STT 430, Summer 2006
Lecture 5
(part II)
Materials Covered: Chapter 8
Suggested Exercises: 8.17, 8.36, 8.37, 8.40, 8.54, 8.55,
8.5 Confidence Interval
Definition: Suppose that ˆL and ˆU are two functions of the sample,
 is a parameter, if
P(ˆL    ˆU ) =1-.
Then the resulting interval defined by [ ˆL , ˆU ] is called a two-sided confidence interval. ˆL is
called the lower confidence limit, and ˆU is called the upper confidence limit. The probability 1- is
called the confidence coefficient or confidence level. If ˆU =  , then [ ˆL ,  ) is a lower one-sided
confidence interval; If ˆL =  , then (-  , ˆU ] is an upper one-sided confidence interval.
Pivot method for finding the confidence interval.
This method depends upon finding a pivotal quantity that possesses two characteristics:
(1) It is a function of the sample and the unknown parameter  , where  is the only unknown
parameter.
(2) Its probability distribution does not depend upon the parameter.
Example 8.4: Suppose that we are to obtain a single observation Y from an exponential distribution
with mean  , Use Y to form a confidence interval for  with confidence coefficient 0.90.
STT 430, Summer 2006
Example 8.5: Suppose that we take a sample of size n=1 from a uniform distribution defined on the
interval [0,  ], where  is unknown. Find a 95% lower confidence bound for  .
8.6 Large-Sample Confidence Intervals
Basic Idea: Suppose  is the parameter of interest, and ˆ is an estimator. In some cases, for large
samples, Z 
ˆ  
possesses approximately a standard normal distribution. Consequently, Z forms
 ˆ
(at least approximately) a pivotal quantity, and the pivotal method can be employed to develop
confidence intervals for the target parameter  .
Example 8.6 Let ˆ be a statistic that is normally distributed with mean  and the standard error  ˆ .
Find a confidence interval for  with confidence coefficient 1-.
Example 8.7 The shopping times of n=64 randomly selected customers at a local supermarket were
recorded. The average and variance of the 64 shopping times were 33 minutes and 256, respectively.
Estimate , the true average shopping time per customer, with confidence coefficient of 1-=0.90.
Example 8.8 Two brands of refrigerators, denoted by A and B, are each guaranteed for 1 year. In a
random sample of 50 refrigerators of brand A, 12 were observed to fail before the guarantee period
ended. An independent random sample of 60 brand B refrigerators also revealed 12 failures during the
guarantee period. Estimate the true difference p1-p2 between proportions of failures during the
guarantee period, with confidence coefficient 0.98.
STT 430, Summer 2006
8.7 Selecting the Sample Size
8.8