Transcript Document

Single Sample: Estimating the Mean
(σ unknown, n not large)
• Given:
– σ is unknown and X is the mean of a random sample
of size n (where n is not large),
• Then,
– the (1 – α)100% confidence interval for μ is given by
X  t / 2,n 1(
-5
-4
-3
-2
s
s
)    X  t / 2,n 1(
)
n
n
-1
0
1
2
3
4
5
Recall Our Example
A traffic engineer is concerned about the delays at an
intersection near a local school. The intersection is
equipped with a fully actuated (“demand”) traffic light
and there have been complaints that traffic on the main
street is subject to unacceptable delays.
To develop a benchmark, the traffic engineer randomly
samples 25 stop times (in seconds) on a weekend day.
The average of these times is found to be 13.2 seconds,
and the sample variance, s2, is found to be 4 seconds2.
Based on this data, what is the 95% confidence interval
(C.I.) around the mean stop time during a weekend day?
Example (cont.)
X = ______________
s = _______________
α = ________________
α/2 = _____________
t0.025,24 = _____________
__________________ < μ < ___________________
Your turn
A thermodynamics professor gave a physics
pretest to a random sample of 15 students who
enrolled in his course at a large state university.
The sample mean was found to be 59.81 and
the sample standard deviation was 4.94.
Find a 99% confidence interval for the mean on
this pretest.
Solution
X = ______________
s = _______________
α = ________________
α/2 = _____________
(draw the picture)
T___ , ____ = _____________
__________________ < μ < ___________________
Standard Error of a Point Estimate
• Case 1: σ known
– The standard deviation, or standard error of X is

n
• Case 2: σ unknown, sampling from a normal
distribution
– The standard deviation, or (usually) estimated
standard error of X is
______
9.6: Prediction Interval
• For a normal distribution of unknown mean μ, and
standard deviation σ, a 100(1-α)% prediction
interval of a future observation, x0 is
1
1
X  z / 2 1   x0  X  z / 2 1 
n
n
if σ is known, and
X  t / 2,n 1s 1 
if σ is unknown
1
1
 x0  X  t / 2,n 1s 1 
n
n
9.7: Tolerance Limits
• For a normal distribution of unknown mean μ,
and unknown standard deviation σ, tolerance
limits are given by
x + ks
where k is determined so that one can assert
with 100(1-γ)% confidence that the given limits
contain at least the proportion 1-α of the
measurements.
• Table A.7 gives values of k for (1-α) = 0.9, 0.95,
0.99; γ = 0.05, 0.01; and for selected values of
n.
Summary
• Confidence interval  population mean μ
• Prediction interval 
a new observation x0
• Tolerance interval 
a (1-α) proportion of
the measurements
can be estimated with
a 100(1-γ)%
confidence
Estimating the Difference Between Two
Means
• Given two independent random samples, a point
estimate the difference between μ1 and μ2 is
given by the statistic
x1  x 2
We can build a confidence interval for μ1 - μ2
(given σ12 and σ22 known) as follows:
( x 1  x 2 )  z / 2
 12
n1

 22
n2
 1  2  ( x 1  x 2 )  z / 2
 12
n1

 22
n2
An example
• Look at example 9.8, pg. 248
Differences Between Two Means:
Variances Unknown
• Case 1: σ12 and σ22 unknown but equal
( x 1  x 2 )  t / 2,n1 n2 2Sp
Where,
1 1
1 1

 1  2  ( x1  x 2 )  t / 2,n1 n2 2Sp

n1 n2
n1 n2
(n1  1)S12  (n2  1)S22
S 
n1  n2  2
2
p
Differences Between Two Means:
Variances Unknown
• Case 2: σ12 and σ22 unknown and not equal
( x 1  x 2 )  t / 2,
Where,
s12 s22
s12 s22

 1  2  ( x 1  x 2 )  t / 2,

n1 n2
n1 n2
(S12 / n1  S22 / n2 )2

 S2 / n 2   S2 / n 2 
 1 1  2 2 
 n1  1   n2  1 



