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Class 5 Estimating Confidence Intervals Estimation of • Imagine that we do not know what is, so we would like to estimate it. • In order to get a point estimate of , we would take a sample and compute x . Is there a better way? Should we use our sample to compute something else that would yield better guesses of ? • If you take a sample and (1) multiple the sample values by any amount that you like, and (2) add the results together, you can not do better than x . Estimation (cont.) • For example, take a sample of size 4 from a normal population and compute W X1 X 2 X 2 X 3 , 4 • where Xi is the ith sample value. Then W has a normal distribution, E(W) = , but Var(W ) Var( X ). • What does this mean about W? Estimation (cont.) • An estimator, Y, of is said to be unbiased if E(Y) = . Thus, W and X are unbiased. • In fact, it can be shown that X is the best linear unbiased estimator (BLUE) of in the sense that, among all linear unbiased estimators, it has the smallest variance. Building Interval Estimates: The Confidence Interval • We do not know , but if we did (and if we had a large enough sample), we would know exactly how X was distributed. • This tells us where X will probably fall. But we have a different problem: we see X. Where does probably fall? • For a given probability, this is called a confidence interval. The Confidence Interval • Let z be the point on the standard normal distribution that cuts off % in the upper What information comes tail. from the sample? • A 100(1-)% confidence interval for • when the normality of X is justifiable, and • is known: x z / 2 X ( x z / 2 X , x z / 2 X ) ( x z / 2 / n , x z / 2 / n ) Probability Statements About the Sampling Error There is a 1 - probability that the value of a sample mean will provide a sampling error of z / 2 x or less. Sampling distribution of X /2 1x - of all values /2 Example • Incomes in a community are known to be normally distributed with = $2000. In order to compute a 90% confidence interval for , you take a sample of 400 incomes and determine that x = $24,000. • What is z/2? Then Example (cont.) • What is true about this interval? • How would it change if we contructed a 95% confidence interval? Example (cont.) • Now assume that you wish to construct a 95% confidence interval using a sample of 1600. • If x = $24,000, then our interval is (23,902, 24,098). • We would like our interval to be small. What are the two things we can change? Confidence Intervals for - Unknown • If you did not know , what would you use to estimate ? • To construct a 100(1-)% Confidence Interval for when is unknown, compute: x t / 2,n 1 s n Where t/2,n-1 is the value on the t distribution with n-1 degrees of freedom that cuts off /2 of the distribution in the upper tail. t distributions • The t distribution is a family of similar probability distributions. • A specific t distribution depends on a parameter known as the degrees of freedom. • As the number of degrees of freedom increases, the difference between the t distribution and the standard normal probability distribution becomes smaller and smaller. • A t distribution with more degrees of freedom has less dispersion. t distributions t Value: Assume a sample of size 10. At 95% confidence, 1 - = .95, = .05, and /2 = .025. t.025,9 is based on n - 1 = 10 - 1 = 9 degrees of freedom. In the t distribution table we see that t.025,9 = 2.262. Degrees of Freedom . 7 8 9 10 . .10 . 1.415 1.397 1.383 1.372 . Area in Upper Tail .05 .025 .01 . 1.895 . 2.365 . 2.998 1.860 1.833 2.306 2.262 2.896 2.821 1.812 . 2.228 . 2.764 . .005 . 3.499 3.355 3.250 3.169 . t distributions (Once again) • A t distribution is mound shaped and symmetric about 0. • There is a different t distribution for every different degree of freedom. • The t distribution approaches the z (standard normal) distribution as n. • t.1,20 = t.05,10 = • t.025, = t.025,30 = Example • Incomes in a community are normally distributed. A sample of size 4 is taken, and the following incomes are found: 21,000 24,500 22,500 22,000 • Estimate the mean income of the community with a 95% confidence interval. x t / 2,n 1 s Let’s begin by doing this in EXCEL. n Our example in EXCEL • Input the data into an EXCEL column. • Select tools/data analysis/descriptive statistics. • Input data range and output range. • Select summary statistics and confidence level for mean. • How was the standard error computed? • What is the ratio of the confidence level and the standard error? Summary Large n Small n x z / 2 x z / 2 known n n ( population normal ) n x t / 2,n1 s x t / 2,n 1 s unknown x z / 2 s n n ( population normal ) Note: The expression s n is frequently referred to as the standard error.