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)

Examples Population Samples Students pursuing undergraduate engineering degrees Cars capable of speeds in excess of 160 mph. Potato chips produced at the Frito-Lay plant in Kathleen Freshwater lakes and rivers Samples 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. 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 OTHERS?

Basic Statistics (review) 1. Sample Mean: Example: 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: = ___________________ XQ = 87.5 XS = 85 Q S 92 85 95 88 75 78

Basic Statistics (review) 1. Sample Variance: For our example: SQ2 = ___________________ SS2 = ___________________ Q S 92 85 95 88 75 78 S2Q = 7.593857 S2S = 7.25718

Your Turn Work in groups of 4 or 5. Find the mean, variance, and standard deviation for your group of the (approximate) number of hours spent working on homework each week.

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:

Central Limit Theorem Given: Then, 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 is _________________________ The standard normal distribution n(z;0,1)

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.

Example: The time to respond to a request for information from a customer help line is uniformly distributed between 0 and 2 minutes. In one month 48 requests are randomly sampled and the response time is recorded. What is the probability that response time is between 0.9 and 1.1 minutes? μ =______________ σ2 = ________________ μX =__________ σX2 = ________________ Z1 = _____________ Z2 = _______________ P(0.9 < X < 1.1) = _____________________________ f(x) = ½, 0<x<2 (uniform dist) μ = (b-a)/2 = 1 σ2 = (b-a)2/12 = 1/3 μx = 1 σx2 = (1/3)/48 = 1/144 Z1 = (.9-1)/(1/12) = -1.2 Z2 = (1.1-1)/(1/12) = 1.2 P(Z2) – P(Z1) = 0.8849-.1151 =0.7698