Transcript 6.1 Confidence Intervals for the Mean

6.1 Confidence Intervals for the Mean ( known)
• Key Concepts:
– Point Estimates
– Building and Interpreting Confidence Intervals
– Margin of Error
– Relationship Between Confidence Level and
Precision
– Determining the Minimum Sample Size
6.1 Confidence Intervals for the Mean ( known)
• A simple random sample of 20 recent U.S. weddings
yielded the following data on wedding costs, in dollars:
12,113
16,406
10,929
7,171
11,077
20,423
13,820
21,905
26,698
20,513
22,715
5,977
25,795
35,263
16,670
24,886
33,023
27,667
13,700
12,127
– Use the data to obtain a point estimate for the population mean
wedding cost of all recent U.S. weddings.
– If we assume wedding costs are normally distributed with σ =
$8100, find the 95% confidence interval for the mean cost of all
recent U.S. weddings.
6.1 Confidence Intervals for the Mean ( known)
• What exactly is a point estimate?
– a single value estimate for a population
parameter.
– If we want to estimate a population mean, the
best statistic to use is the sample mean.
– Since point estimates are single values, it is
unlikely they will actually equal the population
parameter.
• It would be better to build an interval estimate for
the parameter.
6.1 Confidence Intervals for the Mean ( known)
• Confidence Intervals
– Review the Sampling Distribution of Sample Means,
the Central Limit Theorem, and the Empirical Rule.
• General Form of a z-Interval Estimate for µ:

  
  
 X  Zc 
 , X  Zc 

 n
 n 

• Let’s go back to the U.S. Weddings example to see how this all
works.
6.1 Confidence Intervals for the Mean ( known)
• What do we mean by sampling error?
– Sampling error is defined as the difference between
the point estimate and the population parameter.
– Since we do not know the population parameter, we
look for a maximum value of our sampling error. We
call that maximum value the Margin of Error or the
Maximum Error of our estimate.
  
E  Zc 

 n
Haven’t we seen this before?
6.1 Confidence Intervals for the Mean ( known)
• We can re-express our confidence interval using
margin of error notation:

  
  
 X  Zc 
 , X  Zc 

 n
 n 

 X  E,
X  E
6.1 Confidence Intervals for the Mean ( known)
• Practice working with confidence intervals:
#36 p. 306 (Sodium Chloride Concentration)
• Note how the higher confidence level resulted in a wider
interval. The amount of precision in our estimate drops as
we increase the confidence level. There is an inverse
relationship between precision and level of confidence.
#38 p. 306 (Repair Costs: Refrigerators)
6.1 Confidence Intervals for the Mean ( known)
• If we’re told what level of confidence to use and
the maximum amount of sampling error that will
be tolerated in a study, how do we determine the
appropriate sample size?
 Zc  
n

 E 
2
Note: if σ is unknown, we can estimate it using the sample
standard deviation, s, as long as n ≥ 30.
#52 p. 308 (Water Dispensing Machine)