Page 293 - Applied statistics and probability for engineers
P. 293
8
Statistical Intervals for
a Single Sample
Introduction
Chapter Outline
8-1 Conidence Interval on the Mean of a Engineers are often involved in estimating parameters.
Normal Distribution, Variance Known For example, there is an ASTM Standard E23 that dei nes
8-1.1 Development of the Coni dence a technique called the Charpy V-notch method for notched
Interval and Its Basic Properties bar impact testing of metallic materials. The impact energy
8-1.2 Choice of Sample Size is often used to determine whether the material experiences
8-1.3 One-Sided Coni dence Bounds a ductile-to-brittle transition as the temperature decreases.
8-1.4 General Method to Derive a Suppose that we have tested a sample of 10 specimens of a
Coni dence Interval particular material with this procedure. We know that we can
8-1.5 Large-Sample Coni dence Interval use the sample average X to estimate the true mean impact
for μ energy μ. However, we also know that the true mean impact
8-2 Conidence Interval on the Mean of a energy is unlikely to be exactly equal to your estimate.
Normal Distribution, Variance Unknown Reporting the results of your test as a single number is unap-
8-2.1 t Distribution pealing because nothing inherent in X provides any informa-
8-2.2 Conidence Interval on μ tion about how close it is to μ. Our estimate could be very
close, or it could be considerably far from the true mean.
8-3 Conidence Interval on the Variance
And Standard Deviation of a Normal A way to avoid this is to report the estimate in terms of a
Distribution range of plausible values called a coni dence interval. A
conidence interval always speciies a conidence level, usu-
8-4 Large-Sample Conidence Interval for a ally 90%, 95%, or 99%, which is a measure of the reliabil-
Population Proportion ity of the procedure. So if a 95% conidence interval on the
8-5 Guidelines for Constructing Coni dence impact energy based on the data from our 10 specimens has
Intervals a lower limit of 63.84 J and an upper limit of 65.08 J, then
8-6 Bootstrap Coni dence Interval we can say that at the 95% level of conidence any value of
mean impact energy between 63.84 J and 65.08 J is a plau-
8-7 Tolerance and Prediction intervals
8-7.1 Prediction Interval for a Future sible value. By reliability, we mean that if we repeated this
experiment over and over again, 95% of all samples would
Observation
8-7.2 Tolerance Interval for a Normal produce a coni dence interval that contains the true mean
Distribution
271
