CABT SHS Statistics & Probability - The Standard Normal Distribution

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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 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



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….)


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.


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


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 probability function of a random variable Z with a

standard normal distribution by is given by

2

212

z

y p z e



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 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


In the table, the area A gives the PROBABILITY that the value of Z lies between 0 and a constant z0; i.e.


00A P Z z



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



1Find the area that

corresponds toz = 1.

The area is A = 0.3413CABT Statistics & Probability – Grade 11 Lecture Presentation



2Find the area that

corresponds toz = 2.58.

The area is A = 0.4951




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.




3The area that corresponds toz = 2.58 is the same as the area that corresponds to z = 2.58, which isA = 0.4951.



The Normal DistributionProbabilities and Areas Under the Standard Normal Curve

5Find the area that corresponds toz = 1.15.



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



Do you have any QUESTIONs?


Finding areas of other regions


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



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



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



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)



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



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



8Find the area to the left of z = 1.5.


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



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



CASE 2: Finding areas between two values of z with the SAME SIGN


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



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


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


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


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



CASE 3: Finding areas between two values of z with DIFFERENT SIGNS


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



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



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


• Look up the z score to get the area.

• Add 0.5 to the area.


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



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



Check your understanding



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.


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.


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?

Direct table valueCase 1Case 4Case 2Case 3



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?


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


Check your understanding



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!

CABT SHS Statistics & Probability - The Standard Normal Distribution

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Transcript of CABT SHS Statistics & Probability - The Standard Normal Distribution