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6-1Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
6-2Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
Statistics for Business and
Economics
Chapter 6
Inferences Based on a Single Sample
6-3Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
Content
1. Identifying and Estimating the Target
Parameter
2. Confidence Interval for a Population Mean:
Normal (z) Statistic
3. Confidence Interval for a Population Mean:
Student’s t-Statistic
4. Large-Sample Confidence Interval for a
Population Proportion
5. Determining the Sample Size
6. Finite Population Correction for Simple
Random Sampling
7. Confidence Interval for a Population Variance
6-4Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
Learning Objectives
1. Estimate a population parameter (means,
proportion, or variance) based on a large
sample selected from the population
2. Use the sampling distribution of a statistic to
form a confidence interval for the population
parameter
3. Show how to select the proper sample size
for estimating a population parameter
6-5Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
Thinking Challenge
Suppose you’re
interested in the
average amount of
money that students
in this class (the
population) have on
them. How would
you find out?
6-6Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
Statistical Methods
Statistical
Methods
Estimation
Hypothesis
Testing
Inferential
Statistics
Descriptive
Statistics
6-7Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
6.1
Identifying and Estimating
the Target Parameter
6-8Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
Estimation Methods
Estimation
Interval
Estimation
Point
Estimation
6-9Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
Target Parameter
The unknown population parameter (e.g., mean or
proportion) that we are interested in estimating is
called the target parameter.
6-10Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
Target Parameter
Determining the Target Parameter
Parameter Key Words of Phrase Type of Data
µ Mean; average Quantitative
p Proportion; percentage
fraction; rate Qualitative
6-11Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
Point Estimator
A point estimator of a population parameter is a
rule or formula that tells us how to use the sample
data to calculate a single number that can be used
as an estimate of the target parameter.
6-12Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
Point Estimation
1. Provides a single value
• Based on observations from one sample
1. Gives no information about how close the
value is to the unknown population
parameter
3. Example: Sample mean x = 3 is the
point estimate of the unknown
population mean
6-13Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
Interval Estimator
An interval estimator (or confidence interval) is
a formula that tells us how to use the sample data
to calculate an interval that estimates the target
parameter.
6-14Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
Interval Estimation
1. Provides a range of values
• Based on observations from one sample
1. Gives information about closeness to unknown
population parameter
• Stated in terms of probability
– Knowing exact closeness requires knowing
unknown population parameter
1. Example: Unknown population mean lies between
50 and 70 with 95% confidence
6-15Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
6.2
Confidence Interval for a
Population Mean:
Normal (z) Statistic
6-16Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
Estimation Process
Mean, µ, is
unknown
Population








Sample


Random Sample
 
I am 95%
confident that µ
is between 40 &
60.


Mean
x = 50
6-17Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
Key Elements of
Interval Estimation
Sample statistic
(point estimate)
Confidence interval
Confidence limit
(lower)
Confidence limit
(upper)
A confidence interval provides a range of
plausible values for the population parameter.
6-18Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
Confidence Interval
According to the Central Limit Theorem, the
sampling distribution of the sample mean is
approximately normal for large samples. Let us
calculate the interval estimator:
x ±1.96σx = x ±
1.96σ
n
That is, we form an interval from 1.96 standard
deviations below the sample mean to 1.96 standard
deviations above the mean. Prior to drawing the
sample, what are the chances that this interval will
enclose µ, the population mean?
6-19Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
Confidence Interval
If sample measurements yield a value of that falls
between the two lines on either side of µ, then the
interval will contain µ.
The area under the
normal curve between
these two boundaries
is exactly .95. Thus,
the probability that a
randomly selected
interval will contain µ
is equal to .95.
x
x ±1.96σx
6-20Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
Confidence Coefficient
The confidence coefficient is the probability that
a randomly selected confidence interval encloses
the population parameter - that is, the relative
frequency with which similarly constructed
intervals enclose the population parameter when
the estimator is used repeatedly a very large
number of times. The confidence level is the
confidence coefficient expressed as a percentage.
6-21Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
95% Confidence Level
If our confidence level is 95%, then in the long run,
95% of our confidence intervals will contain µ and 5%
will not.
For a confidence coefficient of 95%, the area in the
two tails is .05. To choose a different confidence
coefficient we increase or decrease the area (call it α)
assigned to the tails. If we place α/2 in each tail
and zα/2 is the z-value, the
confidence interval with
coefficient (1 – α) is
x ± zα 2( )σx .
6-22Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
Conditions Required for a Valid
Large-Sample
Confidence Interval for µ
1. A random sample is selected from the target
population.
2. The sample size n is large (i.e., n ≥ 30). Due to
the Central Limit Theorem, this condition
guarantees that the sampling distribution of is
approximately normal. Also, for large n, s will be
a good estimator of σ.
x
6-23Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
Large-Sample (1 – α)% Confidence
Interval for µ
where zα/2 is the z-value with an area α/2 to its right
and in the standard normal distribution. The
parameter σ is the standard deviation of the
sampled population, and n is the sample size.
Note: When σ is unknown and n is large (n ≥ 30),
the confidence interval is approximately equal to
where s is the sample standard deviation.
x ± zα 2( )σx = x ± zα 2
σ
n




x ± zα 2
s
n




6-24Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
Thinking Challenge
You’re a Q/C inspector for
Gallo. The σ for 2-liter bottles
is .05 liters. A random
sample of 100 bottles showed
x = 1.99 liters. What is the
90% confidence interval
estimate of the true mean
amount in 2-liter bottles?
2 liter
© 1984-1994 T/Maker Co.
2 liter
6-25Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
Confidence Interval
Solution*
x − zα/2
⋅
σ
n
≤ µ ≤ x + zα/2
⋅
σ
n
1.99 −1.645⋅
.05
100
≤ µ ≤1.99 +1.645⋅
.05
100
1.982 ≤ µ ≤1.998
6-26Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
6.3
Confidence Interval for a
Population Mean:
Student’s t-Statistic
6-27Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
Small Sample σ Unknown
Instead of using the standard normal statistic
use the t–statistic
z =
x − µ
σx
=
x − µ
σ n
t =
x − µ
s n
in which the sample standard deviation, s, replaces
the population standard deviation, σ.
6-28Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
Student’s t-Statistic
The t-statistic has a sampling distribution very
much like that of the z-statistic: mound-shaped,
symmetric, with mean 0.
The primary
difference between
the sampling
distributions of t and
z is that the t-
statistic is more
variable than the z-
statistic.
6-29Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
Degrees of Freedom
The actual amount of variability in the sampling
distribution of t depends on the sample size n. A
convenient way of expressing this dependence is
to say that the t-statistic has (n – 1) degrees of
freedom (df).
6-30Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
z
t
Student’s t Distribution
0
t (df = 5)
Standard
Normal
t (df = 13)
Bell-Shaped
Symmetric
‘Fatter’ Tails
6-31Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
t - Table
6-32Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
t-value
If we want the t-value with an area of .025 to its
right and 4 df, we look in the table under the
column t.025 for the entry in the row corresponding to
4 df. This entry is t.025 = 2.776. The corresponding
standard normal z-score is z.025 = 1.96.
6-33Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
Small-Sample
Confidence Interval for µ
where ta/2 is based on (n – 1) degrees of freedom.
x ± tα 2
s
n




6-34Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
Conditions Required for a
Valid Small-Sample
Confidence Interval for µ
1. A random sample is selected from the target
population.
2. The population has a relative frequency
distribution that is approximately normal.
6-35Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
Estimation Example
Mean (σ Unknown)
x − tα/2 ⋅
s
n
≤ µ ≤ x + tα/2 ⋅
s
n
50 − 2.064 ⋅
8
25
≤ µ ≤ 50 + 2.064 ⋅
8
25
46.70 ≤ µ ≤ 53.30
A random sample of n = 25 has = 50 and s = 8.
Set up a 95% confidence interval estimate for µ.
x
6-36Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
Thinking Challenge
You’re a time study analyst
in manufacturing. You’ve
recorded the following task
times (min.):
3.6, 4.2, 4.0, 3.5, 3.8, 3.1.
What is the 90% confidence
interval estimate of the
population mean task time?
6-37Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
Confidence Interval Solution*
• x = 3.7
• s = 3.8987
• n = 6, df = n – 1 = 6 – 1 = 5
• t.05 = 2.015
3.7 − 2.015⋅
.38987
6
≤ µ ≤ 3.7 + 2.015⋅
.38987
6
.492 ≤ µ ≤ 6.908
6-38Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
6.4
Large-Sample Confidence
Interval for a Population
Proportion
6-39Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
Sampling Distribution of
1. The mean of the sampling distribution of is p;
that is, is an unbiased estimator of p.
ˆp
ˆp
3. For large samples, the sampling distribution of
is approximately normal. A sample size is
considered large if both nˆp ≥ 15 and nˆq ≥ 15.
ˆp
2. The standard deviation of the sampling
distribution of is ; that is,
where q = 1–p.
pq nˆp σ =ˆp pq n
ˆp
6-40Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
Large-Sample Confidence
Interval for
where
ˆp ± zα 2σ ˆp = ˆp ± zα 2 ⋅
pq
n
≈ ˆp ± zα 2 ⋅
ˆpˆq
n
ˆp =
x
n
and ˆq = 1− ˆp.
Note: When n is large, can approximate the
value of p in the formula for .
ˆp
σ ˆp
ˆp
6-41Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
2. The sample size n is large. (This condition will be
satisfied if both . Note that
and are simply the number of successes and
number of failures, respectively, in the sample.).
nˆp ≥ 15 and nˆq ≥ 15 nˆp
nˆq
Conditions Required for a
Valid Large-Sample
Confidence Interval for p
1. A random sample is selected from the target
population.
6-42Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
Estimation Example
Proportion
A random sample of 400 graduates showed
32 went to graduate school. Set up a 95%
confidence interval estimate for p.
( ) ( )
/2 /2
ˆ ˆ ˆ ˆ 32
ˆ ˆ ˆ 0.08
400
.08 .92 .08 .92
.08 1.96 .08 1.96
400 400
.053 .107
α α− ⋅ ≤ ≤ + ⋅ = =
− ⋅ ≤ ≤ + ⋅
≤ ≤
pq pq
p Z p p Z p
n n
p
p
6-43Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
Thinking Challenge
You’re a production
manager for a newspaper.
You want to find the %
defective. Of 200
newspapers, 35 had
defects. What is the 90%
confidence interval
estimate of the population
proportion defective?
6-44Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
Confidence Interval
Solution*
/2 /2
ˆ ˆ ˆ ˆ
ˆ ˆ
.175(.825) .175(.825)
.175 1.645 .175 1.645
200 200
.1308 .2192
α α
× ×
− × ≤ ≤ + ×
− × ≤ ≤ + ×
≤ ≤
p q p q
p z p p z
n n
p
p
6-45Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
Adjusted (1 – α)100%
Confidence Interval for a
Population Proportion, p
where is the adjusted sample proportion
of observations with the characteristic of interest, x
is the number of successes in the sample, and n is
the sample size.
( )
2
1
4
α
−
±
+
% %
%
p p
p z
n
%p =
x + 2
n + 4
6-46Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
6.5
Determining the Sample Size
6-47Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
Sampling Error
In general, we express the reliability associated
with a confidence interval for the population mean
µ by specifying the sampling error within which
we want to estimate µ with 100(1 – )% confidence.
The sampling error (denoted SE), then, is equal to
the half-width of the confidence interval.
α
6-48Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
Sample Size Determination
for 100(1 – α) %
Confidence Interval for µ
In order to estimate µ with a sampling error (SE)
and with 100(1 – α)% confidence, the required
sample size is found as follows:
zα 2
σ
n



 = SE
The solution for n is given by the equation
n =
zα 2( )
2
σ( )2
SE( )2
6-49Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
Sample Size Example
What sample size is needed to be 90%
confident the mean is within ± 5? A pilot
study suggested that the standard deviation
is 45.
n =
(zα 2
)2
σ 2
(SE) 2
=
1.645( )
2
45( )
2
5( )
2
= 219.2 ≅ 220
6-50Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
Sample Size Determination
for 100(1 – α) %
Confidence Interval for p
In order to estimate p with a sampling error SE and
with 100(1 – α)% confidence, the required sample
size is found by solving the following equation for
n:
zα 2
pq
n
= SE
The solution for n can be written as follows:
n =
zα 2( )
2
pq( )
SE( )2
Note: Always round n
up to the nearest
integer value.
6-51Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
Sample Size Example
What sample size is needed to estimate p
within .03 with 90% confidence?
.03
.015
2 2
width
SE = = =
n =
(Zα 2 )2
pq( )
(SE) 2
=
1.645( )2
.5⋅.5( )
.015( )2 = 3006.69 ≅ 3007
6-52Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
Thinking Challenge
You work in Human
Resources at Merrill Lynch.
You plan to survey employees
to find their average medical
expenses. You want to be
95% confident that the
sample mean is within ± $50.
A pilot study showed that σ
was about $400. What
sample size do you use?
6-53Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
Sample Size Solution*
n =
(zα 2
)2
σ2
(SE)2
=
1.96( )
2
400( )
2
50( )
2
= 245.86 ≅ 246
6-54Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
6.6
Finite Population Correction
for Simple Random Sample
6-55Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
Finite Population Correction Factor
In some sampling situations, the sample size n
may represent 5% or perhaps 10% of the total
number N of sampling units in the population.
When the sample size is large relative to the
number of measurements in the population (see
the next slide), the standard errors of the
estimators of µ and p should be multiplied by a
finite population correction factor.
6-56Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
Rule of Thumb for Finite
Population Correction Factor
Use the finite population correction factor
when n/N > .05.
6-57Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
Simple Random Sampling with
Finite Population of Size N
Estimation of the Population Mean
Estimated standard error:
Approximate 95% confidence interval:
ˆσx =
s
n
N − n
N
ˆ2σ± xx
6-58Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
Simple Random Sampling with
Finite Population of Size N
Estimation of the Population Proportion
Estimated standard error:
Approximate 95% confidence interval:ˆp ± 2 ˆσ ˆp
ˆσ ˆp =
ˆp(1− ˆp)
n
N − n
N
6-59Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
Finite Population Correction
Factor Example
You want to estimate a population mean, μ, where
x =115, s =18, N =700, and n = 60. Find an
approximate 95% confidence interval for μ.
is greater than .05 use the finite correction
factor
086.
700
60 ==
N
n
Since
6-60Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
Finite Population Correction
Factor Example
You want to estimate a population mean, μ, where
x =115, s =18, N =700, and n = 60. Find an
approximate 95% confidence interval for μ.
x ± 2
s
n
N − n
N
= 115 ± 2 ⋅
18
60
700 − 60
700
= 115 ± 4.4
= 110.6, 119.4( )
6-61Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
6.7
Confidence Interval for a
Population Variance
6-62Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
Confidence Interval for a
Population Variance
6-63Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
Conditions Required for a Valid
Confidence Interval for σ2
1. A random sample is selected from the
target population.
2. The population of interest has a relative
frequency distribution that is
approximately normal.
6-64Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
Thinking Challenge
You’re a marketing
manager for a 5K race. You
take a random sample of
the times of 292 runners
from the last race, with
mean of 28.5 minutes and
standard deviation of 8.3
minutes. What is the 95%
confidence interval
estimate of the population
variance?
6-65Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
Confidence Interval
Solution*
( ) ( )
( )
( )( ) ( ) ( )
2
2 2
2
2 2
1
2
2 2
2
2
1 1
292 1 8.3 292 1 8.3
349.874 253.912
57.30 78.95
n s n s
α α
σ
χ χ
σ
σ
−
− −
≤ ≤
− −
≤ ≤
≤ ≤
df = 292 − 1 = 291 (use 300 df) .025
2
α
=
6-66Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
Key Ideas
Population Parameters, Estimators, and
Standard Errors
Parameter Estimator Standard
Error of
Estimator
Estimated
Std Error
Mean, µ
Proportion, p ˆpˆq n
σ ˆθ( )
pq nˆp
s nσ nx
ˆθ( )θ( )
ˆσ ˆθ( )
6-67Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
Key Ideas
Population Parameters, Estimators, and
Standard Errors
Confidence Interval: An interval that encloses
an unknown population parameter with a certain
level of confidence (1 – α)
Confidence Coefficient: The probability (1 – α)
that a randomly selected confidence interval
encloses the true value of the population
parameter.
6-68Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
Key Ideas
Key Words for Identifying the Target
Parameter
µ – Mean, Average
p – Proportion, Fraction, Percentage, Rate,
Probability
σ2
- Variance
6-69Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
Key Ideas
Commonly Used z-Values for a Large-
Sample Confidence Interval
90% CI: (1 – α) = .10 z.05 = 1.645
95% CI: (1 – α) = .05 z.025 = 1.96
98% CI: (1 – α) = .02 z.005 = 2.326
99% CI: (1 – α) = .01 z.005 = 2.575
6-70Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
Key Ideas
Determining the Sample Size n
( ) ( ) ( )α σ=
2 22
2 MEn zEstimating µ:
Estimating p: ( ) ( ) ( )α=
2 2
2 MEn z pq
6-71Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
Key Ideas
Finite Population Correction Factor
Required when n/N > .05
6-72Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
Key Ideas
Confidence Interval for Population
Variance
Uses chi-square (χ2
) distribution
Need to know and df.
2
α
6-73Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
Key Ideas
Illustrating the Notion of “95%
Confidence”
6-74Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
Key Ideas
Illustrating the Notion of “95%
Confidence”

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Msb12e ppt ch06

  • 1. 6-1Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
  • 2. 6-2Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Statistics for Business and Economics Chapter 6 Inferences Based on a Single Sample
  • 3. 6-3Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Content 1. Identifying and Estimating the Target Parameter 2. Confidence Interval for a Population Mean: Normal (z) Statistic 3. Confidence Interval for a Population Mean: Student’s t-Statistic 4. Large-Sample Confidence Interval for a Population Proportion 5. Determining the Sample Size 6. Finite Population Correction for Simple Random Sampling 7. Confidence Interval for a Population Variance
  • 4. 6-4Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Learning Objectives 1. Estimate a population parameter (means, proportion, or variance) based on a large sample selected from the population 2. Use the sampling distribution of a statistic to form a confidence interval for the population parameter 3. Show how to select the proper sample size for estimating a population parameter
  • 5. 6-5Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Thinking Challenge Suppose you’re interested in the average amount of money that students in this class (the population) have on them. How would you find out?
  • 6. 6-6Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Statistical Methods Statistical Methods Estimation Hypothesis Testing Inferential Statistics Descriptive Statistics
  • 7. 6-7Copyright © 2014, 2011, and 2008 Pearson Education, Inc. 6.1 Identifying and Estimating the Target Parameter
  • 8. 6-8Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Estimation Methods Estimation Interval Estimation Point Estimation
  • 9. 6-9Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Target Parameter The unknown population parameter (e.g., mean or proportion) that we are interested in estimating is called the target parameter.
  • 10. 6-10Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Target Parameter Determining the Target Parameter Parameter Key Words of Phrase Type of Data µ Mean; average Quantitative p Proportion; percentage fraction; rate Qualitative
  • 11. 6-11Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Point Estimator A point estimator of a population parameter is a rule or formula that tells us how to use the sample data to calculate a single number that can be used as an estimate of the target parameter.
  • 12. 6-12Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Point Estimation 1. Provides a single value • Based on observations from one sample 1. Gives no information about how close the value is to the unknown population parameter 3. Example: Sample mean x = 3 is the point estimate of the unknown population mean
  • 13. 6-13Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Interval Estimator An interval estimator (or confidence interval) is a formula that tells us how to use the sample data to calculate an interval that estimates the target parameter.
  • 14. 6-14Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Interval Estimation 1. Provides a range of values • Based on observations from one sample 1. Gives information about closeness to unknown population parameter • Stated in terms of probability – Knowing exact closeness requires knowing unknown population parameter 1. Example: Unknown population mean lies between 50 and 70 with 95% confidence
  • 15. 6-15Copyright © 2014, 2011, and 2008 Pearson Education, Inc. 6.2 Confidence Interval for a Population Mean: Normal (z) Statistic
  • 16. 6-16Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Estimation Process Mean, µ, is unknown Population         Sample   Random Sample   I am 95% confident that µ is between 40 & 60.   Mean x = 50
  • 17. 6-17Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Key Elements of Interval Estimation Sample statistic (point estimate) Confidence interval Confidence limit (lower) Confidence limit (upper) A confidence interval provides a range of plausible values for the population parameter.
  • 18. 6-18Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Confidence Interval According to the Central Limit Theorem, the sampling distribution of the sample mean is approximately normal for large samples. Let us calculate the interval estimator: x ±1.96σx = x ± 1.96σ n That is, we form an interval from 1.96 standard deviations below the sample mean to 1.96 standard deviations above the mean. Prior to drawing the sample, what are the chances that this interval will enclose µ, the population mean?
  • 19. 6-19Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Confidence Interval If sample measurements yield a value of that falls between the two lines on either side of µ, then the interval will contain µ. The area under the normal curve between these two boundaries is exactly .95. Thus, the probability that a randomly selected interval will contain µ is equal to .95. x x ±1.96σx
  • 20. 6-20Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Confidence Coefficient The confidence coefficient is the probability that a randomly selected confidence interval encloses the population parameter - that is, the relative frequency with which similarly constructed intervals enclose the population parameter when the estimator is used repeatedly a very large number of times. The confidence level is the confidence coefficient expressed as a percentage.
  • 21. 6-21Copyright © 2014, 2011, and 2008 Pearson Education, Inc. 95% Confidence Level If our confidence level is 95%, then in the long run, 95% of our confidence intervals will contain µ and 5% will not. For a confidence coefficient of 95%, the area in the two tails is .05. To choose a different confidence coefficient we increase or decrease the area (call it α) assigned to the tails. If we place α/2 in each tail and zα/2 is the z-value, the confidence interval with coefficient (1 – α) is x ± zα 2( )σx .
  • 22. 6-22Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Conditions Required for a Valid Large-Sample Confidence Interval for µ 1. A random sample is selected from the target population. 2. The sample size n is large (i.e., n ≥ 30). Due to the Central Limit Theorem, this condition guarantees that the sampling distribution of is approximately normal. Also, for large n, s will be a good estimator of σ. x
  • 23. 6-23Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Large-Sample (1 – α)% Confidence Interval for µ where zα/2 is the z-value with an area α/2 to its right and in the standard normal distribution. The parameter σ is the standard deviation of the sampled population, and n is the sample size. Note: When σ is unknown and n is large (n ≥ 30), the confidence interval is approximately equal to where s is the sample standard deviation. x ± zα 2( )σx = x ± zα 2 σ n     x ± zα 2 s n    
  • 24. 6-24Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Thinking Challenge You’re a Q/C inspector for Gallo. The σ for 2-liter bottles is .05 liters. A random sample of 100 bottles showed x = 1.99 liters. What is the 90% confidence interval estimate of the true mean amount in 2-liter bottles? 2 liter © 1984-1994 T/Maker Co. 2 liter
  • 25. 6-25Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Confidence Interval Solution* x − zα/2 ⋅ σ n ≤ µ ≤ x + zα/2 ⋅ σ n 1.99 −1.645⋅ .05 100 ≤ µ ≤1.99 +1.645⋅ .05 100 1.982 ≤ µ ≤1.998
  • 26. 6-26Copyright © 2014, 2011, and 2008 Pearson Education, Inc. 6.3 Confidence Interval for a Population Mean: Student’s t-Statistic
  • 27. 6-27Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Small Sample σ Unknown Instead of using the standard normal statistic use the t–statistic z = x − µ σx = x − µ σ n t = x − µ s n in which the sample standard deviation, s, replaces the population standard deviation, σ.
  • 28. 6-28Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Student’s t-Statistic The t-statistic has a sampling distribution very much like that of the z-statistic: mound-shaped, symmetric, with mean 0. The primary difference between the sampling distributions of t and z is that the t- statistic is more variable than the z- statistic.
  • 29. 6-29Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Degrees of Freedom The actual amount of variability in the sampling distribution of t depends on the sample size n. A convenient way of expressing this dependence is to say that the t-statistic has (n – 1) degrees of freedom (df).
  • 30. 6-30Copyright © 2014, 2011, and 2008 Pearson Education, Inc. z t Student’s t Distribution 0 t (df = 5) Standard Normal t (df = 13) Bell-Shaped Symmetric ‘Fatter’ Tails
  • 31. 6-31Copyright © 2014, 2011, and 2008 Pearson Education, Inc. t - Table
  • 32. 6-32Copyright © 2014, 2011, and 2008 Pearson Education, Inc. t-value If we want the t-value with an area of .025 to its right and 4 df, we look in the table under the column t.025 for the entry in the row corresponding to 4 df. This entry is t.025 = 2.776. The corresponding standard normal z-score is z.025 = 1.96.
  • 33. 6-33Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Small-Sample Confidence Interval for µ where ta/2 is based on (n – 1) degrees of freedom. x ± tα 2 s n    
  • 34. 6-34Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Conditions Required for a Valid Small-Sample Confidence Interval for µ 1. A random sample is selected from the target population. 2. The population has a relative frequency distribution that is approximately normal.
  • 35. 6-35Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Estimation Example Mean (σ Unknown) x − tα/2 ⋅ s n ≤ µ ≤ x + tα/2 ⋅ s n 50 − 2.064 ⋅ 8 25 ≤ µ ≤ 50 + 2.064 ⋅ 8 25 46.70 ≤ µ ≤ 53.30 A random sample of n = 25 has = 50 and s = 8. Set up a 95% confidence interval estimate for µ. x
  • 36. 6-36Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Thinking Challenge You’re a time study analyst in manufacturing. You’ve recorded the following task times (min.): 3.6, 4.2, 4.0, 3.5, 3.8, 3.1. What is the 90% confidence interval estimate of the population mean task time?
  • 37. 6-37Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Confidence Interval Solution* • x = 3.7 • s = 3.8987 • n = 6, df = n – 1 = 6 – 1 = 5 • t.05 = 2.015 3.7 − 2.015⋅ .38987 6 ≤ µ ≤ 3.7 + 2.015⋅ .38987 6 .492 ≤ µ ≤ 6.908
  • 38. 6-38Copyright © 2014, 2011, and 2008 Pearson Education, Inc. 6.4 Large-Sample Confidence Interval for a Population Proportion
  • 39. 6-39Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Sampling Distribution of 1. The mean of the sampling distribution of is p; that is, is an unbiased estimator of p. ˆp ˆp 3. For large samples, the sampling distribution of is approximately normal. A sample size is considered large if both nˆp ≥ 15 and nˆq ≥ 15. ˆp 2. The standard deviation of the sampling distribution of is ; that is, where q = 1–p. pq nˆp σ =ˆp pq n ˆp
  • 40. 6-40Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Large-Sample Confidence Interval for where ˆp ± zα 2σ ˆp = ˆp ± zα 2 ⋅ pq n ≈ ˆp ± zα 2 ⋅ ˆpˆq n ˆp = x n and ˆq = 1− ˆp. Note: When n is large, can approximate the value of p in the formula for . ˆp σ ˆp ˆp
  • 41. 6-41Copyright © 2014, 2011, and 2008 Pearson Education, Inc. 2. The sample size n is large. (This condition will be satisfied if both . Note that and are simply the number of successes and number of failures, respectively, in the sample.). nˆp ≥ 15 and nˆq ≥ 15 nˆp nˆq Conditions Required for a Valid Large-Sample Confidence Interval for p 1. A random sample is selected from the target population.
  • 42. 6-42Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Estimation Example Proportion A random sample of 400 graduates showed 32 went to graduate school. Set up a 95% confidence interval estimate for p. ( ) ( ) /2 /2 ˆ ˆ ˆ ˆ 32 ˆ ˆ ˆ 0.08 400 .08 .92 .08 .92 .08 1.96 .08 1.96 400 400 .053 .107 α α− ⋅ ≤ ≤ + ⋅ = = − ⋅ ≤ ≤ + ⋅ ≤ ≤ pq pq p Z p p Z p n n p p
  • 43. 6-43Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Thinking Challenge You’re a production manager for a newspaper. You want to find the % defective. Of 200 newspapers, 35 had defects. What is the 90% confidence interval estimate of the population proportion defective?
  • 44. 6-44Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Confidence Interval Solution* /2 /2 ˆ ˆ ˆ ˆ ˆ ˆ .175(.825) .175(.825) .175 1.645 .175 1.645 200 200 .1308 .2192 α α × × − × ≤ ≤ + × − × ≤ ≤ + × ≤ ≤ p q p q p z p p z n n p p
  • 45. 6-45Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Adjusted (1 – α)100% Confidence Interval for a Population Proportion, p where is the adjusted sample proportion of observations with the characteristic of interest, x is the number of successes in the sample, and n is the sample size. ( ) 2 1 4 α − ± + % % % p p p z n %p = x + 2 n + 4
  • 46. 6-46Copyright © 2014, 2011, and 2008 Pearson Education, Inc. 6.5 Determining the Sample Size
  • 47. 6-47Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Sampling Error In general, we express the reliability associated with a confidence interval for the population mean µ by specifying the sampling error within which we want to estimate µ with 100(1 – )% confidence. The sampling error (denoted SE), then, is equal to the half-width of the confidence interval. α
  • 48. 6-48Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Sample Size Determination for 100(1 – α) % Confidence Interval for µ In order to estimate µ with a sampling error (SE) and with 100(1 – α)% confidence, the required sample size is found as follows: zα 2 σ n     = SE The solution for n is given by the equation n = zα 2( ) 2 σ( )2 SE( )2
  • 49. 6-49Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Sample Size Example What sample size is needed to be 90% confident the mean is within ± 5? A pilot study suggested that the standard deviation is 45. n = (zα 2 )2 σ 2 (SE) 2 = 1.645( ) 2 45( ) 2 5( ) 2 = 219.2 ≅ 220
  • 50. 6-50Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Sample Size Determination for 100(1 – α) % Confidence Interval for p In order to estimate p with a sampling error SE and with 100(1 – α)% confidence, the required sample size is found by solving the following equation for n: zα 2 pq n = SE The solution for n can be written as follows: n = zα 2( ) 2 pq( ) SE( )2 Note: Always round n up to the nearest integer value.
  • 51. 6-51Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Sample Size Example What sample size is needed to estimate p within .03 with 90% confidence? .03 .015 2 2 width SE = = = n = (Zα 2 )2 pq( ) (SE) 2 = 1.645( )2 .5⋅.5( ) .015( )2 = 3006.69 ≅ 3007
  • 52. 6-52Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Thinking Challenge You work in Human Resources at Merrill Lynch. You plan to survey employees to find their average medical expenses. You want to be 95% confident that the sample mean is within ± $50. A pilot study showed that σ was about $400. What sample size do you use?
  • 53. 6-53Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Sample Size Solution* n = (zα 2 )2 σ2 (SE)2 = 1.96( ) 2 400( ) 2 50( ) 2 = 245.86 ≅ 246
  • 54. 6-54Copyright © 2014, 2011, and 2008 Pearson Education, Inc. 6.6 Finite Population Correction for Simple Random Sample
  • 55. 6-55Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Finite Population Correction Factor In some sampling situations, the sample size n may represent 5% or perhaps 10% of the total number N of sampling units in the population. When the sample size is large relative to the number of measurements in the population (see the next slide), the standard errors of the estimators of µ and p should be multiplied by a finite population correction factor.
  • 56. 6-56Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Rule of Thumb for Finite Population Correction Factor Use the finite population correction factor when n/N > .05.
  • 57. 6-57Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Simple Random Sampling with Finite Population of Size N Estimation of the Population Mean Estimated standard error: Approximate 95% confidence interval: ˆσx = s n N − n N ˆ2σ± xx
  • 58. 6-58Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Simple Random Sampling with Finite Population of Size N Estimation of the Population Proportion Estimated standard error: Approximate 95% confidence interval:ˆp ± 2 ˆσ ˆp ˆσ ˆp = ˆp(1− ˆp) n N − n N
  • 59. 6-59Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Finite Population Correction Factor Example You want to estimate a population mean, μ, where x =115, s =18, N =700, and n = 60. Find an approximate 95% confidence interval for μ. is greater than .05 use the finite correction factor 086. 700 60 == N n Since
  • 60. 6-60Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Finite Population Correction Factor Example You want to estimate a population mean, μ, where x =115, s =18, N =700, and n = 60. Find an approximate 95% confidence interval for μ. x ± 2 s n N − n N = 115 ± 2 ⋅ 18 60 700 − 60 700 = 115 ± 4.4 = 110.6, 119.4( )
  • 61. 6-61Copyright © 2014, 2011, and 2008 Pearson Education, Inc. 6.7 Confidence Interval for a Population Variance
  • 62. 6-62Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Confidence Interval for a Population Variance
  • 63. 6-63Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Conditions Required for a Valid Confidence Interval for σ2 1. A random sample is selected from the target population. 2. The population of interest has a relative frequency distribution that is approximately normal.
  • 64. 6-64Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Thinking Challenge You’re a marketing manager for a 5K race. You take a random sample of the times of 292 runners from the last race, with mean of 28.5 minutes and standard deviation of 8.3 minutes. What is the 95% confidence interval estimate of the population variance?
  • 65. 6-65Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Confidence Interval Solution* ( ) ( ) ( ) ( )( ) ( ) ( ) 2 2 2 2 2 2 1 2 2 2 2 2 1 1 292 1 8.3 292 1 8.3 349.874 253.912 57.30 78.95 n s n s α α σ χ χ σ σ − − − ≤ ≤ − − ≤ ≤ ≤ ≤ df = 292 − 1 = 291 (use 300 df) .025 2 α =
  • 66. 6-66Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Key Ideas Population Parameters, Estimators, and Standard Errors Parameter Estimator Standard Error of Estimator Estimated Std Error Mean, µ Proportion, p ˆpˆq n σ ˆθ( ) pq nˆp s nσ nx ˆθ( )θ( ) ˆσ ˆθ( )
  • 67. 6-67Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Key Ideas Population Parameters, Estimators, and Standard Errors Confidence Interval: An interval that encloses an unknown population parameter with a certain level of confidence (1 – α) Confidence Coefficient: The probability (1 – α) that a randomly selected confidence interval encloses the true value of the population parameter.
  • 68. 6-68Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Key Ideas Key Words for Identifying the Target Parameter µ – Mean, Average p – Proportion, Fraction, Percentage, Rate, Probability σ2 - Variance
  • 69. 6-69Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Key Ideas Commonly Used z-Values for a Large- Sample Confidence Interval 90% CI: (1 – α) = .10 z.05 = 1.645 95% CI: (1 – α) = .05 z.025 = 1.96 98% CI: (1 – α) = .02 z.005 = 2.326 99% CI: (1 – α) = .01 z.005 = 2.575
  • 70. 6-70Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Key Ideas Determining the Sample Size n ( ) ( ) ( )α σ= 2 22 2 MEn zEstimating µ: Estimating p: ( ) ( ) ( )α= 2 2 2 MEn z pq
  • 71. 6-71Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Key Ideas Finite Population Correction Factor Required when n/N > .05
  • 72. 6-72Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Key Ideas Confidence Interval for Population Variance Uses chi-square (χ2 ) distribution Need to know and df. 2 α
  • 73. 6-73Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Key Ideas Illustrating the Notion of “95% Confidence”
  • 74. 6-74Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Key Ideas Illustrating the Notion of “95% Confidence”

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