Survey

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Document related concepts

no text concepts found

Transcript

Populations and Samples Population: “a group of individual persons, objects, or items from which samples are taken for statistical measurement” Sample: “a finite part of a statistical population whose properties are studied to gain information about the whole” (Merriam-Webster Online Dictionary, http://www.m-w.com/, October 5, 2004) EGR 252 - Ch. 8 8th edition Spring 2008 Slide 1 Examples Population Samples Students pursuing undergraduate engineering degrees 1000 engineering students selected at random from all engineering programs in the US Cars capable of speeds in excess of 160 mph. 50 cars selected at random from among those certified as having achieved 160 mph or more during 2003 EGR 252 - Ch. 8 8th edition Spring 2008 Slide 2 Examples (cont.) Population Potato chips produced at the FritoLay plant in Kathleen Freshwater lakes and rivers EGR 252 - Ch. 8 8th edition Samples 10 chips selected at random every 5 minutes as the conveyor passes the inspector 4 samples taken from randomly selected locations in randomly selected and representative freshwater lakes and rivers Spring 2008 Slide 3 Basic Statistics (review) n Sample Mean: X X i 1 i n At the end of a team project, team members were asked to give themselves and each other a grade on their contribution to the group. The results for two team members were as follows: Q S 92 85 95 88 85 75 78 92 EGR 252 - Ch. 8 8th edition X Q = ___________________ 87.5 X S = ___________________ 85.0 Spring 2008 Slide 4 Basic Statistics (review) 1. Sample Variance: n S2 ( X i 1 i X )2 n 1 n n i 1 i 1 n X i2 ( X i )2 n(n 1) For our example: Q S 92 85 95 88 85 75 78 92 SQ2 = ___________________ SS2 = ___________________ S2Q = 7.593857 S2S = 7.25718 EGR 252 - Ch. 8 8th edition Spring 2008 Slide 5 Sampling Distributions If we conduct the same experiment several times with the same sample size, the probability distribution of the resulting statistic is called a sampling distribution Sampling distribution of the mean: if n observations are taken from a normal population with mean μ and variance σ2, then: ... x n 2 2 2 2 2 ... 2 x 2 n n EGR 252 - Ch. 8 8th edition Spring 2008 Slide 6 Central Limit Theorem Given: X : the mean of a random sample of size n taken from a population with mean μ and finite variance σ2, Then, the limiting form of the distribution of X Z , n / n is the standard normal distribution n(z;0,1) EGR 252 - Ch. 8 8th edition Spring 2008 Slide 7 Central Limit Theorem If the population is known to be normal, the sampling distribution of X will follow a normal distribution. Even when the distribution of the population is not normal, the sampling distribution of X is normal when n is large. NOTE: when n is not large, we cannot assume the distribution of X is normal. EGR 252 - Ch. 8 8th edition Spring 2008 Slide 8 Sampling Distribution of the Difference Between Two Averages Given: Two samples of size n1 and n2 are taken from two populations with means μ1 and μ2 and variances σ12 and σ22 Then, X X 1 2 1 X2 2 1 2 1 X 2 n1 2 2 n2 and Z ( X 1 X 2 ) ( 1 2 ) 1 2 n1 EGR 252 - Ch. 8 8th edition 2 2 n2 Spring 2008 Slide 9 Sampling Distribution of S2 Given: If S2 is the variance of of a random sample of size n taken from a population with mean μ and finite variance σ2, Then, 2 (n 1)s 2 2 n i 1 ( X i X )2 2 has a χ2 distribution with ν = n - 1 EGR 252 - Ch. 8 8th edition Spring 2008 Slide 10 χ2 Distribution χ2 χα2 represents the χ2 value above which we find an area of α, that is, for which P(χ2 > χα2 ) = α. EGR 252 - Ch. 8 8th edition Spring 2008 Slide 11 Example Look at example 8.10, pg. 256: A manufacturer of car batteries guarantees that his batteries will last, on average, 3 years with a standard deviation of 1 year. A sample of five of the batteries yielded a sample variance of 0.815. Does the manufacturer have reason to suspect the standard deviation is no longer 1 year? μ = 3 σ = 1 n = 5 s2 = 0.815 2 (n 1) s 2 2 (4)(0.815) 3.26 1 If the χ2 value fits within an interval that covers 95% of the χ2 values with 4 degrees of freedom, then the estimate for σ is reasonable. (See Table A.5, pp. 755-756) χ2 Χ20.025 =11.143 EGR 252 - Ch. 8 8th edition Χ20.975 = 0.484 Spring 2008 Slide 12 Your turn … If a sample of size 7 is taken from a normal population (i.e., n = 7), what value of χ2 corresponds to P(χ2 < χα2) = 0.95? (Hint: first determine α.) χ2 12.592 EGR 252 - Ch. 8 8th edition Spring 2008 Slide 13 t- Distribution Recall, by CLT: X Z / n is n(z; 0,1) Assumption: _____________________ (Generally, if an engineer is concerned with a familiar process or system, this is reasonable, but …) EGR 252 - Ch. 8 8th edition Spring 2008 Slide 14 What if we don’t know σ? New statistic: X T S/ n Where, n X i 1 Xi n ( Xi X ) and S n 1 i 1 n 2 follows a t-distribution with ν = n – 1 degrees of freedom. EGR 252 - Ch. 8 8th edition Spring 2008 Slide 15 Characteristics of the t-Distribution Look at fig. 8.11, pg. 221*** Note: Shape: _________________________ Effect of ν: __________________________ See table A.4, pp. 753-754 EGR 252 - Ch. 8 8th edition Spring 2008 Slide 16 Comparing Variances of 2 Samples Given two samples of size n1 and n2, with sample means X1 and X2, and variances, s12 and s 22 … Are the differences we see in the means due to the means or due to the variances (that is, are the differences due to real differences between the samples or variability within each samples)? See figure 8.16, pg. 226 EGR 252 - Ch. 8 8th edition Spring 2008 Slide 17 F-Distribution Given: S12 and S22, the variances of independent random samples of size n1 and n2 taken from normal populations with variances σ12 and σ22, respectively, Then, S12 / 12 22S12 F 2 2 2 2 S2 / 2 1 S2 has an F-distribution with ν1 = n1 - 1 and ν2 = n2 – 1 degrees of freedom. (See table A.6, pp. 757-760) EGR 252 - Ch. 8 8th edition Spring 2008 Slide 18

Related documents