An important measure of the performance of a locomotive is its "adhesion," which is the
locomotive's pulling force as a multiple of its weight. The adhesion of one 4400-horsepower
diesel locomotive model varies in actual use according to a Normal distribution with mean μ
= 0.34 and standard deviation σ = 0.049
What proportion of adhesions (± 0.001) measured in use are higher than 0.47?
Z = (0.47-0.34) / 0.049 = 2.653
Area to the right of z = 2.65 is 0.0040
What proportion of adhesions (± 0.001) are between 0.47 and 0.49?
The new z-score is (0.49-0.34)/0.049 = 3.06
To find the area between the two z-scores, we find the difference in the areas to the left of
each.
Area left of 3.06 = 0.9989
Area left of 2.65 = 0.9960
Area between = 0.9989-0.9960 = 0.0029
What do you do if your z-value is bigger than the table values?
Use the last value on the table. We know that the probability to the right of z = 3.939 is
smaller than the area to the right for 3.49 (or whatever the last value on the table is). For a zscore of 6.2978, you can safely put down 0 or 1 (whichever side is appropriate) and be
correct to within rounding.
CHAPTER 6
BPS - 5TH ED.
1
CHAPTER 6
Two-Way Tables
CHAPTER 6
BPS - 5TH ED.
2
CATEGORICAL VARIABLES
• In this chapter we will study the
relationship between two categorical
variables (variables whose values fall in groups
or categories).
• To analyze categorical data, use the counts
or percents of individuals that fall into
various categories.
CHAPTER 6
BPS - 5TH ED.
3
TWO-WAY TABLE
• When there are two categorical variables, the
data are summarized in a two-way table
• each row in the table represents a value of the row variable
• each column of the table represents a value of the column
variable
• The number of observations falling into each
combination of categories is entered into each cell of
the table
CHAPTER 6
BPS - 5TH ED.
4
MARGINAL DISTRIBUTIONS
• A distribution for a categorical variable tells how
often each outcome occurred
• totaling the values in each row of the table gives the
marginal distribution of the row variable (totals
are written in the right margin)
• totaling the values in each column of the table gives
the marginal distribution of the column variable
(totals are written in the bottom margin)
CHAPTER 6
BPS - 5TH ED.
5
MARGINAL DISTRIBUTIONS
• It is usually more informative to display each marginal distribution in terms
of percents rather than counts
•
each marginal total is divided by the table total to give the percents
• A bar graph could be used to graphically display marginal distributions for
categorical variables
CHAPTER 6
BPS - 5TH ED.
6
CASE STUDY
Age and Education
(Statistical Abstract of the United States, 2001)
Data from the U.S. Census Bureau
for the year 2000 on the level of
education reached by Americans
of different ages.
CHAPTER 6
BPS - 5TH ED.
7
CASE STUDY
Age and Education
Variables
Marginal distributions
CHAPTER 6
BPS - 5TH ED.
8
CASE STUDY
Age and Education
Variables
15.9%
33.1%
25.4%
25.6%
21.6%
46.5%
32.0%
Marginal distributions
CHAPTER 6
BPS - 5TH ED.
9
CASE STUDY
Age and Education
Marginal Distribution
for Education Level
CHAPTER 6
Not HS grad
15.9%
HS grad
33.1%
College 1-3 yrs
25.4%
College ≥4 yrs
25.6%
BPS - 5TH ED.
10
CONDITIONAL DISTRIBUTIONS
• Relationships between categorical variables are described by
calculating appropriate percents from the counts given in the
table
• prevents misleading comparisons due to unequal sample
sizes for different groups
CHAPTER 6
BPS - 5TH ED.
11
CASE STUDY
Age and Education
Compare the 25-34
age group to the 35-54
age group in terms of
success in completing
at least 4 years of
college:
Data are in thousands, so we have that 11,071,000 persons in the 25-34 age
group have completed at least 4 years of college, compared to 23,160,000
persons in the 35-54 age group.
The groups appear greatly different, but look at the group totals.
CHAPTER 6
BPS - 5TH ED.
12
CASE STUDY
Age and Education
Compare the 25-34
age group to the 35-54
age group in terms of
success in completing
at least 4 years of
college:
Change the counts to percents:
Now, with a fairer comparison
using percents, the groups appear
very similar.
CHAPTER 6
11,071
= .293 (29.3%) for 25 - 34 age group
37,786
23,160
= .284 (28.4%) for 35 - 54 age group
81,435
BPS - 5TH ED.
13
CASE STUDY
Age and Education
If we compute the percent completing at least four
years of college for all of the age groups, this would
give us the conditional distribution of age, given
that the education level is “completed at least 4
years of college”:
CHAPTER 6
Age:
25-34
35-54
55 and over
Percent with
≥ 4 yrs college:
29.3%
28.4%
18.9%
BPS - 5TH ED.
14