Chapter 22
What Is a Test of Significance?
Thought Question 1
Suppose 60% (0.60) of the population are in favor
of new tax legislation. A random sample of 265
people results in 175, or 0.66, who are in favor.
From the Rule for Sample Proportions, we know
the potential sample proportions in this situation
follow an approximately normal distribution, with a
mean of 0.60 and a standard deviation of 0.03.
Find the standardized score for the observed value
of 0.66, then find the probability of observing a
standardized score at least that large or larger.
Chapter 22
2
Thought Question 1: Bell-Shaped Curve
of Sample Proportions (n=265)
0.66  0.60
z
 2.0
0.03
mean = 0.60
S.D. = 0.03
2.27%
0.51
0.54
0.57
0.60
Chapter 22
0.63
0.66
0.69
3
Thought Question 2
Suppose that in the previous question we do not know
for sure that the proportion of the population who favor
the new tax legislation is 60%. Instead, this is just the
claim of a politician. From the data collected, we have
discovered that if the claim is true, then the sample
proportion observed falls at the 97.73 percentile (about
the 98th percentile) of possible sample proportions for
that sample size.
Should we believe the claim and conclude that we just
observed strange data, or should we reject the claim?
What if the result fell at the 85th percentile?
At the 99.99th percentile?
Chapter 22
4
Thought Question 2: Bell-Shaped Curve
of Sample Proportions (n=265)
99.99th
98th
85th
0.51
0.54
0.57
0.60
Chapter 22
0.63
0.66
0.69
5
Why do we need significance tests?
“Significance” in the statistical sense does not
mean “important”. It means simply “not likely
to happen just by chance”.
An outcome is very unlikely if it provides good
evidence that a claim is not true. (If we took
many samples and the claim were true, we
would rarely get a result like this).
Chapter 22
6
Is the coffee fresh?
People of taste are supposed to prefer freshbrewed coffee to the instant variety. But
perhaps many coffee drinkers just need their
caffeine fix. A skeptic claims that coffee drinkers
can’t tell the difference.
Let’s do an experiment to test this claim.
Chapter 22
7
Is the coffee fresh?
Chapter 22
8
Is the coffee fresh?
Chapter 22
9
Is the coffee fresh?
Chapter 22
10
Is the coffee fresh?
Figure 22.2 The sampling distribution of the proportion of 50 coffee drinkers who
prefer fresh-brewed coffee if the truth about all coffee drinkers is that 50% prefer
fresh coffee. The shaded area is the probability that the sample proportion is 56%
or greater.
Chapter 22
11
Is the coffee fresh?
Chapter 22
12
Is the coffee fresh?
Chapter 22
13
Chapter 22
14
Chapter 22
15
Chapter 22
16
Statistical significance
If the P-value is as small or smaller than α, we
say that the data are statistically significant at
level α.
Chapter 22
17
The Five Steps to Significance
Testing
• Determine the two hypotheses.
• Compute the sampling distribution based on the null
hypothesis.
• Collect and Summarize the data.
(Calculate the observed test statistic.)
• Determine how unlikely the test statistic is if the null
hypothesis is true.
(Calculate the P-value.)
• Make a decision/draw a conclusion.
(based on the p-value, is the result statistically significant?)
Chapter 22
18
Count Buffon’s coin
Chapter 22
19
Chapter 22
20
Figure 22.3 The sampling distribution of the proportion of heads in 4040 tosses
of a balanced coin. Count Buffon’s result, proportion 0.507 heads, is marked.
Chapter 22
21
Chapter 22
22
Figure 22.4 The P-value for testing whether Count Buffon’s coin was balanced. This is
the probability, calculated assuming a balanced coin, of a sample proportion as far or farther
from 0.5 as Buffon’s result of 0.507.
Chapter 22
23
22.13 Unemployment
The national unemployment rate in a recent
month was 4.9%. You think the rate may be
different in your city, so you plan a sample survey
that will ask the same questions as the Current
Population Survey. To see if the local rate differs
significantly from 4.9%, what hypotheses will you
test?
Chapter 22
24
Answer for 22.13
Chapter 22
25
22.16 Using the Internet
In 2006, 75.9% of first-year college students
responding to a national survey said that they
used the Internet frequently for research or
homework. A state university finds that 130 of an
SRS of 200 of its first-year students said that they
used the Internet frequently for research or
homework. We wonder if the proportion at this
university differs from the national value, 75.9%
Chapter 22
26
Chapter 22
27
Answer for 22.16
Chapter 22
28
22.26 Do chemists have more girls?
Some people think that chemists are more likely than
other parents to have female children. The Washington
State Department of Health lists the parents’
occupations on birth certificates. Between 1980 and
1990, 555 children were born to fathers who were
chemists. Of these births, 273 were girls. During this
period, 48.8% of all births in Washington State were girls.
Is there evidence that proportion of girls born to chemists
is higher than the state proportion?
Chapter 22
29
Answer for 22.26
Chapter 22
30
22.27 Speeding
It often appears that most drivers on the road are driving
faster than the posted speed limit. Situations differ, of
course, but here is one set of data. Researchers studied
the behavior of drivers on a rural interstate highway in
Maryland where the speed limit was 55 miles per hour.
They measured speed with an electronic device hidden in
the pavement and, to eliminate large trucks, considered
only vehicles less than 20 feet long. They found that 5690
out of 12931 vehicles were exceeding the speed limits. Is
this good evidence that fewer than half of all drivers are
speeding?
Chapter 22
31
Answer for 22.27
Chapter 22
32