Chapter 8: Fundamental Sampling Distributions and Data Descriptions Introduction to random sampling and statistical inference Populations and samples Sampling distribution of means Central Limit Theorem Other distributions S2 t-distribution F-distribution Samples: 1000 engineering students selected at random from all engineering programs in the US. 50 cars selected at random from among those certified as having achieved 160 mph or more during 2003. OTHERS? EGR 252 - Ch. 8 9th edition JMB 2019

Populations and Samples Population: “a group of individual persons, objects, or items from which samples are taken for statistical measurement” 1 Sample: “a finite part of a statistical population whose properties are studied to gain information about the whole” 1 Population – the totality of the observations with which we are concerned 2 Sample – a subset of the population 2 1 (Merriam-Webster Online Dictionary, http://www.m-w.com/, October 5, 2004) 2 Walpole, Myers, Myers, and Ye (2007) Probability and Statistics for Engineers and Scientists EGR 252 - Ch. 8 9th edition JMB 2019

Examples Population Sample 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 9th edition JMB 2019

More Examples Population Sample Potato chips produced at the Frito-Lay plant in Kathleen 10 chips selected at random every 5 minutes as the conveyor passes the inspector Freshwater lakes and rivers 4 samples taken from randomly selected locations in randomly selected and representative freshwater lakes and rivers EGR 252 - Ch. 8 9th edition JMB 2019

Basic Statistics (review) Sample Mean: A class project involved the formation of three 10-person teams (Team Q, Team R and Team S). At the end of the project, team members were asked to give themselves and each other a grade on their contribution to the group. A random sample from two of the teams yielded the following results: = 87.5 = 85.0 Q R 92 85 95 88 75 78 XQ = 87.5 XR = 85 EGR 252 - Ch. 8 9th edition JMB 2019

Basic Statistics (review) Sample variance equation: For our example: Calculate the sample standard deviation (s) for each sample. SQteam = 7.59386 and SRteam = 7.25718 Q team sample R team sample 92 85 95 88 75 78 EGR 252 - Ch. 8 9th edition JMB 2019

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: EGR 252 - Ch. 8 9th edition JMB 2019

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, the limiting form of the distribution of is the standard normal distribution n(z;0,1) Note that this equation for Z applies when we have sample data. Compare to the Z equation for the population (Ch6). EGR 252 - Ch. 8 9th edition JMB 2019

Central Limit Theorem-Distribution of X 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 9th edition JMB 2019

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, See example 8.8, pg 249 and example 8.9, pg 250 EGR 252 - Ch. 8 9th edition JMB 2019

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, has a χ 2 distribution with ν = n - 1 EGR 252 - Ch. 8 9th edition JMB 2019

Chi-squared (χ2) Distribution α 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 9th edition JMB 2019

Example Look at example 8.7, pg. 245: 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 Degrees of freedom (v) = n-1 s2 = 0.815 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. 739-740) from book (& excel) – s2 = 0.815 χ2 = (n-1)s2 / σ2 = (4)(0.815)/1 = 3.26 looking for X2 values that cover 95%, meaning α values between 0.025 and 0.975 from table A.5 => Χ0.0252 =11.143 Χ0.9752 = 0.484 For alpha = 0.025, Χ2 =11.143 The Χ2 value for alpha = 0.975 is 0.484. 0.484 3.26 11.143 EGR 252 - Ch. 8 9th edition JMB 2019

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.) 95% NOTE the figure associated with table A.5!! (These values cover areas > the X2 value …) Degrees of freedom = ν= n-1 7-1 = 6; α = 0.05 X2 = 12.592 12.592 EGR 252 - Ch. 8 9th edition JMB 2019

t- Distribution Recall, by Central Limit Theorem: is n(z; 0,1) Assumption: σ known ( Z ) (Generally, if an engineer is concerned with a familiar process or system, this is reasonable, but …) assumption: EGR 252 - Ch. 8 9th edition JMB 2019

What if we don’t know σ? (sigma unknown) New statistic: Where, 2 and follows a t-distribution with ν = n – 1 degrees of freedom. EGR 252 - Ch. 8 9th edition JMB 2019

Characteristics of the t-Distribution Look at Figure 8.8, pg. 248 Note: Shape: _________________________ Effect of ν: __________________________ See table A.4, pp. 737-738 Note that the table yields the right tail of the distribution. shape – symmetrical about 0 effect of ν – variance, as seen in the width of the curve, depends on sample size note – as ν increases, curve looks more like a normal distribution (hence, the CLT) Table A.4 – critical values of t for several values of α and df. EGR 252 - Ch. 8 9th edition JMB 2019

F-Distribution Given: Then, 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, has an F-distribution with ν1 = n1 - 1 and ν2 = n2 – 1 degrees of freedom. (See table A.6, pp. 741-744) EGR 252 - Ch. 8 9th edition JMB 2019