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CHAPTER OUTLINE
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8-1 INTRODUCTION
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Statistical Intervals for a Single Sample
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8-3.1 The t Distribution 8-3.2 Development of the t Distribution (CD Only) 8-3.3 The t Con dence Interval on 8-4 CONFIDENCE INTERVAL ON THE VARIANCE AND STANDARD DEVIATION OF A NORMAL POPULATION 8-5 A LARGE-SAMPLE CONFIDENCE INTERVAL FOR A POPULATION PROPORTION 8-6 A PREDICTION INTERVAL FOR A FUTURE OBSERVATION 8-7 TOLERANCE INTERVALS FOR A NORMAL DISTRIBUTION
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8-2 CONFIDENCE INTERVAL ON THE MEAN OF A NORMAL DISTRIBUTION, VARIANCE KNOWN 8-2.1 Development of the Con dence Interval and its Basic Properties 8-2.2 Choice of Sample Size 8-2.3 One-Sided Con dence Bounds 8-2.4 General Method to Derive a Con dence Interval 8-2.5 A Large-Sample Con dence Interval for 8-2.6 Bootstrap Con dence Intervals (CD Only) 8-3 CONFIDENCE INTERVAL ON THE MEAN OF A NORMAL DISTRIBUTION, VARIANCE UNKNOWN
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LEARNING OBJECTIVES
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After careful study of this chapter, you should be able to do the following: 1. Construct con dence intervals on the mean of a normal distribution, using either the normal distribution or the t distribution method 2. Construct con dence intervals on the variance and standard deviation of a normal distribution 3. Construct con dence intervals on a population proportion 4. Construct prediction intervals for a future observation 5. Construct a tolerance interval for a normal population
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CHAPTER 8 STATISTICAL INTERVALS FOR A SINGLE SAMPLE
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6. Explain the three types of interval estimates: con dence intervals, prediction intervals, and tolerance intervals 7. Use the general method for constructing a con dence interval CD MATERIAL 8. Use the bootstrap technique to construct a con dence interval Answers for many odd numbered exercises are at the end of the book. Answers to exercises whose numbers are surrounded by a box can be accessed in the e-Text by clicking on the box. Complete worked solutions to certain exercises are also available in the e-Text. These are indicated in the Answers to Selected Exercises section by a box around the exercise number. Exercises are also available for some of the text sections that appear on CD only. These exercises may be found within the e-Text immediately following the section they accompany.
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INTRODUCTION
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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. For example, suppose that we estimate the mean viscosity of a chemical product to be x 1000. Now because of sampling variability, it is almost never the case that x. The point estimate says nothing about how close is to . Is the process mean likely to be between 900 and 1100 Or is it likely to be between 990 and 1010 The answer to these questions affects our decisions regarding this process. Bounds that represent an interval of plausible values for a parameter are an example of an interval estimate. Surprisingly, it is easy to determine such intervals in many cases, and the same data that provided the point estimate are typically used. An interval estimate for a population parameter is called a con dence interval. We cannot be certain that the interval contains the true, unknown population parameter we only use a sample from the full population to compute the point estimate and the interval. However, the con dence interval is constructed so that we have high con dence that it does contain the unknown population parameter. Con dence intervals are widely used in engineering and the sciences. A tolerance interval is another important type of interval estimate. For example, the chemical product viscosity data might be assumed to be normally distributed. We might like to calculate limits that bound 95% of the viscosity values. For a normal distribution, we know that 95% of the distribution is in the interval 1.96 , 1.96 (8-1)
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However, this is not a useful tolerance interval because the parameters and are unknown. Point estimates such as x and s can be used in Equation 8-1 for and . However, we need to account for the potential error in each point estimate to form a tolerance interval for the distribution. The result is an interval of the form x ks, x ks (8-2)
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where k is an appropriate constant (that is larger than 1.96 to account for the estimation error). As for a con dence interval, it is not certain that Equation 8-2 bounds 95% of the distribution, but the interval is constructed so that we have high confidence that it does. Tolerance intervals are widely used and, as we will subsequently see, they are easy to calculate for normal distributions.
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