Review: Week 2
Read: Chapters 1-7 in the text book
Review: Linear transformations and the z-score
A linear transformation changes the variable x into a new variable, z, using the equation
z = a + bx .
Linear transformations do not change the shape of a distribution. However, a linear
transformation multiplies a measure of spread by |b| and changes a measure of center m into
bm+a.
In statistics we are often interested in measuring how many standard deviations away from the
mean an observation lays. We can obtain this value by using a linear transformation. Suppose y is
an observation on a variable with mean y and standard deviation s.
The standardized value, or z-score, is z =
y−y
.
s
(Here a = −
y
1
and b = )
s
s
The z-score measures the distance of each data value from the mean in standard deviations. Using
standardized values makes it possible to compare values that are measured on different scales,
units or populations.
Standardizing the data does not change the shape of the distribution (for b>0). However the mean
of the resulting data is always 0 and the standard deviation is always 1.
Exercises:
Exercise 1: Linear transformations
(a) A student received x points on an exam. The instructor gave the student 5 extra credit
points. What is the student’s new score after receiving extra credit?
(b) Change a weight x in lbs into the metric system value y in kg (1 kg = 2.2 lbs). How much
is 80 lbs in kg?
Exercise 2: Z-scores
In the United States, men’s heights have mean 69.1 inches and standard deviation 2.9 inches,
while female’s heights have mean 63.7 inches and standard deviation 2.7 inches.
(a) Calculate the standardized height (z-score) of a U.S. male who is 70 inches tall.
(b) Calculate the z-score of a U.S. male who is 64 inches tall.
(c) Calculate the z-score of a U.S. female who is 64 inches tall.
(d) Calculate your standardized height.
Exercise 3: Linear transformations
A student can score anywhere between 0 and 40 on an exam. The instructor wants to rescale the
exam so that the scores lay between 100 and 200.
(a) How would the instructor convert the scores?
(b) Suppose a student get a 36 on the exam. What is the equivalent score using the new
scale?
(c) Suppose the mean on the exam was 30 and the standard deviation was 4. What is the
corresponding mean and standard deviation for the rescaled data?
Review: The Normal Model
A mathematical model is an idealized description of a real world phenomenon.
We are interested in constructing a description of the distribution that explains the overall pattern
but ignores irregularities that are present in the actual data. The area under the curve and between
any ranges of values represents the relative frequency of all observations that fall within that
range.
If the distribution is bell-shaped, single peaked and roughly symmetric it is appropriate to use a
normal model.
Numbers that describe a model are called parameters. We use μ and σ to describe the mean and
standard deviation of the model. A normal model with mean μ and standard deviation σ is often
written as N(μ, σ).
Suppose we model the data using a N(μ, σ) model. If we standardize the data using μ and σ, then
the z-score
x−μ
z=
σ
follows the N(0,1) model, or the standard normal distribution.
The 68-95-99.7 Rule:
68% of the data lies within ±1 σ from μ.
95% of the data lies within ±2 σ from μ.
99.7% of the data lies within ±3 σ from μ.
Table z on pages A-56-57 can be used to find the area under the curve for more general ranges.
A normal probability plot can show us if a Normal model is appropriate. If the plot shows a
straight line, then the data follows a normal model.
Exercises:
Exercise 1: Normal model - The scores on a Statistics exam follow a normal model with mean 70
and standard deviation 13. Use the 68-95-99.7 rule to answer the following questions.
(a) Within which range of scores did 95% of the class fall?
(b) What score did a student need to get to be in the top 2.5% in the class?
Exercise 2: Normal model - In the United States, men’s heights follow a normal model with
mean 69.1 inches and standard deviation 2.9 inches.
(a) What percentage of U.S. males are shorter than 70 inches?
(b) What percentage of U.S. males are taller than 70 inches?
(c) What percentage of U.S. males are shorter than 67 inches?
(d) What percentage of U.S. males are between 67 and 70 inches tall?
(e) Suppose that men whose heights are in the top 15% are considered tall. What is the
shortest a male can be and still be considered tall?
Normal Model Drill
Suppose the results of an exam, taken by a class with a large number of students, follow a normal
model with mean 60 and standard deviation 8.
Answer the following questions regarding the exam:
•
What proportion of the class had scores between 52 and 68?
•
What proportion of the class had scores between 36 and 84?
•
What proportion of the class had scores below 36?
•
What proportion of the class had scores above 76?
•
What score do you need to ensure that you are in the top 2.5% of the class?
•
What score do you need to ensure that you are not in the bottom 2.5% of the class?
•
What proportion of the class had scores between 50 and 70?
•
What proportion of the class had scores between 70 and 80?
•
Suppose you scored a 72 on the test. How many percent of the students had better scores?
How many percent had worse scores?
•
Suppose you scored a 25 on the test. How many percent of the students had better scores?
How many percent had worse scores?
•
What score do you need to ensure that you are in the top 10% of the class?
•
What score do you need to ensure that you are in the top 20% of the class?