Rule of thumb: If the normal approximation will be satisfactory regardless of the shape of the population. (If n is evaluated from more than 30 values, we will consider hat s is a good point estimate for o. ) Otherwise, we need to use a different shape of probability distribution function that will consider an error on o. Lecture 12 3 Confidence Interval on the mean, small sample () & Variance unknown Let XSL, XX, ,Oxen be a random sample from a normal distribution with unknown mean and variance.
The random variable: Has a t distribution with n-l degrees of freedom. 4 Confidence Interval on the mean, small sample (insurance unknown) 5 Operating Characteristic curves (probabilities for errors on t-test) d= 6 Confidence Interval on the mean, variance unknown. Double handed: – TA /2,n-1 < p /2,n-1 Single Handed, upper confidence bound: Single Handed, lower confidence bound p2x-ta ,n-l 7 Course Outline in Engineering (Chapter 2. Data Summary and Presentaon 2) 3.
Random Variables Distributions 3) 4. Decision Making Single Sample 4) 5. Decision Two samples 5) 6. Building Empirical Models 6) 7. Design of Experiments 7) Book sections Inference on the variance of a normal population 199-204 Chapter 5. Inference on the mean of two populations, variances known 232-238 9 Confidence Interval on the Variance of a normal population Let XSL, XX,… Oxen be a random sample from a normal striation with mean p and variance o, and let SO be the sample variance.
The random variable: Has a chi-square (V) distribution with n-l degrees of freedom Hypothesis testing on the variance The confidence interval the variance is given by X a /2,n-1 X I-a /2,n-1 11 Problem The variance for one machine preparing ibuprofen pills is problematic. You have the following dataset : 185, 199, 205, 183, 208, 201, 195, 196, 196, 203. 0=0. 05) a) b) c) d) Calculate the sample variance. Is the standard deviation statistically higher than 7 MGM? Calculate the P-Value on your test. What is the double sided 95% Confidence interval on the ample variance? 13 )Tuttle .
Staccatos 1) presentation . Random probability Distributions for TWO Hypothesis Testing on Two-samples Simple comparative experiments, I. E. Study the difference in two parameters e. G. Comparing two conditions (= treatments) to determine whether either condition produces a significant effect on the response 0 2 different catalysts prepared by different methods – does one perform significantly better than the other one? Mean and Variance of the difference between the mean of two Random Samples The difference between between the mean of two random samples, D = X -X is a normal variable with: Mean .