Chapter 21
What Is a Confidence Interval?
Chapter 15
1
Recall from previous chapters:
Parameter
fixed, unknown number that describes the population
Statistic
known value calculated from a sample
a statistic is used to estimate a parameter
Sampling Variability
different samples from the same population may yield
different values of the sample statistic
estimates from samples will be closer to the true values
in the population if the samples are larger.
Chapter 21
2
The Rule for Sample Proportions
Chapter 21
3
Rule Conditions and Illustration
For rule to be valid, must have
1. Random sample
2. ‘Large’ sample size
Chapter 21
4
Case Study: Fingerprints
A Question
Assume that the proportion of all men
who have leftward asymmetry is 15%.
Is it unusual to observe a sample
of 66 men with a sample
proportion (p̂) of 30% if the true
population proportion (p) is 15%?
Chapter 21
5
Case Study: Fingerprints
Answer to Question
Where should about 95% of the sample
proportions lie?
mean plus or minus two standard deviations
0.15  2(0.044) = 0.062
0.15 + 2(0.044) = 0.238
95% should fall between 0.062 & 0.238
Chapter 21
6
Formula for a 95% Confidence
Interval
• sample proportion plus or minus two
standard deviations of
p(1 p)
the sample proportion: p  2
n

Chapter 21
7
Formula for a 95% Confidence
Interval
pˆ (1  pˆ )
pˆ  2
n
standard error (estimated standard deviation of p̂ )
Chapter 21
8
Margin of Error
pˆ (1  pˆ )
2
(plus or minus part of C.I.)
n

2
0.5(1  0.5)
n
Chapter 21

1
n
9
Formula for a C-level (%) Confidence
Interval for the Population Proportion
pˆ (1  pˆ )
pˆ  z *
n
where z* is the critical value of the standard
normal distribution for confidence level C
Chapter 21
10
Common Values of z*
Confidence Level
C
Critical Value
z*
50%
0.67
60%
0.84
68%
1
70%
1.04
80%
1.28
90%
1.64
95%
1.96 (or 2)
99%
2.58
99.7%
3
99.9%
3.29
Chapter 21
11
Gambling on Sports
A December 2007 Gallup Poll con-sisting
of a random sample of 1027 adult Americans
found that 17% had gambled on sports in the
last 12 months. Find a 95% confidence interv
for the proportion of all adult Americans who
gambled on sports in this time period. How
would you interpret this interval?
Chapter 21
12
A 99% confidence interval
The BRFSS random sample of 2166
college
graduates in California in 2006 found that
279 had engaged in binge drinking in the
past
year. We want a 99% confidence interval
for
the proportion p of all college graduates in
California who engaged in binge drinking in
the past year.
Chapter 15
13
Our statistical software has a “random
number generator” that is supposed to
produce numbers scattered at random
between 0 to 1. If this is true, the numbers
generated come from a population with μ =
0.5. A command to generate 100 random
numbers gives outcomes with mean x =
0.536 and s = 0.312. Give a 90% confidence
interval for the mean of all numbers produced
by the software.
Chapter 15
14
We would like to estimate the mean GPA of
the population of all WSU undergraduates.
We would like to compute a 90 percent
confidence interval for this mean, and we
would like our interval to have margin of
error no greater than 0.1. The Registrar's
Office says that a reliable guess of the
population standard deviation is 0.5. What is
the smallest number of students we should
recruit to achieve our goals?
Chapter 15
15