Chapters 8 - 9

1. Chapters 8 - 9 Estimation Mat og metlar

2. Estimator and EstimateMetill og mat An estimator of a population parameter is a random variable that depends on the sample information and whose value provides approximations to this unknown parameter. A specific value of that random variable is called an estimate. Metill fyrir þýðisstika er hending sem er háð úrtaksupplýsingum og gildi metilsins sem kallast mat gefur nálgun á hinn óþekkta þýðisstika.

3. Point Estimator and Point Estimate Punktmetill og punktmat Let  represent a population parameter (such as the population mean  or the population proportion ). A point estimator, , of a population parameter, , is a function of the sample information that yields a single number called a point estimate. For example, the sample mean, , is a point estimator of the population mean , and the value that assumes for a given set of data is called the point estimate. Þýðisstiki (population parameter)

4. UnbiasednessÓhneigður (óbjagaður) The point estimator is said to be an unbiased estimator of the parameter  if the expected value, or mean, of the sampling distribution of is ; that is, Punktmetill er sagður óhneigður metill fyrir stikann  ef vongildi líkindadreifingar úrtaks fyrir er ; þ.e.,

5. Probability Density Functions for unbiased and Biased EstimatorsÞéttifall fyrir hneigðan og óhneigðan metil(Figure 8.1)

6. Bias Bjögun (skekkja) Let be an estimator of . The bias in is defined as the difference between its mean and ; that is It follows that the bias of an unbiased estimator is 0. Látum vera metil fyrir . Bjögun í er skilgreind sem mismunur milli vongildis metilsins og ; þ.e. Samkvæmt þessu er bjögun (bias) fyrir óhneigðan metil 0.

7. Most Efficient Estimator and Relative EfficiencySkilvirkasti metillinn og hlutfallsleg skilvirkni Suppose there are several unbiased estimators of . Then the unbiased estimator with the smallest variance is said to be the most efficient estimator or to be the minimum variance unbiased estimator of . Let and be two unbiased estimators of , based on the same number of sample observations. Then, a) is said to be more efficient than if b) The relative efficiency of with respect to is the ratio of their variances; that is, hlutfallsleg skilvirkni

8. Point Estimators of Selected Population Parameters(Table 8.1)

9. Confidence Interval EstimatorMetill fyrir öryggismörk A confidence interval estimator for a population parameter  is a rule for determining (based on sample information) a range, or interval that is likely to include the parameter. The corresponding estimate is called a confidence interval estimate. Metill fyrir öryggismörk á þýðisstika  er til að ákvarða (byggt á úrtaksgögnum) spönn, eða bil sem líklegt er til að ná utan um hinn sanna stika. Samsvarandi mat köllum við mat fyrir öryggismörk eða bara öryggismörk.

10. Confidence Interval and Confidence Level Let  be an unknown parameter. Suppose that on the basis of sample information, random variables A and B are found such that P(A <  < B) = 1 - , where  is any number between 0 and 1. If specific sample values of A and B are a and b, then the interval from a to b is called a 100(1 - )% confidence interval of . The quantity of (1 - ) is called the confidence level of the interval. If the population were repeatedly sampled a very large number of times, the true value of the parameter  would be contained in 100(1 - )% of intervals calculated this way. The confidence interval calculated in this manner is written as a <  < b with 100(1 - )% confidence. Látum  vera óþekktan stika. Hugsum okkur á að á grunni úrtaksupplýsinga séu hendingar A og B reiknaðar þannig að P(A <  < B) = 1 - , þar sem  er einhver tala milli 0 og 1. Ef ákveðin gildi A og B eru a and b, þá er bilið frá a til b kallað 100(1 - )% öryggismörk fyrir . Stærðin (1 - ) er kallað öryggsstig bilsins. Ef endurtekin úrtök væru tekin úr þýðinu mjög oft þá myndi 100(1 - )% allra þeirra bila sem reiknuð væri út innihalda hinn sanna stika . Öryggismörkin sem reiknuð eru á þennan hátt eru skrifuð sem a <  < b með 100(1 - )% vissu.

11. P(-1.96 < Z < 1.96) = 0.95, where Z is a Standard Normal Variable(Figure 8.3) 0.95 = P(-1.96 < Z < 1.96) 0.025 0.025 -1.96 1.96

12. Notation Táknmálsnotkun Látum Z/2 vera tölu sem Þar sem hendingin Z fylgir staðlaðri normaldreifingu Let Z/2 be the number for which where the random variable Z follows a standard normal distribution.

13. Selected Values Z/2 from the Standard Normal Distribution Table(Table 8.2)

14. Confidence Intervals for the Mean of a Population that is Normally Distributed: Population Variance KnownÖryggismörk fyrir meðaltal þýðis sem er normaldreift og með þekkta dreifni Consider a random sample of n observations from a normal distribution with mean  and variance 2. If the sample mean is X, then a 100(1 - )% confidence interval for the population mean with known variance is given by or equivalently, where the margin of error (also called the sampling error, the bound, or the interval half width) is given by

15. Basic Terminology for Confidence Interval for a Population Mean with Known Population VarianceOrðnotkun fyrir öryggismörk þýðismeðaltals með þekktri dreifni(Table 8.3)

16. Student’s t Distribution Given a random sample of n observations, with mean X and standard deviation s, from a normally distributed population with mean , the variable t follows the Student’s t distribution with (n - 1) degrees of freedom and is given by Hugsum okkur slembið úrtak n athugana með úrtaksmeðaltal X og úrtaksstaðalfrávik s, úrtakið er fengið úr þýði sem er normaldreift með vongildi , breytan t er sögð fylgja Student’s t dreifingu með (n - 1) frígráður og er gefin af

17. Notation Táknmálsnotkun A random variable having the Student’s t distribution with v degrees of freedom will be denoted tv. The tv,/2 is defined as the number for which Slembin breyta sem hefur Student’s t dreifingu með v frelsisgráður verður táknuð með tv. Stærðin tv,/2 er skilgreind sem stærðin sem

18. Confidence Intervals for the Mean of a Normal Population: Population Variance Unknown Öryggismörk fyrir vongildi í normaldreifðu þýði með óþekktri dreifni Suppose there is a random sample of n observations from a normal distribution with mean  and unknown variance. If the sample mean and standard deviation are, respectively, X and s, then a 100(1 - )% confidence interval for the population mean, variance unknown, is given by or equivalently, where the margin of error, the sampling error, or bound, B, is given by and tn-1,/2 is the number for which The random variable tn-1 has a Student’s t distribution with v=(n-1) degrees of freedom.

19. Confidence Intervals for Population Proportion (Large Samples) Öryggismörk fyrir þýðishlutfall (Stór úrtök) Let p denote the observed proportion of “successes” in a random sample of n observations from a population with a proportion  of successes. Then, if n is large enough that (n)()(1- )>9, then a 100(1 - )% confidence interval for the population proportion is given by or equivalently, where the margin of error, the sampling error, or bound, B, is given by and Z/2, is the number for which a standard normal variable Z satisfies

20. Notation Táknmálsnotkun A random variable having the chi-square distribution with v = n-1 degrees of freedom will be denoted by 2v or simply 2n-1. Define as 2n-1, the number for which Hending með chi-square dreifingu þar sem v = n-1 frelsisgráður er táknuð með 2v eða 2n-1. Skilgreinum 2n-1, sem töluna sem um gildir að

21. The Chi-Square Distribution(Figure 8.17) 1 -   0 2n-1,

22. The Chi-Square Distribution for n – 1 and (1-)% Confidence Level(Figure 8.18) /2 /2 1 -  2n-1,1- /2 2n-1,/2

23. Confidence Intervals for the Variance of a Normal Population Öryggismörk fyrir dreifni í normaldreifðu þýði Suppose there is a random sample of n observations from a normally distributed population with variance 2. If the observed variance is s2 , then a 100(1 - )% confidence interval for the population variance is given by Hugsum okkur slembið úrtak n gagna úr normaldreifðu þýði með dreifni 2. Ef úrtaksdreifni er s2 , þá eru 100(1 - )% öryggismörk fyrir þýðisdreifni gefin sem where 2n-1,/2 is the number for which and 2n-1,1 - /2 is the number for which And the random variable 2n-1 follows a chi-square distribution with (n – 1) degrees of freedom. Og hendingin 2n-1 fylgir chi-square dreifingu með (n – 1) frelsisgráður

24. Confidence Intervals for Two Means: Matched Pairs Öryggismörk fyrir tvö vongildi : Pör (Matched Pairs) Suppose that there is a random sample of n matched pairs of observations from a normal distributions with means X and Y . That is, x1, x2, . . ., xn denotes the values of the observations from the population with mean X ; and y1, y2, . . ., yn the matched sampled values from the population with mean Y . Let d and sd denote the observed sample mean and standard deviation for the n differences di = xi – yi . If the population distribution of the differences is assumed to be normal, then a 100(1 - )% confidence interval for the difference between means (d = X - Y) is given by or equivalently,

25. Confidence Intervals for Two Means: Matched Pairs(continued) Where the margin of error, the sampling error or the bound, B, is given by And tn-1,/2 is the number for which The random variable tn – 1, has a Student’s t distribution with (n – 1) degrees of freedom.

26. Confidence Intervals for Difference Between Means: Independent Samples (Normal Distributions and Known Population Variances) Öryggismörk fyrir mismun vongilda: Óháð úrtök Suppose that there are two independent random samples of nx and ny observations from normally distributed populations with means X and Y and variances 2x and 2y . If the observed sample means are X and Y, then a 100(1 - )% confidence interval for (X - Y) is given by or equivalently, where the margin of error is given by

27. Confidence Intervals for Two Means: Unknown Population Variances that are Assumed to be EqualÖryggismörk fyrir mismun vongilda: Óþekkt dreifni en dreifnin er eins skv. Forsendu. Suppose that there are two independent random samples with nx and ny observations from normally distributed populations with means X and Y and a common, but unknown population variance. If the observed sample means are X and Y, and the observed sample variances are s2X and s2Y, then a 100(1 - )% confidence interval for (X - Y) is given by or equivalently, where the margin of error is given by

28. Confidence Intervals for Two Means: Unknown Population Variances that are Assumed to be Equal(continued) The pooled sample variance, s2p, is given by is the number for which The random variable, T, is approximately a Student’s t distribution with nX + nY –2 degrees of freedom and T is given by,

29. Confidence Intervals for Two Means: Unknown Population Variances, Assumed Not Equal Suppose that there are two independent random samples of nx and ny observations from normally distributed populations with means X and Y and it is assumed that the population variances are not equal. If the observed sample means and variances are X, Y, and s2X , s2Y, then a 100(1 - )% confidence interval for (X - Y) is given by where the margin of error is given by

30. Confidence Intervals for Two Means: Unknown Population Variances, Assumed Not Equal(continued) The degrees of freedom, v, is given by If the sample sizes are equal, then the degrees of freedom reduces to

31. Confidence Intervals for the Difference Between Two Population Proportions (Large Samples) Öryggismörk fyrir mismun þýðishlutfalla (stór úrtök) Let pX, denote the observed proportion of successes in a random sample of nX observations from a population with proportion X successes, and let pY denote the proportion of successes observed in an independent random sample from a population with proportion Y successes. Then, if the sample sizes are large (generally at least forty observations in each sample), a 100(1 - )% confidence interval for the difference between population proportions (X - Y) is given by Where the margin of error is

32. Sample Size for the Mean of a Normally Distributed Population with Known Population Variance Gagnasafn fyrir vongildi normaldreifðs þýðis með þekktri þýðisdreifni Suppose that a random sample from a normally distributed population with known variance 2 is selected. Then a 100(1 - )% confidence interval for the population mean extends a distance B (sometimes called the bound, sampling error, or the margin of error) on each side of the sample mean, if the sample size, n, is

33. Sample Size for Population Proportion Stærð gagnasafns fyrir þýðishlutfall Suppose that a random sample is selected from a population. Then a 100(1 - )% confidence interval for the population proportion, extending a distance of at most B on each side of the sample proportion, can be guaranteed if the sample size, n, is

34. Bias Bound Confidence interval: For mean, known variance For mean, unknown variance For proportion For two means, matched For two means, variances equal For two means, variances not equal For variance Confidence Level Estimate Estimator Interval Half Width Lower Confidence Limit (LCL) Margin of Error Minimum Variance Unbiased Estimator Most Efficient Estimator Point Estimate Point Estimator Key Words

35. Relative Efficiency Reliability Factor Sample Size for Mean, Known Variance Sample Size for Proportion Sampling Error Student’s t Unbiased Estimator Upper Confidence Limit (UCL) Width Key Words(continued)