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:

– 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.

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

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.

, c cX Z X Zn n

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:

• 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.

Haven’t we seen this before?

cE Zn

6.1 Confidence Intervals for the Mean ( known)

• We can re-express our confidence interval using margin of error notation:

, X E X E

, c cX Z X Zn n

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?

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)

2

cZnE