Msb12e ppt ch06

Statistics for Business and
Economics
Chapter 6
Inferences Based on a Single Sample

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

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

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?

Statistical Methods
Statistical
Methods
Estimation
Hypothesis
Testing
Inferential
Statistics
Descriptive
Statistics

6.1
Identifying and Estimating
the Target Parameter

Estimation Methods
Estimation
Interval
Estimation
Point
Estimation

Target Parameter
The unknown population parameter (e.g., mean or
proportion) that we are interested in estimating is
called the target parameter.

Target Parameter
Determining the Target Parameter
Parameter Key Words of Phrase Type of Data
µ Mean; average Quantitative
p Proportion; percentage
fraction; rate Qualitative

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.

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

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.

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

6.2
Confidence Interval for a
Population Mean:
Normal (z) Statistic

Estimation Process
Mean, µ, is
unknown
Population








Sample


Random Sample
 
I am 95%
confident that µ
is between 40 &
60.


Mean
x = 50

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.

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?

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

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.

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 .

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

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





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

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

6.3
Population Mean:

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.

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.

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).

z
t
Student’s t Distribution
0
t (df = 5)
Standard
Normal
t (df = 13)
Bell-Shaped
Symmetric
‘Fatter’ Tails

t - Table

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.

Small-Sample
where ta/2 is based on (n – 1) degrees of freedom.
x ± tα 2
s
n





Conditions Required for a
Valid Small-Sample
population.
2. The population has a relative frequency
distribution that is approximately normal.

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

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?

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

6.4
Large-Sample Confidence
Interval for a Population
Proportion

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

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

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
population.

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

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?

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

Adjusted (1 – α)100%
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

6.5
Determining the Sample Size

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

Sample Size Determination
for 100(1 – α) %
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

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

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.

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

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?

Sample Size Solution*
n =
(zα 2
)2
σ2
(SE)2
=
1.96( )
2
400( )
2
50( )
2
= 245.86 ≅ 246

6.6
Finite Population Correction
for Simple Random Sample

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.

Rule of Thumb for Finite
Population Correction Factor
Use the finite population correction factor
when n/N > .05.

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

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

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

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( )

6.7
Population Variance

Population Variance

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.

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?

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
=

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( )

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.

Key Ideas
Key Words for Identifying the Target
Parameter
µ – Mean, Average
p – Proportion, Fraction, Percentage, Rate,
Probability
σ2
- Variance

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

Key Ideas
Determining the Sample Size n
( ) ( ) ( )α σ=
2 22
2 MEn zEstimating µ:
Estimating p: ( ) ( ) ( )α=
2 2
2 MEn z pq

Key Ideas
Finite Population Correction Factor
Required when n/N > .05

Key Ideas
Confidence Interval for Population
Variance
Uses chi-square (χ2
) distribution
Need to know and df.
2
α

Key Ideas
Illustrating the Notion of “95%
Confidence”

Msb12e ppt ch06

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