copyright ©2013 pearson education, inc. publishing as prentice hall 8-1 chapter 8
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Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall8-1
Chapter 8
Confidence Intervals
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Population Mean
σ Unknown
ConfidenceIntervals
PopulationProportion
σ Known
Section 8.2
Section 8.4
Section 8.3
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• A confidence interval for the mean
– interval (or range) around the sample mean ( ) within which true (population) mean (µ) is expected to be
– µ = 362.3 ± 1.96 * 15 =
• 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
8.2 Calculating Confidence Intervals for the Mean when the Standard Deviation (σ) of a Population is Known
2α / xx z σ
x
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
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n
σσ x
whereσ = Population standard deviationn = Sample size
• Formulas for the Confidence Interval for the Mean (σ Known)
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Confidence Intervals for the Mean, σ Known
xα/x
xα/x
σzxLCL
σzxUCL
2
2
mean the of error standard The
score- critical The
mean sample The
where
x
α/
σ
zz
x
2
• is called the critical z-score
• The variable is known as the significance level
• Example: if = .10, then = 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– The total area to the left
of the left-hand
boundary is 0.05
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2/z
05.02/ zz
α/2 = 0.050.90
z0-1.645 1.645
Confidence Intervals for the Mean, σ Known
α/2 = 0.05
0.95
0.05
Calculating the Margin of Error
• The Margin of Error ME is the amount added and subtracted to the point estimate to form the confidence interval
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Point EstimateLower Confidence Limit
UpperConfidence Limit
Width of confidence interval
x
x
xα/xx
MEx
σzxLCLUCL
2 , xα/x σzME 2
Margin of Error Margin of Error
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
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xx MEσn
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)
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xμμ
Confidence Intervals
Each interval extends from
to
But varies from sample to sample
For 90% confidence, 90% of intervals constructed contain μ ; 10% do not
xα/ σzx 2x1
x2
/2 /21
xα/ σzx 2
x
Interpreting a Confidence Interval
x
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)
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• encloses 95% of the area under the
curve, with 2.5% in each tail
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• Consider a 95% confidence interval:
0.951
0
1.96z
Changing Confidence Levels
0.025/2 0.025/2
-1.96/2 z -1.96/2 z0.975
Confidence Level:
Significance Level:
Critical z-score:
80%
90%
95%
98%
99%
20%
10%
5%
2%
1%
z0.10 = ±1.28
z0.05 = ±1.645
z0.025 = ±1.96
z0.01 = ±2.33
z0.005 = ±2.575
• z-scores for the most commonly used confidence levels are shown in this table:
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)%100(1 )%(100 2/z
Changing Confidence Levels
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
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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
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• 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):
8.3 Calculating Confidence Intervals when the Population Standard Deviation (σ) is Unknown
11
2
n
)x(xs
n
ii
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
Confidence Intervals
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Population Mean
σ Unknown
ConfidenceIntervals
PopulationProportion
σ Known
Section 8.2
Section 8.4
Section 8.3
Using the Student’s t-distribution
• Formula for the Approximate Standard Error of the Mean
• 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
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n
sσ x ˆ
Using the Student’s t-distribution
• Formulas for the Confidence Interval for the Mean (σ Unknown)
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xα/x
xα/x
σtxLCL
σtxUCL
ˆ
ˆ
2
2
where:
= The sample mean
= The critical t-score
= The approximate standard error of the meanx
α/
σ
t
x
ˆ2
Using the Student’s t-distribution
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t
t (df = 5)
t (df = 13)
t-distributions are bell-shaped and symmetric, but have ‘fatter’ tails than the normal
Normal distribution
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
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• Comparing t-scores and z-scores:
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Confidence t t t z Level (10 df ) (20 df ) (30 df )
.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
Using the Student’s t-distribution
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
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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)
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where: alpha ( ) = The significance level of the confidence interval degrees of freedom = n - 1 n = Sample size
Confidence Intervals
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Population Mean
σ Unknown
ConfidenceIntervals
PopulationProportion
σ Known
Section 8.2
Section 8.4
Section 8.3
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• 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
8.4 Calculating ConfidenceIntervals for Proportions
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
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n
xp
Formula for the Standard Error of the Proportion:
Formula for the Sample Proportion:
nσ p
)1(
where: π = The population proportion x = The number of observations of interest in the sample (successes) n = The sample size
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:
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n
ppσ p
)1(ˆ
where: p = The sample proportion n = The sample size
Calculating Confidence Intervals for Proportions
• Formulas for the Confidence Interval for a Proportion:
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where: p = The sample proportion = The critical z-score
= The approximate standard error of the proportion
p
p
σzpLCL
σzpUCL
α/p
α/p
ˆ
ˆ
2
2
pσ
zα/ˆ2
Calculating Confidence Intervals for Proportions
• Formula for the Margin of Error for a Confidence Interval for the Proportion
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p
p
MEp
σzpLCLUCL α/pp
ˆ, 2
pα/p σzME ˆ2
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:
2. Find for 98%:
3. Calculate the interval endpoints: (next slide)
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2/z 2.33 01.02/ zz
Calculating Confidence Intervals for Proportions
0.041100
0.22)0.22(1
n
ppσ p
)1(ˆ0.22
100
22
n
xp
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• 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
8.5 Determining the Sample Size
Calculating the Sample Size to Estimate a Population Mean
• Solving for n:
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x
α/
α/x
xα/x
ME
σzn
n
σzME
σzME
2
2
2
so
2
222
)(
)(
x
α/
ME
σzn
Formula for the Sample Size Needed to Estimate a Population Mean:
Calculating the Sample Size Needed to Estimate a Population Proportion
• Solving for n:
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p
α/
α/p
pα/p
ME
ppzn
n
ppzME
σzME
)1(
)1(
ˆ
2
2
2
so
Formula for the Sample Size Needed to Estimate a Population Proportion:
2
22
)(
)1()(
p
α/
ME
ppzn
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• 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)
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 for 95%:
2.Calculate the required sample size:
so use a sample of size n = 264 (always round up)
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2/z 1.96 025.02/ zz
Calculating the Sample Size Needed to Estimate a Population Proportion
263.69(0.05)
0.22)(0.22)(1(1.96)2
2
2
22
)(
)1()(
p
α/
ME
ppzn
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