Statistical Methods II&III: Confidence Intervals ChE 477 (UO Lab) Lecture 5 Larry Baxter, William Hecker, & Ron Terry Brigham Young University

Population vs. Sample Statistics Population statistics –Characterizes the entire population, which is generally the unknown information we seek –Mean generally designated as  –Variance & standard deviation generally designated as  , and , respectively Sample statistics –Characterizes the random sample we have from the total population –Mean generally designated –Variance and standard deviation generally designated as s 2 and s, respectively

Overall Approach Use sample statistics to estimate population statistics Use statistical theory to indicate the accuracy with which the population statistics have been estimated Use trends indicated by theory to optimize experimental design

Data Come From pdf

Histogram Approximates a pdf

All Statistical Info Is in pdf Probabilities are determined by integration. Moments (means, variances, etc.) Are obtained by simple means. Most likely outcomes are determined from values.

Gaussian or Normal pdf Pervasive

Properties of a Normal pdf About 68.26%, 95.44%, and 99.74% of data lie within 1, 2, and 3 standard deviations of the mean, respectively. When mean is zero and standard deviation is 1, it is referred to as a standard normal distribution. Plays fundamental role in statistical analysis because of the Central Limit Theorem.

Lognormal Distributions Used for non-negative random variables. –Particle size distributions. –Drug dosages. –Concentrations and mole fractions. –Duration of time periods. Similar to normal pdf when variance is < 0.04.

Student’s t Distribution Widely used in hypothesis testing and confidence intervals Equivalent to normal distribution for large sample size

Central Limit Theorem Possibly most important single theory in applied statistics Deals with distributions of normalized sample and population means Not quite applicable because it assumes population mean and variance are known

Central Limit Theorem Distribution of means calculated from data from most distributions is approximately normal –Becomes more accurate with higher number of samples –Assumes distributions are not peaked close to a boundary

Student’s t Distribution Used to compute confidence intervals according to Assumes mean and variance estimated by sample values

Values of Student’s t Distribution Depends on both confidence level being sought and amount of data. Degrees of freedom generally n -1, with n = number of data points (assumes mean and variance are estimated from data and estimation of population mean only). This table assumes two- tailed distribution of area.

Sample Size Is Important Confidence interval decreases proportional to inverse of square root of sample size and proportional to decrease in t value. Limit of t value is normal distribution. Limit of confidence interval is 0.

Theory Can Be Taken Too Far Accuracy of measurement ultimately limits confidence interval to something greater than 0. Not all sample means are appropriately treated using central limit theorem and t distribution.

Typical Numbers Two-tailed analysis Population mean and variance unknown Estimation of population mean only Calculated for 95% confidence interval Based on number of data points, not degrees of freedom

An Example Five data points with sample mean and standard deviation of and 107.8, respectively. The estimated population mean and 95% confidence interval is:

Point vs. Model Estimation Point estimation –Characterizes a single, usually global value –Generally simple mathematics and statistical analysis –Procedures are unambiguous Model development –Characterizes a function of dependent variables –Complexity of parameter estimation and statistical analysis depend on model complexity –Parameter estimation and especially statistics somewhat ambiguous

Overall Approach Assume model Estimate parameters Check residuals for bias or trends Estimate parameter confidence intervals Consider alternative models

Data Come From pdf

Histogram Approximates a pdf

General Confidence Interval Degrees of freedom generally = n-p, where n is number of data points and p is number of parameters Confidence interval for parameter given by

Linear Fit Confidence Interval For intercept: For slope:

Definition of Terms

Confidence Interval for Y at a Given X