School

California State University, Northridge **We aren't endorsed by this school

Course

BIO 391

Subject

Statistics

Date

Aug 10, 2023

Pages

6

Uploaded by MegaTree8411 on coursehero.com

NORMAL DISTRIBUTIONS,
SAMPLING DISTRIBUTION OF THE
MEAN, & Z-SCORES
John Pineda, MPH
HSCI 390: Biostatistics
1
Objectives
◦
Know the properties of the normal distribution.
◦
Know how to calculate z-scores and determine the
proportion of scores
◦
Greater than/equal to a given z-score
◦
Less than/equal to a given z-score
◦
Between two z-scores
◦
Understand the standard normal distribution table: looking
up z-scores based on given probability, and probability
based on the z-score.
◦
Understand the central limit theorem and how it applies
to sample distributions.
2
Normal Distribution - Properties
◦
Symmetry
: Same shape on
both halves, if you folded the
distribution in half lengthwise
(mirror image).
◦
Unimodality
: Only one mode,
and it equals the mean and
median.
◦
Shape
: It has a very distinct bell
curve, where specific parts of
the bell curve always represent
a fixed proportion of the entire
distribution.
This applies for continuous variables, not discrete!
3
Z-Score Distribution
◦
It's a version of the normal distribution (called the
standard
normal distribution) where the scores
(answers) are changed such that the
mean always = 0 & standard deviation always = 1
.
◦
Why use the z-distribution instead of std. deviation?
◦
It can tell you
exactly
how far away or how close a given score is
to the mean.
◦
You can calculate probabilities & percentiles
.
4

Z-Score Calculation
◦
A given score on a
normal distribution is
expressed as the
number of
standard
deviation units
away
from the mean.
◦
Known as Z-scores,
or standardized
scores
◦
x = Individual Score
◦
̅࠵? ࠵?࠵? ࠵?
= Mean
◦
࠵? ࠵?࠵? ࠵? =
Standard Deviation
࠵? =
!" ̅!
$
or
࠵? =
!"%
&
**Doesn't matter which formula you use, however
it will matter later for hypothesis testing!!
5
Example #1
A television cable company receives numerous phone calls throughout
the day from customers reporting service troubles and from would-be
subscribers to the cable network. Most of these callers are put "on
hold" until a company operator is free to help them. The company has
determined that the length of time a caller is on hold is normally
distributed with a mean of 3.1 minutes and a standard deviation 0.9
minutes. Company experts have decided that if as many as 5% of the
callers are put on hold for 4.8 minutes or longer, more operators should
be hired.
What percentage of the company's callers are put on hold for
more than 4.8 minutes?
Should the company hire more operators?
Show these probabilities on a sketch of the normal curve.
6
Example #1 - Continued
Given
◦
x = 4.8 minutes,
̅࠵?
= 3.1 minutes, s = 0.9 minutes
** Write out the given information, and draw the curve!!
࠵?
࠵? > 4.8
= ࠵?
࠵? >
!"!
#
= ࠵?
࠵? >
$.&"'.(
).*
= ࠵?
࠵? >
(.+
).*
= ࠵?(࠵? > 1.89)
Since it is "greater," you subtract the probability you
get from 1:
1- 0.9706 =
0.0294
[Percentage: 0.0294 x 100 = 2.94%]
7
Example #2
A public health researcher at St. Jude's Hospital wanted to
know more about a patient's hypertension problems. Suppose
a mild hypertensive is defined as a person whose diastolic
blood pressure (DBP) is between 90 and 100 mm Hg inclusive,
and the subjects are 35- to 44-year-old men whose blood
pressures are normally distributed with mean 80 and variance
144.
What is the probability that a randomly selected person
from this population will be a mild hypertensive?
8

Example #2 - Continued
࠵?
90 < ࠵? < 100
= ࠵?
࠵? −
̅࠵?
࠵?
< ࠵? <
࠵? −
̅࠵?
࠵?
= ࠵?
90 − 80
12
< ࠵? <
100 − 80
12
= ࠵?
0.83 < ࠵? < 1.67
◦
You find the probability that corresponds to both 0.83 and 1.67; 0.83 = 0.7967
and 1.67 = 0.9525
◦
Find the range based on those: 0.9525 - 0.7967 =
0.1558
[percentage: 0.1558 x 100 = 15.58%]
Given
◦
x
1
= 90 mm Hg, X
2
= 100 mm Hg,
࠵?
= 80 mm Hg, s
2
= 144 mm Hg
9
Example #3
A treatment trial is proposed to test the
efficacy of vitamin E as a preventive agent
for cancer. One problem with such a study
is how to assess compliance among
participants. A small pilot study is
undertaken to establish criteria for
compliance with the proposed study
agents. In this study 10 patients are given
400 IU/day of vitamin E and 10 patients are
given similar-sized tablets of placebo over
a 3-month period. Their serum vitamin-E
levels are measured before and after the 3-
month period and the change 3-month
baseline is shown.
What percentage of the vitamin E group
would be expected to show a change
of not more than 0.30 mg/dL?
10
Example #3 - Continued
Given
◦
x = 0.30 mg/dL,
࠵?
= 0.80 mg/dL, s = 0.48 mg/dL
࠵?
࠵? < 0.30
= ࠵?
࠵? <
࠵? −
̅࠵?
࠵?
= ࠵?
࠵? <
0.30 − 0.80
0.48
= ࠵?
࠵? <
"#.%#
#.&'
= ࠵?(࠵? < −1.04)
Since it is "Less," the probability you find is the answer:
0.1492
[Percentage: 0.1492 x 100 = 14.92%]
11
Finding Original Score
࠵? = ࠵? +
࠵?
࠵?
or
࠵? =
̅࠵? +
࠵?
࠵?
◦
x = Individual Score
◦
̅࠵? ࠵?࠵? ࠵?
= Mean
◦
࠵? ࠵?࠵? ࠵? =
Standard Deviation
◦
If you're given a percentage
or proportion, convert that to a
z-score. Look at the table!
12

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