Z-Scores
Continued
Ch. 6 Day 2
AP Statistics
EQ: What is the normal distribution and
why do we use it?
The larger the z-score, the more unusual it is.
For symmetric data, the _______________
is usually a bit ___________ than the IQR.
When is a z-score big?
• Are appropriate for distributions that
are roughly symmetric or unimodal
• We write a normal model as
_____________.
• Z-scores use the Standard Normal
Model
Normal Models
Parameter
Statistics
Parameters vs. Statistics
A cereal manufacturer has a machine that fills the boxes.
Boxes are labeled “16 ounces”, so the company wants
to have that much cereal in each box, but since no
packaging process is perfect, there will be minor
variations. If the machine is set at exactly 16 ounces
and the Normal model applies, then about ½ the boxes
will be underweight, making consumers unhappy and
exposing the company to bad publicity and possible
lawsuits. To prevent underweight boxes, the
manufacturer has to set the mean a little higher than
16 ounces.
Based on their experience with the
packaging machine, the company believes
that the amount of cereal in the boxes fit a
Normal model with a standard deviation
of 0.2 ounces. The manufacturer decides
to set the machine to put an average of
16.3 ounces in each box. What fraction of
the boxes will be underweight?
The company’s lawyers say that 6.7% is
too high. They insist that no more
than 4% of the boxes can be
underweight. So the company needs
to set the machine to put a little more
cereal in each box. What mean setting
do they need?
The company president vetoes that plan, saying
the company should give away less free cereal,
not more. Her goal is to set the machine no
higher than 16.2 ounces and still have only 4%
underweight boxes. The only way to
accomplish this is to reduce the standard
deviation. What standard deviation must the
company achieve, and what does that mean
about the machine?
Many professional schools require
applicants to take a standardized test.
Suppose that 1000 students take such a test.
Several weeks after the test, Pete receives
his score report: he got a 63, which placed
him in the 73rd percentile. What does that
mean?
If the heights of American men follow a
normal distribution, and 99.7% have
heights between 5 feet and 7 feet, what
is your estimate of the standard
deviation of the height of American
men?
4 inches
The scores on a university examination are
normally distributed with a mean of 62 and
a standard deviation of 11. If the bottom
5% of students will fail the course, what is
the lowest mark that a student can have and
still be awarded a passing grade?
44
Let X denote the time taken for a computer link
to be made between the terminal in an
executive’s office and the computer at a remote
factory site. It is known that X has a normal
distribution with a mean of 15 seconds and a
standard deviation of 3 seconds. On 90% of the
occasions the computer link is made in less
than???
18.84 seconds
P.129 # 18, 20, 24, 36,
40, 47
Homework