 < Z < 0.99?
e. 2.99 < Z < 0.2?
Direct table valueCase 1Case 4Case 2Case 3
CABT Statistics & Probability – Grade 11 Lecture Presentation
The Normal Distribution
14If Z is a standard normal random variable, what is the probability that
a. 0 < Z < 0.33?
b. Z > 2? c. Z < 1.67?
d. 1.03 < Z < 0.99?
e. 2.99 < Z < 0.2?
Probabilities and Areas Under the Standard Normal Curve
A = 0.1293A = 0. 5 – 0.4772
= 0.0228A = 0. 5 +
0.4525= 0.9525A = 0. 3485 –
0.3389= 0.0096A = 0. 4986 +
0.0793= 0.5779
CABT Statistics & Probability – Grade 11 Lecture Presentation
Check your understanding
The Normal Distribution
CABT Statistics & Probability – Grade 11 Lecture Presentation
Given the random variable Z with a standard normal distribution, determine the following probabilities:1. P(– 0.75 < Z <
0)2. P(Z > 1.92)3. P(Z < 1.11) 4. P(0.33 < Z <
0.99)5. P(Z > 0.2)
0.27340.5 + 0.4726 = 0.97260.5 + 0.3665 = 0.86650.3389 0.1293 = 0.20960.5 0.0793 = 0.4207
Do you have any QUESTIONs?
Summingit up!
Thank you!
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