Download Chapter 8 - Confidence Intervals

Chapter
Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall
8-1
8
Confidence Intervals
Confidence
Intervals
Population
Mean
Population
Proportion
Section 8.4
σ Known
σ Unknown
Section 8.2
Section 8.3
Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall
8-2
8.2 Calculating Confidence Intervals for
the Mean when the Standard Deviation
(σ) of a Population is Known
• A confidence interval for the mean
– interval (or range) around the sample mean ( x ) within
which true (population) mean (µ) is expected to be
– µ = 362.3 ± 1.96 * 15 =
x  zα/ 2σ x
• A confidence level (dependent on Z(α))
– probability that the interval estimate will include the
population parameter of interest (1-α)
– Here α = .05 => 95% confidence
Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall
8-3
Confidence Intervals for the Mean,
σ Known
• Assumptions for section 8.2:
– The sample size is at least 30 (n ≥ 30)
– The population standard deviation (σ) is known
• Recall from Chapter 7:
– Formula for the Standard Error of the Mean
σ
σx 
n
where
σ = Population standard deviation
n = Sample size
Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall
8-4
Confidence Intervals for the Mean,
σ Known
• Formulas for the Confidence Interval for the
Mean (σ Known)
UCLx  x  zα/ 2σ x
LCLx  x  zα/ 2σ x
where
x  The sample mean
zα/ 2  The critical z - score
σ x  The standard error of the mean
Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall
8-5
Confidence Intervals for the Mean,
σ Known
•
z / 2 is called the critical z-score
• The variable  is known as the significance level
• Example: if  = .10, then z / 2  z0.05 = 1.645 is the
value that encloses 90% of the area under the normal
distribution and leaves 5% in each tail
– The total area to the left of the
right-hand boundary is
0.90 + 0.05 = 0.95
α/2 = 0.05
– The total area to the left
of the left-hand
-1.645
boundary is 0.05
0.90
0.95
0.05
Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall
8-6
0
α/2 = 0.05
1.645
z
Calculating the Margin of Error
• The Margin of Error MEx is the amount added and
subtracted to the point estimate to form the confidence
interval
UCLx , LCLx  x  zα/ 2 σ x
 x  ME x
Margin of Error
Lower
Confidence
Limit
ME x  zα/ 2σ x
Margin of Error
Point Estimate
Width of confidence interval
Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall
8-7
Upper
Confidence
Limit
Calculating the Margin of Error
• Increasing the sample size while keeping the confidence
level constant will reduce the margin of error, resulting in
a narrower (more precise) confidence interval
n 
σ x   ME x 
Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall
8-8
Interpreting a Confidence Interval
• We are 90% confident that the true mean is between
137.10 and 153.90
– Population mean (µ) may/may not be in this interval
90% of the sample means drawn from samples of
this population will produce confidence intervals that
include that population’s mean
• An incorrect interpretation is that there is 90%
probability that this interval contains the true population
mean
– (This interval either does or does not contain the true
mean, there is no probability for a single interval)
Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall
8-9
Interpreting a Confidence Interval
/2
Each interval
extends from
x  zα/ 2σ x
to
x  zα/ 2σ x
1 
/2
x
μ  μx
x1
x2
For 90% confidence,
90% of intervals
constructed contain
μ ; 10% do not
But x varies from
sample to sample
Confidence Intervals
Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall
8-10
Changing Confidence Levels
• The significance level,  ,
– Probability any given confidence interval will not
contain the true population mean
• The confidence level of an interval is the
complement to the significance level, 1 ─ 
– i.e., a 100(1 –  )% confidence interval has a
significance level equal to 
• The confidence interval gets wider if the
confidence level increases (as Z(α) increases)
Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall
8-11
Changing Confidence Levels
• Consider a 95% confidence interval:
1   0.95
/2  0.025
/2  0.025
z/2  -1.96
0
z/2  -1.96
0.975
•
z  1.96 encloses 95% of the area under the
curve, with 2.5% in each tail
Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall
8-12
Changing Confidence Levels
• z-scores for the most commonly used
confidence levels are shown in this table:
Confidence
Level:
Significance
Level:
100(1  )%
100( )%
Critical
z-score:
z / 2
80%
20%
z0.10 = ±1.28
90%
10%
z0.05 = ±1.645
95%
5%
z0.025 = ±1.96
98%
2%
z0.01 = ±2.33
99%
1%
z0.005 = ±2.575
Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall
8-13
Using Excel to Determine Confidence
Intervals for the Mean (σ Known)
• Excel’s CONFIDENCE function calculates the
margin of error for confidence intervals
• The CONFIDENCE function has the following
characteristics:
=CONFIDENCE (alpha, standard_dev, size)
where:
alpha = The significance level of the confidence interval
standard_dev = The standard deviation of the population
size = Sample size
Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall
8-14
Confidence Intervals for the Mean with
Small Samples when σ is Known
• When the sample size is less than 30 and sigma
is known, the population must be normally
distributed to calculate a confidence interval
– With n < 30 the Central Limit Theorem cannot be
applied, so we can’t say the sampling distribution will
be approximately normal…
– …but the sampling distribution is always normal
(regardless of sample size) if the population is
normally distributed
Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall
8-15
8.3 Calculating Confidence Intervals when
the Population Standard Deviation (σ) is
Unknown
• Population standard deviation is unknown,
– Use s, the sample standard deviation, in its place
• calculate the standard error (of the mean)
• Formula for the Sample Standard Deviation (recall from Chapter 3):
n
s
2
(x

x
)
 i
i 1
n 1
where:
x = The sample mean
n = The sample size (number of data values)
(xi – x ) = The difference between each data value
and the sample mean
Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall
8-16
Confidence Intervals
Confidence
Intervals
Population
Mean
Population
Proportion
Section 8.4
σ Known
σ Unknown
Section 8.2
Section 8.3
Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall
8-17
Using the Student’s t-distribution
• Formula for the Approximate Standard Error of
the Mean
s
σˆ x 
n
• The Student’s t-distribution
– used instead of normal probability distribution when
the sample standard deviation, s, is used in place of
the population standard deviation, σ
Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall
8-18
Using the Student’s t-distribution
• Formulas for the Confidence Interval for the
Mean (σ Unknown)
UCLx  x  tα/ 2 σˆ x
LCLx  x  tα/ 2 σˆ x
where:
x = The sample mean
tα/ 2 = The critical t-score
σˆ x = The approximate standard error of the mean
Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall
8-19
Using the Student’s t-distribution
Normal
distribution
t (df = 13)
t-distributions are bell-shaped
and symmetric, but have
‘fatter’ tails than the normal
t (df = 5)
t
Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall
8-20
Using the Student’s t-distribution
• The t-distribution is a continuous probability
distribution with the following properties:
– It is bell-shaped and symmetrical around the mean
– The shape of the curve depends on the degrees of
freedom (df), df = n – 1
– The area under the curve is equal to 1.0
– The t-distribution is flatter and wider than the normal
distribution
– The critical score for the t-distribution is greater than
the critical z-score for the same confidence level
Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall
8-21
Using the Student’s t-distribution
• Comparing t-scores and z-scores:
Confidence
t
Level
(10 df )
t
(20 df )
t
(30 df )
z
.80
1.372
1.325
1.310
1.28
.90
1.812
1.725
1.697
1.645
.95
2.228
2.086
2.042
1.96
.99
3.169
2.845
2.750
2.575
Note: t
z as n increases
Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall
8-22
Using the Student’s t-distribution
• The t-distribution is actually a family of
distributions. As the number of degrees of
freedom increases, the shape of the t-distribution
becomes similar to the normal distribution
– With more than 100 degrees of freedom (a sample size
of more than 100), the two distributions are practically
identical
Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall
8-23
Using Excel and PHStat2 to Determine
Confidence Intervals for the Mean (σ Unknown)
• The critical t-score can be found with Excel’s
TINV function, which has the following
characteristics:
=TINV (alpha, degrees of freedom)
where:
alpha () = The significance level of the confidence interval
degrees of freedom = n - 1
n = Sample size
Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall
8-24
Confidence Intervals
Confidence
Intervals
Population
Mean
Population
Proportion
Section 8.4
σ Known
σ Unknown
Section 8.2
Section 8.3
Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall
8-25
8.4 Calculating Confidence
Intervals for Proportions
• Proportion data follow the binomial distribution,
which can be approximated by the normal
distribution under the following conditions:
nπ ≥ 5
and
n(1 – π) ≥ 5
where:
π = The probability of a success in the population
n = The sample size
Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall
8-26
Calculating Confidence Intervals
for Proportions
• The confidence interval for the proportion is
an interval estimate around a sample proportion
that provides us with a range of where the true
population proportion lies
Formula for the Sample
Proportion:
Formula for the Standard Error
of the Proportion:
x
p
n
σp 
 (1   )
n
where:
π = The population proportion
x = The number of observations of interest in the sample (successes)
n = The sample size
Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall
8-27
Calculating Confidence Intervals
for Proportions
• Since the population proportion π is unknown, it
is estimated using the sample proportion, p
• Formula for the Approximate Standard Error of
the Proportion:
σˆ p 
p(1  p)
n
where:
p = The sample proportion
n = The sample size
Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall
8-28
Calculating Confidence Intervals
for Proportions
• Formulas for the Confidence Interval for a
Proportion:
UCLp  p  zα/ 2 σˆ p
LCL p  p  zα/ 2 σˆ p
where:
p = The sample proportion
z α/ 2 = The critical z-score
σˆ p = The approximate standard error of the proportion
Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall
8-29
Calculating Confidence Intervals
for Proportions
• Formula for the Margin of Error for a
Confidence Interval for the Proportion
UCLp , LCLp  p  zα/ 2 σˆp
 p  MEp
ME p  zα/ 2 σˆ p
Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall
8-30
Calculating Confidence Intervals
for Proportions
Example: From a random sample of U.S. citizens, 22 of
100 people are found to have blue eyes.
Calculate a 98% confidence interval for the
population proportion of blue eyes for U.S. citizens
1. Calculate the sample proportion and the approximate standard error
of the proportion:
x 22
p 
 0.22
n 100
2. Find z / 2 for 98%:
σˆ p 
p(1  p)
0.22(1  0.22)

 0.041
n
100
z / 2  z0.01  2.33
3. Calculate the interval endpoints: (next slide)
Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall
8-31
8.5 Determining the Sample Size
• Increasing the sample size, holding all else
constant, reduces the margin of error and
provides a narrower confidence interval
• The sample size needed to achieve a specific
margin of error can be calculated, given the
following information:
– The confidence level
– The population standard deviation
Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall
8-32
Calculating the Sample Size to
Estimate a Population Mean
• Solving for n:
ME x  zα/ 2 σ x
σ
n
ME x  zα/ 2
so
Formula for the Sample
Size Needed to
Estimate a Population
Mean:
( zα/ 2 ) 2 σ 2
n
( ME x ) 2
zα/ 2 σ
n
ME x
Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall
8-33
Calculating the Sample Size Needed to
Estimate a Population Proportion
• Solving for n:
ME p  zα/ 2 σˆ p
ME p  zα/ 2
p(1  p)
n
Formula for the Sample
Size Needed to
Estimate a Population
Proportion:
so
( zα/ 2 ) 2 p (1  p )
n
2
( ME p )
zα/ 2 p(1  p)
n
ME p
Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall
8-34
Calculating the Sample Size Needed to
Estimate a Population Proportion
• In order to calculate the required sample size
to estimate π, the population proportion, we
need to know p, the sample proportion
– Select a pilot sample and use the sample
proportion, p
– If it is not possible to select a pilot sample,
set p = 0.5
• Setting it equal to 0.50 provides the most conservative
estimate for a sample size (a sample size that is at least
large enough to satisfy the margin-of-error requirement)
Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall
8-35
Calculating the Sample Size Needed to
Estimate a Population Proportion
Example: What sample size is needed to estimate with
95% confidence the population proportion of U.S. citizens
with blue eyes within a margin of error of ± 5%? Assume a
pilot sample of 100 people found 22 with blue eyes.
1. Find
z / 2 for 95%:
z / 2  z0.025  1.96
2. Calculate the required sample size:
( zα/ 2 ) 2 p(1  p) (1.96) 2 (0.22)(1  0.22)
n

 263.69
2
2
( ME p )
(0.05)
so use a sample of size n = 264 (always round up)
Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall
8-36