8-1 Introduction In the previous chapter we illustrated how a parameter can be estimated from sample data. However, it is important to understand how.

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Presentation transcript:

8-1 Introduction In the previous chapter we illustrated how a parameter can be estimated from sample data. However, it is important to understand how good is the estimate obtained. Bounds that represent an interval of plausible values for a parameter are an example of an interval estimate. Three types of intervals will be presented: Confidence intervals Prediction intervals Tolerance intervals

8-2.1 Development of the Confidence Interval and its Basic Properties 8-2 Confidence Interval on the Mean of a Normal Distribution, Variance Known

8-2.1 Development of the Confidence Interval and its Basic Properties 8-2 Confidence Interval on the Mean of a Normal Distribution, Variance Known

8-2.1 Development of the Confidence Interval and its Basic Properties The endpoints or bounds l and u are called lower- and upper- confidence limits, respectively. Since Z follows a standard normal distribution, we can write: 8-2 Confidence Interval on the Mean of a Normal Distribution, Variance Known

8-2.1 Development of the Confidence Interval and its Basic Properties Definition 8-2 Confidence Interval on the Mean of a Normal Distribution, Variance Known

Example Confidence Interval on the Mean of a Normal Distribution, Variance Known

Interpreting a Confidence Interval The confidence interval is a random interval The appropriate interpretation of a confidence interval (for example on  ) is: The observed interval [l, u] brackets the true value of , with confidence 100(1-  ). Examine Figure 8-1 on the next slide. 8-2 Confidence Interval on the Mean of a Normal Distribution, Variance Known

Figure 8-1 Repeated construction of a confidence interval for .

Confidence Level and Precision of Error The length of a confidence interval is a measure of the precision of estimation. 8-2 Confidence Interval on the Mean of a Normal Distribution, Variance Known Figure 8-2 Error in estimating  with.

8-2.2 Choice of Sample Size Definition 8-2 Confidence Interval on the Mean of a Normal Distribution, Variance Known

Example Confidence Interval on the Mean of a Normal Distribution, Variance Known

8-2.3 One-Sided Confidence Bounds Definition 8-2 Confidence Interval on the Mean of a Normal Distribution, Variance Known

8-2.4 General Method to Derive a Confidence Interval 8-2 Confidence Interval on the Mean of a Normal Distribution, Variance Known

8-2.4 General Method to Derive a Confidence Interval 8-2 Confidence Interval on the Mean of a Normal Distribution, Variance Known

8-2.4 General Method to Derive a Confidence Interval 8-2 Confidence Interval on the Mean of a Normal Distribution, Variance Known

8-2.5 A Large-Sample Confidence Interval for  Definition 8-2 Confidence Interval on the Mean of a Normal Distribution, Variance Known

Example Confidence Interval on the Mean of a Normal Distribution, Variance Known

Example 8-3 (continued) 8-2 Confidence Interval on the Mean of a Normal Distribution, Variance Known

Example 8-3 (continued) 8-2 Confidence Interval on the Mean of a Normal Distribution, Variance Known Figure 8-3 Mercury concentration in largemouth bass (a) Histogram. (b) Normal probability plot

Example 8-3 (continued) 8-2 Confidence Interval on the Mean of a Normal Distribution, Variance Known

A General Large Sample Confidence Interval 8-2 Confidence Interval on the Mean of a Normal Distribution, Variance Known

8-3.1 The t distribution 8-3 Confidence Interval on the Mean of a Normal Distribution, Variance Unknown

8-3.1 The t distribution 8-3 Confidence Interval on the Mean of a Normal Distribution, Variance Unknown Figure 8-4 Probability density functions of several t distributions.

8-3.1 The t distribution 8-3 Confidence Interval on the Mean of a Normal Distribution, Variance Unknown Figure 8-5 Percentage points of the t distribution.

8-3.3 The t Confidence Interval on  8-3 Confidence Interval on the Mean of a Normal Distribution, Variance Unknown One-sided confidence bounds on the mean are found by replacing t  /2,n-1 in Equation 8-18 with t ,n-1.

Example 8-4

Figure 8-6 Box and Whisker plot for the load at failure data in Example 8-4.

Figure 8-7 Normal probability plot of the load at failure data in Example 8-4.

Definition 8-4 Confidence Interval on the Variance and Standard Deviation of a Normal Distribution

Figure 8-8 Probability density functions of several  2 distributions.

Definition 8-4 Confidence Interval on the Variance and Standard Deviation of a Normal Distribution

One-Sided Confidence Bounds 8-4 Confidence Interval on the Variance and Standard Deviation of a Normal Distribution

Example 8-5

Definition 8-5 A Large-Sample Confidence Interval For a Population Proportion

Definition 8-5 A Large-Sample Confidence Interval For a Population Proportion The quantity is called the standard error of the point estimator.

Definition 8-5 A Large-Sample Confidence Interval For a Population Proportion

Example A Large-Sample Confidence Interval For a Population Proportion

Choice of Sample Size The sample size for a specified value E is given by 8-5 A Large-Sample Confidence Interval For a Population Proportion An upper bound on n is given by

Example A Large-Sample Confidence Interval For a Population Proportion

One-Sided Confidence Bounds 8-5 A Large-Sample Confidence Interval For a Population Proportion

Definition 8-6 A Prediction Interval for a Future Observation The prediction interval for X n+1 will always be longer than the confidence interval for .

Example A Prediction Interval for a Future Observation

Definition 8-7 Tolerance Intervals for a Normal Distribution

Example Tolerance Intervals for a Normal Distribution