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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. Confidence Interval on the Mean of a Normal Distribution,Variance Known If X1 , X 2 ,…, X n are normally and independently distributed with unknown mean and known variance 2 X Z has a standard normal / n distribution X P z / 2 z / 2 1 / n P X z / 2 X z / 2 1 n n Confidence interval on the mean, variance known x z / 2 n x z / 2 n 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 x z / 2 n x z / 2 | x | z / 2 n , we have n z / 2 n | x | 2 If x is used as an estimate of , we can be 100(1 )% confident that the error | x | will not exceed a specified amount E when the sample size is z n /2 E 2 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 100(1 )% upper-confidence bound for is u xz n ◦ A 100(1 )% lower-confidence bound for is x z n l 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. method to derive a confidence General interval ◦ We find a statistic g ( X1 , X 2 ,..., X n ; ) that 1. g ( X 1 , X 2 ,..., X n ; ) depends on both the sample and g ( X 1 , X 2 ,..., X n ; ) 2. The probability distribution 2 of does not depend on and any other unknown parameter For example, g ( X1 , X 2 ,..., X n ; ) ( X ) /( / n ) ◦ Find constants CL and CU so that P[CL g ( X 1 , X 2 ,..., X n ; ) CU ] 1 P[ L( X 1 , X 2 ,..., X n ) U ( X 1 , X 2 ,..., X n )] 1 Large-sample confidence interval on the mean When n is large, the quantity X S/ n has an approximate standard normal distribution. Consequently, x z / 2 s s x z / 2 n n is a large-sample confidence interval for , with confidence level of approximately 100(1 )% . 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 ˆ ˆ has an approximate standard normal distribution. Consequently, ˆ z / 2 ˆ ˆ z / 2 ˆ 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 1 . 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, … ◦ n 53 , x 0.5250 , 0.3486 , z0.025 1.96 . ◦ 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 25 hours. A random sample of 20 bulbshas a mean life of x 1014 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. Confidence Interval on the Mean of a Normal Distribution,Variance Unknown Distribution Let X1 , X 2 ,…, X n are normally and independently distributed with unknown mean and unknown variance 2 . The random variable t X T S/ n has a t distribution with of freedom. n 1 degrees PDF of t distribution From Wikipedia, http://www.wikipedia.org. CDF of t distribution From Wikipedia, http://www.wikipedia.org. The t f ( x) probability density function [( k 1) / 2] 1 2 , x ( k 1) / 2 k (k / 2) [( x / k ) 1] k is the number of degrees of freedom Mean : 0 Variance : k /( k 2) for k 2 ◦ Percentage points t , k P(T t ,k ) t1 ,n t ,n 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 t P(t / 2,n 1 T t / 2,n 1 ) 1 P(t / 2,n 1 P( X t / 2,n 1 X t / 2,n 1 ) 1 S/ n S S X t / 2,n 1 ) 1 n n 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 x and s are the mean and standard deviation of a random sample from a normal distribution with unknown variance 2 , a 100(1 )% confidence interval on is given by S S X t / 2,n 1 X t / 2,n 1 n n ◦ where t / 2 ,n 1 is the upper 100 / 2 percentage point of the t distribution with n 1 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 x1 , x2 ,…, xn is arranged as x(1) , x( 2) ,…, x(n ) ,where x(1) is the smallest observation, x( 2) is the second-smallest observation, and so forth. ◦ The ordered observations x( j ) are then plotted against their observed cumulative frequency ( j 0.5) / n on the appropriate probability paper. ◦ Or, plot the standardized normal scores z j against x( j ) , where j 0.5 P( Z z j ) ( z j ) n 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. Percent-percent plot From Wikipedia, http://www.wikipedia.org. Example 8-5 Alloy Adhesion ◦ The load at specimen failure: 19.8, 10.1, … ◦ x 13.71 , s 3.55 , n 22 . ◦ 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, 3.05. ◦ (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. Confidence Interval on the Variance and Standard Deviation of a Normal Distribution 2 Distribution Let X1 , X 2 ,…, X n are normally and independently distributed mean and variance 2 , and let S 2 be the sample variance. The random variable X 2 (n 1) S 2 2 has a chi-square 2 distribution with degrees of freedom. n 1 PDF of 2 distribution From Wikipedia, http://www.wikipedia.org. CDF of 2 distribution From Wikipedia, http://www.wikipedia.org. 2 The probability density function 1 f ( x) k / 2 x ( k / 2)1e x / 2 , x 0 2 (k / 2) k is the number of degrees of freedom Mean : k Variance : 2k ◦ Percentage points 2,k P( X ,k ) 2 2 2 ,k f (u)du 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 X2 (n 1) S 2 2 ◦ is chi-square with n 1 degrees of freedom, we have P( 12 / 2,n1 X 2 2 / 2,n1 ) 1 P( 2 1 / 2 , n 1 P( (n 1) S 2 / 2,n 1 2 (n 1) S 2 2 / 2,n 1 ) 1 2 2 (n 1) S 2 2 1 / 2 , n 1 ) 1 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 s 2 is the sample variance from a random sample of n observations from a normal distribution with unknown variance 2 , then a 100(1 )% confidence interval on 2 is 2 (n 1) S 2 ( n 1 ) S 2 2 2 / 2,n 1 1 / 2,n 1 2 2 ◦ Where / 2,n1 and 1 / 2,n1 are the upper and lower percentage points of the chi-square distribution with ◦ n 1 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 100(1 )% lower and upper confidence bounds on 2 are (n 1) S 2 ,n 1 2 2 and 2 (n 1) S 2 12 ,n 1 ◦ 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 ◦ s 2 13.5 , n 20 . ◦ Find a 95% upper confidence bound on 2 and . Exercise 8-44 ◦ A rivet is to be inserted into a hole. A random sample of n 15 parts is selected, and the hole diameter is measured. The sample standard deviation of the hole diameter measurements is s 0.008 millimeters. Construct a 99% lower confidence bound for 2 . 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 Confidence Interval for a population proportion Normal approximation for a binomial proportion If n is large, the distribution of Z X np np(1 p) pˆ p p (1 p ) n is approximately standard normal. PMF of binomial distribution From Wikipedia, http://www.wikipedia.org. To construct the confidence interval on p , P( z / 2 Z z / 2 ) 1 P( z / 2 pˆ p z / 2 ) 1 p(1 p) n P pˆ z / 2 p(1 p) p pˆ z / 2 n p (1 p) 1 n P pˆ z / 2 pˆ (1 pˆ ) p pˆ z / 2 n pˆ (1 pˆ ) 1 n 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 p̂ is the proportion of observations in a random sample of size n that belongs to a class of interest, an approximate 100(1 )% confidence interval on the proportion p of the population that belongs to this class is pˆ z / 2 pˆ (1 pˆ ) p pˆ z / 2 n pˆ (1 pˆ ) n where z / 2 is the upper / 2 percentage of the standard normal distribution. Required: np 5 and n(1 p) 5 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 E | p pˆ | z / 2 p(1 p) / n Then 2 z n / 2 p (1 p ) E Or 2 z n / 2 (0.25) E 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 100(1 )% lower and upper confidence bounds are pˆ z pˆ (1 pˆ ) p and p pˆ z n pˆ (1 pˆ ) n 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 n 85 , x 10 , and pˆ x / n 10 / 85 0.12 Find a 95% two-sided confidence interval for p . ◦ Example 8-8 Crankshaft Bearings How large a sample is required if we want to be 95% confident that the error in using p̂ to estimate p 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. Tolerance and Prediction Intervals X n1 is a single future observation E ( X n 1 X ) 0 2 1 V ( X n 1 X ) (1 ) n n X n1 X Z Then 1 1 n has a standard normal distribution and 2 2 X n1 X T 1 S 1 n Has a t distribution with n 1 degrees of freedom. Prediction interval A 100(1 )% prediction interval (PI) on a single future observation from a normal distribution is given by 1 1 x t / 2,n1s 1 X n1 x t / 2,n1s 1 n n Tolerance interval A tolerance interval for capturing at least % of the values in a normal distribution with confidence level 100(1 )% is x ks, x ks where k 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 n 22 , x 13.,71 and s 3.55 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)