Download Interval estimation of a population mean

Interval Estimation
•Interval estimation of a population mean:
Large Sample case
•Interval estimation of a population mean:
Small sample case.
Interval Estimation of a Population Mean:
Large-Sample Case
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Sampling Error
Probability Statements about the Sampling Error
Interval Estimation:  Assumed Known
Interval Estimation:  Estimated by s
Sampling Error
The absolute value of the difference between an
unbiased point estimate and the population
parameter it estimates is called the sampling error.
For the case of a sample mean estimating a
population mean, the sampling error is
Sampling Error  x  
Probability Statements
About the Sampling Error
Knowledge of the sampling distribution of x
enables us to make probability statements about
the sampling error even though the population
mean  is not known.
A probability statement about the sampling error is
a precision statement.
Precision Statement
There is a 1   probability that the value of a
sample mean will provide a sampling error of z 
 /2 x
or less.
Sampling
distribution
of x
/2
1 -  of all
x values
z /2  x

/2
x
z /2  x
Interval Estimate of a Population Mean:
Large-Sample Case (n > 30)
Sampling
distribution
of x
/2
interval
does not
include 
1 -  of all
x values
z /2  x

/2
x
z /2  x
interval
includes 
x -------------------------]
[------------------------- x -------------------------]
[------------------------[-------------------------
x -------------------------]
Interval Estimate of a Population Mean:
Large-Sample Case (n > 30)
  Assumed Known
x  z / 2
where:
s
n
x is the sample mean
1 - is the confidence coefficient
z/2 is the z value providing an area of
/2 in the upper tail of the standard
normal probability distribution
 is the population standard deviation
n is the sample size
Interval Estimate of a Population Mean:
Large-Sample Case (n > 30)
  Estimated by s
x  z / 2
s
n
In most applications the value of the population
standard deviation is unknown. We simply use the
value of the sample standard deviation, s, as the point
estimate of the population standard deviation.
Example: Discount Sounds
Discount Sounds has 260 retail outlets
throughout the United States. The firm
is evaluating a potential location for a
new outlet, based in part, on the mean
annual income of the individuals in
the marketing area of the new location.
A sample of size n = 36 was taken.
The sample mean income is $21,100 and the sample
standard deviation is $4,500. The confidence
coefficient to be used in the interval estimate is .95.
D
Precision Statement
95% of the sample means that can be observed
are within + 1.96  x of the population mean .
Using s as an approximation of  , the margin of
error is:
z /2
s
 4,500 
 1.96 
  1, 470
n
 36 
There is a .95 probability that the value of a
sample mean for Discount Sounds will provide
a sampling error of $1,470 or less.
D
Interval Estimate of Population Mean: D
 Estimated by s
Interval estimate of  is:
$21,100 + $1,470
or
$19,630 to $22,570
We are 95% confident that the interval contains the
population mean.
Using Excel to Construct a
Confidence Interval: Large-Sample Case

D
Formula Worksheet
1
2
3
4
5
6
7
8
9
10
11
12
13
14
A
Income
25,600
19,615
20,035
21,735
17,600
26,080
28,925
25,350
15,900
17,550
14,745
20,000
19,250
B
C
Sample Size 36
Sample Mean =AVERAGE(A2:A37)
Sample Std. Deviation =STDEV(A2:A37)
Confidence Coefficient 0.95
Level of Signif. (alpha) =1-C5
z Value =NORMSINV(1-C5/2)
Standard Error =C3/SQRT(C1)
Margin of Error =C7*C9
Point Estimate =C2
Lower Limit =C12-C10
Upper Limit =C12+C10
Note: Rows 15-37 are not shown.
Using Excel to Construct a
Confidence Interval: Large-Sample Case

Value Worksheet
1
2
3
4
5
6
7
8
9
10
11
12
13
14
A
Income
25,600
19,615
20,035
21,735
17,600
26,080
28,925
25,350
15,900
17,550
14,745
20,000
19,250
B
Sample Size 36
Sample Mean 21,100
Sample Std. Deviation 4500
Confidence Coefficient 0.95
Level of Signif. (alpha) 0.05
z Value 1.960
Standard Error 749.992
Margin of Error 1469.955
Point Estimate 21,100.39
Lower Limit 19,630.43
Upper Limit 22,570.34
Note: Rows 15-37 are not shown.
C
D
Interval Estimation of a Population Mean:
Small-Sample Case (n < 30)
• Population is Not Normally Distributed
The only option is to increase the sample size to
n > 30 and use the large-sample interval-estimation
procedures.
Interval Estimation of a Population Mean:
Small-Sample Case (n < 30)
• Population is Normally Distributed:  Assumed
Known
The large-sample interval-estimation procedure can
be used.
x  z /2

n
Interval Estimation of a Population Mean:
Small-Sample Case (n < 30)
• Population is Normally Distributed:
Estimated by s
The appropriate interval estimate is based on a
probability distribution known as the t distribution.
t Distribution
The t distribution is a family of similar probability
distributions.
A specific t distribution depends on a parameter
known as the degrees of freedom.
As the number of degrees of freedom increases, the
difference between the t distribution and the
standard normal probability distribution becomes
smaller and smaller.
A t distribution with more degrees of freedom has
less dispersion.
Degrees of Freedom
Degrees of freedom (df) refers
to the number of independent
pieces of information that go
into the computation of
( xi  x ) 2
Example
Note that x  36.6
xi
( xi  x )
17
-19.6
22
-14.6
34
-2.6
56
19.4
?
17.4
( xi  x )
0
Note that the 5th
value of x must be
54—given the
values of x1, . . .x4.
Thus 5-1 or 4 values
of x are independent.
xi
t Distribution
t distribution
(20 degrees
of freedom)
Standard
normal
distribution
t distribution
(10 degrees
of freedom)
z, t
0
t Distribution
 /2 Area or Probability in the Upper Tail
/2
0
t/2
t
Interval Estimation of a Population Mean:
Small-Sample Case (n < 30) and
Estimated by s
• Interval Estimate
x  t /2
s
n
where: 1 - = the confidence coefficient
t/2 = the t value providing an area of /2
in the upper tail of a t distribution
with n - 1 degrees of freedom
s = the sample standard deviation
Example: Apartment Rents
• Interval Estimation of a Population Mean:
Small-Sample Case (n < 30) with  Estimated by s
A reporter for a student newspaper is writing an
article on the cost of off-campus
housing. A sample of 10
studio apartments within a
half-mile of campus resulted in
a sample mean of $550 per month and a sample
standard deviation of $60.
Example: Apartment Rents

Interval Estimation of a Population Mean:
Small-Sample Case (n < 30) with  Estimated by s
Let us provide a 95% confidence interval
estimate of the mean rent per
month for the population of
studio apartments within a
half-mile of campus. We will
assume this population to be normally distributed.
Interval Estimation of a Population Mean:
Small-Sample Case (n < 30) and  Estimated by s
• Interval Estimate
x  t /2
s
n
where: 1 - = the confidence coefficient
t/2 = the t value providing an area of /2
in the upper tail of a t distribution
with n - 1 degrees of freedom
s = the sample standard deviation
Example: Apartment Rents

Interval Estimation of a Population Mean:
Small-Sample Case (n < 30) with  Estimated by s
Let us provide a 95% confidence interval
estimate of the mean rent per
month for the population of
studio apartments within a
half-mile of campus. We will
assume this population to be normally distributed.
Interval Estimation of a Population Mean:
Small-Sample Case (n < 30) and  Estimated by s
At 95% confidence,  = .05, and /2 = .025.
t.025 is based on n  1 = 10  1 = 9 degrees of freedom.
In the t distribution table we see that t.025 = 2.262.
Degrees
Area in Upper Tail
of Freedom
.10
.05
.025
.01
.005
.
.
.
.
.
.
7
1.415
1.895
2.365
2.998
3.499
8
1.397
1.860
2.306
2.896
3.355
9
1.383
1.833
2.262
2.821
3.250
10
1.372
1.812
2.228
2.764
3.169
.
.
.
.
.
.
Interval Estimation of a Population Mean:
Small-Sample Case (n < 30) and  Estimated by s

Interval Estimate
x  t.025
s
n
60
= 550 + 42.92
550  2.262
10
We are 95% confident that the mean rent per
month for the population of studio apartments within
a half-mile of campus is between $507.08 and $592.92.
Using Excel to Construct a
Confidence Interval:  Estimated by s

Formula Worksheet
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
A
Rent
600
635
550
535
465
625
570
465
550
505
B
C
Sample Size 10
Sample Mean =AVERAGE(A2:A11)
Sample Std. Deviation =STDEV(A2:A11)
Confidence Coefficient
Level of Signif. (alpha)
Degrees of Freedom
t Value
0.95
=1-C5
=C1-1
=TINV(C6,C7)
Standard Error =C3/SQRT(C1)
Margin of Error =C8*C10
Point Estimate =C2
Lower Limit =C13-C11
Upper Limit =C13+C11
Using Excel to Construct a
Confidence Interval:  Estimated by s

Value Worksheet
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
A
Rent
600
635
550
535
465
625
570
465
550
505
B
Sample Size 10
Sample Mean 550.00
Sample Std. Deviation 60.05
Confidence Coefficient
Level of Signif. (alpha)
Degrees of Freedom
t Value
0.95
0.05
9
2.2622
Standard Error 18.988
Margin of Error 42.955
Point Estimate 550.00
Lower Limit 507.05
Upper Limit 592.95
C
Using Excel’s
Descriptive Statistics Tool
Excel’s Descriptive Statistics tool can
also be used to compute the margin of
error when the t distribution is used for a
confidence interval estimate of a
population mean.
Using Excel’s
Descriptive Statistics Tool

Descriptive Statistics Dialog Box
Using Excel’s
Descriptive Statistics Tool

Value Worksheet
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
A
Rent
600
635
550
535
465
625
570
465
550
505
B
C
D
Rent
Mean
Standard Error
Median
Mode
Standard Deviation
Sample Variance
Kurtosis
Skewness
Range
Minimum
Maximum
Sum
Count
Confidence Level(95.0%)
550
18.9883
550
550
60.04628
3605.556
-1.00162
-0.09863
170
465
635
5500
10
42.95455
Exercise 15, p. 334
The following data were collected for a sample
from a normal population: 10, 8, 12, 15, 13,
11, 6, 5
a. What is the point estimate the population
mean?
b. What is the point estimate of the population
standard deviation?
c. What is the 95 percent confidence interval
for the point estimate of the mean?
Exercise 15, p. 334
(a)
x  10
(b)
s  3.464
(c)
s
3.464
x  t.025 
 x  2.365 
n
8
 10  2.897
Pr(7.103  x  12.897)  .95
Summary of Interval Estimation Procedures
for a Population Mean
Yes
s known ?
Yes
n > 30 ?
No
s known ?
Yes
x  z /2
n
Popul.
approx.
normal
?
Yes
Use s to
estimate s

No
x  z /2
s
n
x  t /2

n
No
Use s to
estimate s
x  t /2
s
n
No
Increase n
to > 30