#### Transcript Ch 8A Statistical Intervals

Statistical Intervals for a Single Sample From only one sample, An interval has been found. Because the sample was ample, The results were quite profound! - author unknown circa 2007 Chapter 8A What to Look Forward to this Week Today only! A Diversion – the sampling distributions or distributions arising from the normal If Z1, Z2, ..., Zn are independent standard normal random variables, then Z Z ... Z 2 1 T 2 2 Z 2 / n 2 n ( n) chi-square distribution with n degrees freedom t-distribution with n degrees freedom t ( n) ( n) / n F 2 (m) / m 2 2 F (n, m) F-distribution with n degrees freedom in the numerator and m degrees of freedom in the denominator What will be Chi-square? Let Xi be the ith sample value from a normal population Xi n( , 2 ) Z Xi n(0,1) Therefore n Xi i 1 n 2 X i 1 i However (n 1) S 2 n 2 n 2 2 X i 1 i X 2 2 n 1 The Chi-square Distribution k = degrees of freedom The t-distribution also known as Student’s t f(x) = v = k = df More about Student The t statistic was introduced by William Sealy Gosset for cheaply monitoring the quality of beer brews. "Student" was his pen name. Gosset was a statistician for the Guinness brewery in Dublin, Ireland, and was hired due to Claude Guinness's innovative policy of recruiting the best graduates from Oxford and Cambridge to apply biochemistry and statistics to Guinness' industrial processes. Gosset published the t test in Biometrika in 1908, but was forced to use a pen name by his employer who regarded the fact that they were using statistics as a trade secret. The F-distribution f(x) B(m,n) is the Beta function an interesting property: F-Distribution named after R.A. Fisher Born 17 February 1890(1890-02-17) East Finchley, London , England Died 29 July 1962 (aged 72) Adelaide, Australia Residence England, Australia Nationality British, Field Statistics, Genetics, Natural selection Institutions Rothamsted Experimental Station University College London, Cambridge University Alma mater Cambridge University Academic advisor Sir James Jeans F.J.M. Stratton Notable students C.R. Rao Known for Maximum likelihood Fisher information Analysis of variance Notable prizes Royal Medal (1938) Copley Medal (1955) Three types of Intervals -Confidence Interval – bound population parameter or distribution parameter. -Tolerance Interval – bound a proportion of the distribution at a certain confidence level. - Prediction Interval – bound a single observation => assumptions on population distribution critical here. Overheard at a rest stop I know that my average driving time on this daily route has been 2.3 hours over the last 7 days. However, that is based on a sample and therefore is unlikely to equal my population mean. What I really need is some way to measure how precise this estimate is. A typical prob-stat graduate I am 95% confident that my mean driving time to work is between 37.4 minutes and 41.2 minutes. 41.2 – 37.4 = 3.8 Best estimate is midpoint = 39.3 Error = 1.9 minutes Confidence Interval - A statement consisting of two values between which the population parameter is estimated to lie. Reliability – degree of confidence – the probability with which the population parameter will be “captured by the two values. Precision – the length of the confidence interval (a measure of the error in estimating the parameter. The Big Picture of a Confidence Interval (L,U) is 100(1-)% CI for the population parameter The Bigger Picture of a Confidence Interval General Approach: Estimate our point estimate our precision reliability Factor x Standard Error our confidence The Biggest Picture of a Confidence Interval population parameter Pr l x1 ,..., xn u x1,..., xn 1 Measure of Uncertainty (precision) (Random Variables; i.e. statistics Measure of Risk (1 – )% of the C.I.s constructed this way contain the mean. Watch the interpretation of this concept. The length of a confidence interval is a measure of the precision of estimation. Confidence Interval on Mean of Normal Distribution Variance Known X Z has a standard normal distribution / n X P z /2 z /2 1 / n with a little algebra, P X z /2 / n X z /2 / n 1 This is our 100(1 )% Confidence Interval on . z /2 is the upper /2 percentage point from standard normal Our Very First Real Confidence Interval A sample of 100 batteries are tested for their operating life. They averaged (mean) 10 hours before failing. The manufacturer has assured us that the population variance is 16 hours. Find a 95 percent confidence interval for the mean life of this particular type of battery. N 100, x 10 hr., 4 hr. z.025 1.96 X z /2 4 10 1.96 (9.216,10.784) n 100 Sample Size and Precision P X z / 2 / n X z / 2 / n 1 Equivalent ly : x z / 2 / n Define the error by x E Choose n as below and we can be 100(1 - )% confident that the error will not exceed E. z / 2 n E 2 Think of E as a measure of practical significance. Our Very First Real Confidence Interval Revisited For our battery problem, what sample size is required to reduce the error to .5 hr. with a 99% confidence? 2 z 2.58 x16 /2 n 426.0096 426 2 E .5 2 Problem 8-12 Life of a 75 watt bulb is normally distributed with std dev = 25 hrs. Suppose we want to be 95% confident that the error in estimating mean life is less than 5 hours. Find a sample size. z /2 1.96(25) n 96.04 96 5 E 2 2 Problem 8-10 Diameter of holes for a cable harness is normally distributed with a standard deviation of .01 in. A random sample of 10 yields average diameter of 1.5045 in. Find a 99% two-sided confidence interval. x z 0 .005 / n x z 0 .005 / n 1 . 5045 2 . 58 ( 0 . 01 ) / 10 1 . 5045 2 . 58 ( 0 . 01 ) / 10 1 . 4963 1 . 5127 Interpreting a Confidence Interval The confidence interval is a random interval The appropriate interpretation of a confidence interval (for example on ) is: The observed interval [l, u] brackets the true value of , with confidence 100(1-). Examine Figure 8-1 on the next slide. Figure 8-1 Repeated construction of a confidence interval for Repeated Confidence Intervals, gen. samples Random samples from a standard normal distribution, N(0,1). Generated in Excel as NORMSINV(RAND()) Sample 1 2 3 4 5 6 1.886 1.040 1.091 -1.618 -1.246 0.301 1.014 -1.008 -0.447 0.874 -0.386 -1.002 -1.534 -0.225 1.393 0.484 0.222 1.791 0.192 0.374 1.105 -1.761 -0.326 -0.212 96 97 98 99 100 -0.778 0.277 0.725 -0.227 -0.425 -0.977 0.646 2.182 -1.575 -0.328 0.361 -0.693 -0.855 1.890 0.475 -1.247 -1.306 -0.831 0.497 1.241 0.801 0.168 -0.012 -0.653 0.969 1.403 -0.045 -1.311 0.359 0.211 0.210 -0.429 0.607 -1.986 -0.432 0.225 0.669 -0.213 -0.489 0.295 0.408 1.409 -0.579 -1.439 -1.518 1.695 0.824 -0.071 -1.772 0.743 -1.639 -0.542 0.641 0.647 -1.070 0.749 0.487 -1.450 -0.306 -0.052 -0.313 1.165 1.423 -2.964 0.149 1.355 1.244 -1.589 0.399 1.576 0.666 -0.219 -1.099 1.832 -1.120 1.015 0.994 -1.123 -0.908 0.566 -0.171 1.273 -1.278 0.090 -0.073 0.983 Repeated Confidence Intervals, hits and misses mean std dev 90% lower 90% upper 95% lower 95% upper -0.104 0.375 0.327 0.064 0.335 0.071 -0.346 0.403 0.217 0.717 0.807 0.929 1.117 1.426 1.214 1.080 0.531 1.124 -0.476 -0.044 -0.155 -0.515 -0.405 -0.559 -0.907 0.127 -0.366 0.268 0.793 0.809 0.643 1.074 0.701 0.214 0.678 0.800 -0.549 -0.125 -0.249 -0.628 -0.549 -0.681 -1.016 0.074 -0.480 0.340 0.875 0.903 0.756 1.219 0.823 0.324 0.732 0.914 0.534 0.387 -0.212 0.469 0.305 0.374 -0.024 1.022 0.750 1.253 0.759 0.612 0.745 0.489 0.004 -0.002 -0.862 0.076 -0.012 -0.013 -0.278 1.064 0.776 0.438 0.863 0.622 0.760 0.230 -0.099 -0.078 -0.988 -0.001 -0.074 -0.088 -0.327 1.168 0.852 0.564 0.940 0.684 0.836 0.279 Totals -> 90% 95% misses misses 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 1 0 0 0 14 0 0 0 0 0 0 0 9 Since these are not 10 and 5, respectively, is there an error? Repeated Confidence Intervals, .90 n p 100 How are these probabilities being generated? Let X = RV, number of misses. Then X ~ Bin(100, .1) E[X] = np = 10 misses 0.1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 prob(p) Cum Prob 0.0000 0.0000 0.0003 0.0003 0.0016 0.0019 0.0059 0.0078 0.0159 0.0237 0.0339 0.0576 0.0596 0.1172 0.0889 0.2061 0.1148 0.3209 0.1304 0.4513 0.1319 0.5832 0.1199 0.7030 0.0988 0.8018 0.0743 0.8761 0.0513 0.9274 0.0327 0.9601 0.0193 0.9794 0.0106 0.9900 0.0054 0.9954 0.0026 0.9980 0.0012 0.9992 Repeated Confidence Intervals, .95 n p 100 misses 0.05 0 1 2 3 4 5 6 7 8 9 10 11 12 prob(p) Cum Prob 0.0059 0.0059 0.0312 0.0371 0.0812 0.1183 0.1396 0.2578 0.1781 0.4360 0.1800 0.6160 0.1500 0.7660 0.1060 0.8720 0.0649 0.9369 0.0349 0.9718 0.0167 0.9885 0.0072 0.9957 0.0028 0.9985 Major Point: Can you see how probability helps us assess the risk associated with statistical inference? One-sided Confidence Bounds Our Very First Real Confidence Interval Revisited Again A One-Sided Confidence Interval Based upon the sample of 100 batteries averaging (mean) 10 hours to failure. The manufacturer continues to assure us that the population variance is 16 hours. Find a 95 percent lower confidence interval for the mean life of this particular type of battery. z.05 1.6449 X z 4 10 1.6449 9.342 hr. n 100 A Transition… The previous development of a confidence interval was limited in two ways: - Needed a Normal population - Needed to know the standard deviation of the Normal distribution The Central Limit Theorem eliminates the need to explicitly know the population is normal – Z will still be approximately standard normal We can estimate using the sample standard deviation, s. Confidence Interval on Mean of Normal – Variance Unknown Same form as for the normal – measure of risk is now from the t distribution, and we boldly use the sample standard deviation – protected by the heavy-tailed t distribution-even when sample size is small! Remember we are still assuming that observations from the underlying population are normally distributed. 100(1 ) percent confidence interval : x t / 2,n 1s / n x t / 2,n 1s / n where t / 2,n 1 is the upper 100 / 2 percentage point of a t distributi on with n 1 d.f.' s. Where Does it Come From? Do we care? From earlier: X Z n(0,1) / n (n 1) S 2 X n 1 2 numerator is standard normal denominator is chisquare divided by d.f. (n-1) T X / n X S/ n (n 1) S 2 2 n 1 T has a t distribution with n-1 degrees of freedom. Are we still caring? X P t /2 t /2 1 S/ n with a little algebra, P X t /2 S / n X t /2 S / n 1 This is our 100(1 )% Confidence Interval on . t / 2 is the upper /2 percentage point from t-distribution Confidence Interval on Mean of Normal – Variance Unknown For large samples the distributional assumption is not critical. If sample size is not large use the t distribution X T has a t distributi on with n - 1 degrees of freedom S/ n if X 1 , X 2 ,..., X n are a sample from a normal distributi on with unknown mean and variance. As n ∞, t distribution becomes standard normal. t Distribution Converges to Standard Normal For large sample size, use the normal distribution even if the variance is unknown. The t distribution k 1 / 2 1 f ( x) x ( k 1) / 2 2 k k / 2 x / k 1 t ,k is the value of the random variable T with k degrees of freedom above which we have area (probabili ty) . Usually k is the number of degrees of freedom associated with the sample standard deviation (n - 1). Our Very First CI using the t-distribution Sulfur dioxide and nitrogen oxide are products of fossil fuel consumption. These compounds can be carried long distances and converted to acid before being deposited in the form of “acid rain.” The following sulfur dioxide concentrations (in micrograms per cubic meter) were obtained from different locations in a forest though to have been damaged by acid rain. Estimate the mean 52.7 43.9 41.7 71.5 47.6 55.1 concentration in the forest. 62.2 45.3 52.4 56.5 63.4 38.6 33.4 53.9 46.1 61.8 65.5 44.4 54.3 66.6 60.7 50 70 56.4 n 24, x 53.92, s 2 101.48, t.025,23 2.069 x t /2,n1s / n x t /2, n1s / n 53.92 2.069 Average concentration in undamaged areas is 20 g/m3 10.07 10.07 53.92 2.069 (49.67,58.17) 24 24 It’s Official Now Stay Tuned – next time…