Statistics Sampling Intervals for a Single Sample Contents, figures, and exercises come from the textbook: Applied Statistics and Probability for Engineers, 5th Edition, by Douglas C. Montgomery, John Wiley & Sons, Inc., 2011.

If,,…, are normally and independently distributed with unknown mean and known variance has a standard normal distribution Confidence Interval on the Mean of a Normal Distribution, Variance Known

Confidence interval on the mean, variance known Contents, figures, and exercises come from the textbook: Applied Statistics and Probability for Engineers, 5th Edition, by Douglas C. Montgomery, John Wiley & Sons, Inc., 2011.

From, we have If is used as an estimate of, we can be confident that the error will not exceed a specified amount when the sample size is Contents, figures, and exercises come from the textbook: Applied Statistics and Probability for Engineers, 5th Edition, by Douglas C. Montgomery, John Wiley & Sons, Inc., 2011.

One-sided confidence bounds on the mean, variance known ◦ A upper-confidence bound for is ◦ A lower-confidence bound for is Contents, figures, and exercises come from the textbook: Applied Statistics and Probability for Engineers, 5th Edition, by Douglas C. Montgomery, John Wiley & Sons, Inc., 2011.

General method to derive a confidence interval ◦ We find a statistic that  1. depends on both the sample and  2. The probability distribution of does not depend on and any other unknown parameter  For example, ◦ Find constants and so that

Large-sample confidence interval on the mean  When is large, the quantity  has an approximate standard normal distribution. Consequently,  is a large-sample confidence interval for, with confidence level of approximately. Contents, figures, and exercises come from the textbook: Applied Statistics and Probability for Engineers, 5th Edition, by Douglas C. Montgomery, John Wiley & Sons, Inc., 2011.

Large-sample approximate confidence interval  If the quantity  has an approximate standard normal distribution. Consequently,  is a large-sample approximate confidence interval for Contents, figures, and exercises come from the textbook: Applied Statistics and Probability for Engineers, 5th Edition, by Douglas C. Montgomery, John Wiley & Sons, Inc., 2011.

Example 8-1 Metallic Material Transition ◦ Ten measurements: 64.1, 64.7, 64.5, 64.6, 64.5, 64.3, 64.6, 64.8, 64.2, 64.3 ◦ Assume it is a normal distribution with. Find a 95% CI for. Example 8-2 Metallic Material Transition ◦ Determine how many specimens must be tested to ensure that the 95% CI for has a length of at most 1.0. Example 8-3 One-Sided Confidence Bound ◦ Determine a lower, one-sided 95% CI for.  Contents, figures, and exercises come from the textbook: Applied Statistics and Probability for Engineers, 5th Edition, by Douglas C. Montgomery, John Wiley & Sons, Inc., 2011.

Example 8-4 Mercury Contamination ◦ 53 measurements: 1.230, 0.490, … ◦,,,. ◦ Find a 95% CI for.  Contents, figures, and exercises come from the textbook: Applied Statistics and Probability for Engineers, 5th Edition, by Douglas C. Montgomery, John Wiley & Sons, Inc., 2011.

Exercise 8-14 ◦ The life in hours of a 75-watt light bulb is known to be normally distributed with hours. A random sample of 20 bulbs has a mean life of hours. ◦ (a) Construct a 95% two-sided confidence interval on the mean life. ◦ (b) Construct a 95% lower-confidence bound on the mean life. Compare the lower bound of this confidence interval with the one in part (a).  Contents, figures, and exercises come from the textbook: Applied Statistics and Probability for Engineers, 5th Edition, by Douglas C. Montgomery, John Wiley & Sons, Inc., 2011.

Distribution Let,,…, are normally and independently distributed with unknown mean and unknown variance. The random variable has a distribution with degrees of freedom. Confidence Interval on the Mean of a Normal Distribution, Variance Unknown

From Wikipedia, PDF of distribution

From Wikipedia, CDF of distribution

The probability density function  is the number of degrees of freedom  Mean :  Variance : for ◦ Percentage points   is a large-sample confidence interval for, with confidence level of approximately. Contents, figures, and exercises come from the textbook: Applied Statistics and Probability for Engineers, 5th Edition, by Douglas C. Montgomery, John Wiley & Sons, Inc., 2011.

confidence interval on  Contents, figures, and exercises come from the textbook: Applied Statistics and Probability for Engineers, 5th Edition, by Douglas C. Montgomery, John Wiley & Sons, Inc., 2011.

Confidence interval on the mean, variance unknown ◦ If and are the mean and standard deviation of a random sample from a normal distribution with unknown variance, a confidence interval on is given by ◦ where is the upper percentage point of the distribution with degrees of freedom  Contents, figures, and exercises come from the textbook: Applied Statistics and Probability for Engineers, 5th Edition, by Douglas C. Montgomery, John Wiley & Sons, Inc., 2011.

Normal probability plot ◦ The sample,,…, is arranged as,,…,,where is the smallest observation, is the second-smallest observation, and so forth. ◦ The ordered observations are then plotted against their observed cumulative frequency on the appropriate probability paper. ◦ Or, plot the standardized normal scores against, where  Contents, figures, and exercises come from the textbook: Applied Statistics and Probability for Engineers, 5th Edition, by Douglas C. Montgomery, John Wiley & Sons, Inc., 2011.

From Wikipedia, Percent-percent plot

Example 8-5 Alloy Adhesion ◦ The load at specimen failure: 19.8, 10.1, … ◦,,. ◦ Find a 95% CI on.  Contents, figures, and exercises come from the textbook: Applied Statistics and Probability for Engineers, 5th Edition, by Douglas C. Montgomery, John Wiley & Sons, Inc., 2011.

Exercise 8-41 ◦ An article in Nuclear Engineering International (February 1988, p. 33) describes several characteristics of fuel rods used in a reactor owned by an electric utility in Norway. Measurements on the percentage of enrichment of 12 rods were reported as follows: 2.94, 3.00, 2.90, 2.75, 3.00, 2.95, 2.90, 2.75, 2.95, 2.82, 2.81, ◦ (a) Use a normal probability plot to check the normality assumption. ◦ (b) Find a 99% two-sided confidence interval on the mean percentage of enrichment. Are you comfortable with the statement that the mean percentage of enrichment is 2.95%? Why?  Contents, figures, and exercises come from the textbook: Applied Statistics and Probability for Engineers, 5th Edition, by Douglas C. Montgomery, John Wiley & Sons, Inc., 2011.

Distribution Let,,…, are normally and independently distributed mean and variance, and let be the sample variance. The random variable has a chi-square distribution with degrees of freedom. Confidence Interval on the Variance and Standard Deviation of a Normal Distribution

From Wikipedia, PDF of distribution

From Wikipedia, CDF of distribution

The probability density function  is the number of degrees of freedom  Mean :  Variance : ◦ Percentage points   is a large-sample confidence interval for, with confidence level of approximately. Contents, figures, and exercises come from the textbook: Applied Statistics and Probability for Engineers, 5th Edition, by Douglas C. Montgomery, John Wiley & Sons, Inc., 2011.

◦ Since ◦ is chi-square with degrees of freedom, we have  Contents, figures, and exercises come from the textbook: Applied Statistics and Probability for Engineers, 5th Edition, by Douglas C. Montgomery, John Wiley & Sons, Inc., 2011.

Confidence interval on the variance ◦ If is the sample variance from a random sample of observations from a normal distribution with unknown variance, then a confidence interval on is ◦ Where and are the upper and lower percentage points of the chi-square distribution with ◦ degrees of freedom, respectively. A confidence interval for has lower and upper limits that are the square roots of the corresponding limits in the above equation  Contents, figures, and exercises come from the textbook: Applied Statistics and Probability for Engineers, 5th Edition, by Douglas C. Montgomery, John Wiley & Sons, Inc., 2011.

One-sided confidence bounds on the variance ◦ The lower and upper confidence bounds on are ◦ respectively.  Contents, figures, and exercises come from the textbook: Applied Statistics and Probability for Engineers, 5th Edition, by Douglas C. Montgomery, John Wiley & Sons, Inc., 2011.

Example 8-6 Detergent Filling ◦,. ◦ Find a 95% upper confidence bound on and. Exercise 8-44 ◦ A rivet is to be inserted into a hole. A random sample of parts is selected, and the hole diameter is measured. The sample standard deviation of the hole diameter measurements is millimeters. Construct a 99% lower confidence bound for.  Contents, figures, and exercises come from the textbook: Applied Statistics and Probability for Engineers, 5th Edition, by Douglas C. Montgomery, John Wiley & Sons, Inc., 2011.

Normal approximation for a binomial proportion If is large, the distribution of is approximately standard normal. Large-Sample Confidence Interval for a population proportion

From Wikipedia, PMF of binomial distribution

 To construct the confidence interval on,  Contents, figures, and exercises come from the textbook: Applied Statistics and Probability for Engineers, 5th Edition, by Douglas C. Montgomery, John Wiley & Sons, Inc., 2011.

◦ Approximate confidence interval on a binomial proportion  If is the proportion of observations in a random sample of size that belongs to a class of interest, an approximate confidence interval on the proportion of the population that belongs to this class is  where is the upper percentage of the standard normal distribution.  Required: and  Contents, figures, and exercises come from the textbook: Applied Statistics and Probability for Engineers, 5th Edition, by Douglas C. Montgomery, John Wiley & Sons, Inc., 2011.

◦ Sample size for a specified error on a binomial proportion  Set  Then  Or  Contents, figures, and exercises come from the textbook: Applied Statistics and Probability for Engineers, 5th Edition, by Douglas C. Montgomery, John Wiley & Sons, Inc., 2011.

◦ Approximate one-sided confidence bounds on a binomial proportion  The approximate lower and upper confidence bounds are  respectively.  Contents, figures, and exercises come from the textbook: Applied Statistics and Probability for Engineers, 5th Edition, by Douglas C. Montgomery, John Wiley & Sons, Inc., 2011.

◦ Example 8-7 Crankshaft Bearings ,, and  Find a 95% two-sided confidence interval for. ◦ Example 8-8 Crankshaft Bearings  How large a sample is required if we want to be 95% confident that the error in using to estimate is less than 0.05?  Contents, figures, and exercises come from the textbook: Applied Statistics and Probability for Engineers, 5th Edition, by Douglas C. Montgomery, John Wiley & Sons, Inc., 2011.

◦ Exercise 8-53  The fraction of defective integrated circuits produced in a photolithography process is being studied. A random sample of 350 circuits is tested, revealing 15 defectives.  (a) Calculate a 95% two-sided CI on the fraction of defective circuits produced by this particular tool.  (b) Calculate a 95% upper confidence bound on the fraction of defective circuits.  Contents, figures, and exercises come from the textbook: Applied Statistics and Probability for Engineers, 5th Edition, by Douglas C. Montgomery, John Wiley & Sons, Inc., 2011.

is a single future observation Then has a standard normal distribution and Has a distribution with degrees of freedom. Tolerance and Prediction Intervals

Prediction interval A prediction interval (PI) on a single future observation from a normal distribution is given by

Tolerance interval A tolerance interval for capturing at least of the values in a normal distribution with confidence level is where is a tolerance interval factor found in Appendix Tabel XII. Values are given for = 90%, 95%, and 99% and for 90%, 95%, and 99% confidence.

Example 8-9 Alloy Adhesion,, and Find a 95% prediction interval on the load at failure for a new specimen. Example 8-10 Alloy Adhesion Find a tolerance interval for the load at failure that includes 90% of the values in the population with 95% confidence.

Exercise 8-39(a)