CABT SHS Statistics & Probability - The z-scores and Problems involving Normal Distributions

CABT Statistics & Probability – Grade 11 Lecture Presentation

The Normal Distribution and
Standard Scores
A Grade 11
Statistics & Probability
Lecture
3

The Normal Distribution and Standard Scores
Normally Distributed Random
Variables
A normally distributed random variable X (or X has
a normal distribution) has a continuous, symmetric,
bell-shaped distribution symmetric about the mean  of
X.
If X is normally-distributed with
mean  and standard deviation
, we write
X ~ N(, )
The distance of the values of X
from the mean is expressed in
terms of the standard deviation
.

Properties of the Normal
Distribution
1. 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

Properties of the Normal
Distribution
5. 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
asymptotic to the x-axis

The Distribution of Area
Under the Normal Curve
- a.k.a. the empirical rule or the
“68% - 95% - 99.7%” rule
The 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,

The Distribution of Area
Under the Normal Curve

The Standard Score or z-Scores
The STANDARD NORMAL random
variable Z has a normal distribution with
mean  = 0 and standard deviation  =
1.If Z is normally-distributed with mean 0 and
standard deviation 1, we write
Z ~ N(0, 1)

QUESTION:
What comes to mind when
you hear the words STANDARD
and STANDARDIZED?

What is STANDARD?
STANDARD n. a level of quality or
attainment (high standard of service); an
idea or thing used as a measure, norm,
or model in comparative evaluations (the
wages are low by today's standards)
STANDARD adj. used or accepted as
normal or average (standard score)

The Standard Score or z-Scores
The standard score or z-score is a
measure of relative standing. It
represents the distance between a given
measurement X and the mean,
expressed in standard deviations.

Converting Normally-Distributed
Scores to z-Scores
If the distribution of a random variable X is
normal with mean  and standard deviation ,
the values x of X can be STANDARDIZED or
can be converted to z-scores using this
formula:
x
z




This formula transforms the values of the
variable x into standard units or z values.

Scores to z-Scores
Relationship between x and
z:
For any population, the mean and
the standard deviation are
fixed. Thus, the z formula
matches the z-values one-to-
one with the x values (raw
scores). That is, for every x value
there corresponds a z-value
and for each z-value there is
x
z





Scores to z-Scores
Note on notation:
If we’re talking about an entire
POPULATION that is normally
distributed, we use  for the mean
and  for the standard deviation.
If we’re talking about a SAMPLE of
a population that is normally
distributed, we use for the mean
and s for standard deviation.
x
x

The z-Scores
Scores to z-Scores
Note on notation:
Z-score for
a population
Z-score for
a sample
x
z






x x
z
s
x

Scores to z-Scores
Why standardize scores?
The major purpose of standard scores is to
place scores for any individual on any variable
having any mean and standard deviation on
the same standard scale so that
comparisons can be made. Without some
standard scale, comparisons across individuals
and/or across variables would be difficult to
make.
(http://faculty.virginia.edu/PullenLab/WJIIIDRBModule/WJIIIDRBModule7.html)

Scores to z-Scores
Application: the NCAE
The scores in the NCAE are
reported in Standard Scores
and Percentile Ranks.
Standard Score – where the
mean is 500 and the standard
deviation is 100. The highest
scores are in the 700’s; the
lowest scores are in the 300’s.
http://www.teacherph.com/national-career-assessment-examination-ncae-overview/

Scores to z-Scores
Percentile Rank – shows
the test taker’s position
among all the examinees.
If an examinee scores at
percentile rank 99+, it
means that he scored
above the other 99 percent
of the examinees.

Scores to z-Scores
The normal curve
on the right
represents a
sample plot of a
percentile rank
(PR) in the NCAE.

Scores to z-Scores
Given a normally distributed
population with mean 75 and standard
deviation 4, find the corresponding
standard score of the following:
a. 69 b. 85
   69, 75, 4x
x
z






69 75
4
 1.5z
   85, 75, 4x
x
z






85 75
4
 2.5z

Scores to z-Scores
In a given normal distribution, the
sample mean is 20.5 and the sample
standard deviation is 5.4. Find the
corresponding standard score of the
following:
a. 18.7 b. 21.3
  18.7, 20.5, 5.4x x s


x x
z
s


18.7 20.5
5.4
 0.33z
  21.3, 20.5, 5.4x x s


x x
z
s


21.3 20.5
5.4
 0.15z

Scores to z-Scores
The average score in a Statistics and
Probability Test is 80 with standard
deviation 10. What is the standard
score of the following students?
JM – 97 Eljhie – 86
   1 97, 80, 10x



 1
1
x
z


97 80
10
1 1.7z
   2 86, 80, 10x



 2
2
x
z


86 80
10
2 0.6z

Scores to z-Scores
A sample of bags of potato chips in a
factory has an average net weight of 24.7
grams with standard deviation of 0.35
grams.
What is the standard score for two samples
A and B with the following weights?
A – 22.6 grams B – 25.9 grams
  22.6, 24.7, 1.02Ax x s

 A
A
x x
z
s


22.6 24.7
1.02
 2.06Az
  25.9, 24.7, 1.02Bx x s

 B
B
x x
z
s


25.9 24.7
1.02
1.18Bz

Scores to z-Scores
Scores in the Precalculus Quarter
Exam has mean 76 and standard
deviation 4, while the Gen. Math exam
has mean 75 and standard deviation
5.
If Yayie scored 90 in Precalculus and 91 in
Gen. Math, in which subject is her standing
better? Assume that the scores in both
exams are normally distributed.

Scores to z-Scores
Scores in the Precalculus Quarter Exam has mean 76 and
standard deviation 4, while the Gen. Math exam has mean
75 and standard deviation 5. If Yayie scored 90 in
Precalculus and 91 in Gen. Math, in which subject is her
standing better? Assume that the scores in both exams are
normally distributed.
   90, 76, 4x




x
z


90 76
4
 3.5z
For Precalculus: For Gen. Math
   91, 75, 5x




x
z


91 75
5
 3.2z
Yayie has a better standing in Precalculus than in Gen.
Math.

Scores to z-Scores
If the standard score z is given, the original or
raw score x can be obtained by solving
x
z




for x, yielding the equation
  x z
For a sampling distribution,
 x x zs

Scores to z-Scores
Find the raw score of a standard
score of z = 1.2 in a normally-
distributed population with mean 30
and standard deviation 4.
    1.2, 30, 4z
  x z     30 1.2 4
 25.2x

Probabilities and
Standardized Scores
Recall that areas under normal curves
correspond to probabilities or percent of
scores.PROBABILITY CORRESPONDING AREA
P(X > a) to the right of a
P(X < a) to the left of a
P( a < X < b) between a and b
NOTE: The area won’t change even if “>” and
“<” are replaced by “” and “”, respectively.

Probabilities and
Standardized Scores
To determine the probabilities or percent that
values of X in a normally-distributed population
fall on a particular interval:
STEP 1 – Convert the x scores to z-scores.
STEP 2 – Draw the region defined by the z
scores.
STEP 3 – Locate the areas corresponding to
the z scores using the table.
STEP 4 – Find the area of the indicated region.

Probabilities and
Standardized Scores
In the Oral Comm test given by Sir
Aldous, the mean score is 60 with
standard deviation 6.
Assuming that the scores are normally-
distributed, what is the probability that a
randomly-selected student has a score
a. between 60 and 65? P(60 < X < 65)
b. greater than 65? P(X > 65)
c. between 50 and 65? P(50 < X < 65)

Probabilities and
Standardized Scores
a. Convert x = 60 and x = 65 to z-
scores   1 60, 60, 6x



 1
1
x
z


60 60
6
1 0z
   2 65, 60, 6x



 2
2
x
z


65 60
6
2 0.83z
0.83
The area can be directly read
in the table: 0.2967. Hence,
P(0 < X < 65) = 0.2967

Probabilities and
Standardized Scores
b. Convert x = 65 to z-score
   65, 60, 6x




x
z


65 60
6
 0.83
The area corresponding to z = 0.83 is 0.2967.
P(X < 65) = P(Z < 0.83)
= 0.5 + 0.2967
= 0.79670.83

Probabilities and
Standardized Scores
In a university, the average number of
years a person takes to complete a
master’s degree program is 3 and the
standard deviation is 4 months.
Assume the variable is normally distributed. If an
individual enrolls in the program, find the probability
that it will take
a. more than 4 years to complete the program.
b. less than 3 years to complete the program.
c. between 3.8 and 4.5 years to complete the program
d. between 2.5 and 3.1 years to complete the

Probabilities and
Standardized Scores
The mean age of a population of
10,000 is 56 years old, with standard
deviation of 5 years. If the ages are
normally-distributed, how many
a. have ages between 40 and 45 years?
b. are senior citizens?
c. are teenagers?
d. are ages 56 years old and below?

Standardized Scores and
Percentiles
Recall that
• a percentile is a measure of relative
standing or position. It is a descriptive
measure of the relationship of a
measurement to the rest of the data.
• a score x is in the kth percentile rank
(Pk) if k% of the scores are equal or
below x.

Percentiles
Percentile and z-scores
 A probability value corresponds to an area under
the normal curve.
 In the Table of Areas Under the Normal Curve, the
numbers in the extreme left and across the top are
z-scores, which are the distances along the
horizontal scale. The numbers in the body of the
table are areas or probabilities.
 The z-scores to the left of the mean are negative
values.

Percentiles
Percentile and z-scores
 Recall that a probability corresponds to a
percent or proportion; e.g. a probability of
0.4922 is the same as a probability of
49.22%
 Since x = Pk represents scores LESS that
or equal to x, the region representing a
percentile rank in a normal distribution is
the same as P(X < x).

In an examination, the scores are
normally distributed with mean 20
and the standard deviation 2.
Determine the percentile ranks of
the following scores:
a. 18 (15.87% - 16th percentile)
b. 25 (99.38% - 99th percentile)
Percentiles

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