Transcript 7.1 confidence Intervals for the Mean When SD is Known

7.1 CONFIDENCE INTERVALS
FOR THE MEAN WHEN SD IS
KNOWN
-A point estimate is a specific numerical
value that estimates a parameter.
-The best point estimate of the population
mean µ is the sample mean X.
3 PROPERTIES OF A GOOD ESTIMATOR
1. Unbiased: the mean of estimates is equal to
the parameter being estimated.
2. Consistent: as the sample size increases, the
value of the estimator approaches the value
of the parameter.
3. Relatively Efficient: has the smallest variance
CONFIDENCE INTERVALS
Interval Estimate: an interval (range) used to
estimate the parameter.
-may or may not contain the parameter
Ex. 20 < µ < 25
CONFIDENCE INTERVALS
Confidence Level- probability that the interval estimate
will contain the parameter
-three common levels are 90%, 95%, and 99%
Confidence Interval- specific interval estimate of a
parameter
-determined by using data and specific confidence
levels
-as the percent increases, so does the range of the
interval.
FORMULA FOR THE CONFIDENCE INTERVAL
  
  
X  z / 2 
    X  z / 2 

 n 
 n 
X = sample mean
N = sample size
σ = standard deviation
For 90% confidence interval:
z  / 2  1.65
95% confidence interval:
z  / 2  1.96
99% confidence interval:
z  / 2  2.58
MARGIN OF ERROR
The maximum likely difference
between the point estimate and the
actual value of the parameter.
  
E  z / 2 

 n 
A researcher wishes to estimate the number of days it
takes an automobile dealer to sell a Chevrolet Aveo. A
sample of 50 cars had a mean time on the dealer’s lot of
54 days. Assume the population standard deviation to
be 6.0 days. Find the best point estimate of the
population mean and the 95% confidence interval of the
population mean.
X  54,   6.0, n  50, 95%  z  1.96
X  z
2
  

    X  z
 n 
2
  


 n 
X  54,   6.0, n  50, 95%  z  1.96
  
  
X  z 2 
    X  z 2 

 n 
 n 
 6.0 
 6.0 
54  1.96 
    54  1.96 

 50 
 50 
54  1.7    54  1.7
52.3    55.7
52    56
So, we can say with 95% confidence that the
interval between 52.3 and 55.7 does contain
the population mean.
A survey of 30 emergency room patients found
that the average waiting time for treatment was
174.3 minutes.
Assuming that the population standard deviation
is 46.5 minutes, find the best point estimate of
the population mean and the 99% confidence of
the population mean.
Practice problems: p. 366
Numbers 11, 13, 14, 15, 17
DETERMINING SAMPLE SIZE
 z 2   
n

E


2
E = margin of error
N = sample size
σ = standard deviation
z  2 depends on confidence interval
- A scientist wishes to estimate the average depth of a
river. He wants to be 99% confident that the estimate
is accurate within 2 feet. From a previous study, the
standard deviation of the depths measured was 4.33
feet. How large a sample is required?
99%  z  2.58, E  2,   4.33
2
2
 z 2   
 2 .5 8  4 .3 3 
n 
 3 1 .2  3 2
 

E
2




PRACTICE!
p. 367 numbers 21, 23, 24, 25