Sampling and Sampling Distributions Úrtak og úrtaksdreifingar Chapter 7 Sampling and Sampling Distributions Úrtak og úrtaksdreifingar

Simple Random Sample Einfalt slembiúrtak Suppose that we want to select a sample of n objects from a population of N objects. A simple random sample is selected such that every object has an equal probability of being selected and the objects are selected independently - -the selection of one object does not change the probability of selecting any other objects. Simple random samples are the ideal sample. In a number of real world sampling studies analysts develop alternative sampling procedures to lower the costs of sampling. But the basis for determining if these strategies are acceptable is to determine how closely they approximate a simple random sample. Hugsum okkur að við viljum velja úrtak n hluta úr þýði N hluta. Einfalt slembiúrtak er valið þannig að sérhvert stak hefur jafnmiklar líkur á að vera valið og stökin eru valin óháð hvert öðru. Val á einu staki hefur ekki áhrif á líkurnar á vali annars staks. Einföld slembiúrtök eru eftirsóknarverð úrtök. Í veruleikanum er leitast við að þróa úrtaksaðferðir sem byggja á því að lágmarka kostnað við úrtakið. En grunnurinn til að ákvarða hvort þessar úrtaksaðferðir eru ásættanlegar er að hversu miklu leyti aðferðin nálgast að vera slembiúrtak.

Sampling Distributions Consider a random sample selected from a population to make an inference about some population characteristic, such as the population mean, by using a sample statistic such as a sample mean, . The inference is based on the realization that every random sample would have a different number for and thus is a random variable. The sampling distribution of this statistic is the probability distribution of the values it could take over all possible samples of the same number of observations drawn from the population. Hugsum okkur slembið úrtak úr þýði sem valið er til að álykta um einkenni þýðisins. T.d. þýðismeðaltalið með því að nota einfalda lýsistærð (statistic) eins og úrtaksmeðaltalið . Ályktunin byggir á að fengin niðurstaða byggir á slembnu úrtaki. Og fyrir hvert slembið úrtak kemur ólík niðurstaða fyrir . Þannig er úrtaksmeðaltalið hending. Úrtaksdreifing þessarar lýsistærðar er líkindadreifing þeirra mögulegu gilda sem getur tekið fyrir öll möguleg úrtök sem dregin eru úr þýðinu.

Sampling Distribution of the Sample Means from the Worker Population (Table 7.2) Probability of 4.50 1/15 4.75 2/15 5.00 5.25 5.50 5.75 3/15 6.00 6.25 6.75

Sample Mean Úrtaksmeðaltal Let X1, X2, . . . Xn be a random sample from a population. The sample mean value of these observations is defined as Látum X1, X2, . . . Xn vera slembið úrtak úr þýði. Þá er úrtaksmeðaltalið skilgreint sem:

Standard Normal Distribution for the Sample Mean Stöðluð normaldreifing fyrir úrtaksmeðaltalið Whenever the sampling distribution of the sample mean is a normal distribution we can compute a standardized normal random variable, Z, that has mean 0 and variance 1 Þegar dreifing úrtaksmeðaltals er normaldreifð þá getum við umreiknað stærðina í staðlaða normaldreifða stærð Z.

Results for the Sampling Distribution of the Sample Mean Let X denote the sample mean of a random sample of n observations from a population with a mean X and variance 2. Then The sampling distribution of X has mean Vongildi úrtaksdrefingar The sampling distribution of X has standard deviation This is called the standard error of X. If the sample size is not small compared to the population size N, then the standard error of X is If the population distribution is normal, then the random variable Has a standard normal distribution with mean 0 and variance 1.

Central Limit Theorem Miðleitnisetningin Let X1, X2, . . . , Xn be a set of n independent random variables having identical distributions with mean  and variance 2, with X as the sum and X as the mean of these random variables. As n becomes large, the central limit theorem states that the distribution of approaches the standard normal distribution. Látum X1, X2, . . . , Xn vera mengi n óháðra hendinga sem eru eins dreifðar hver með vongildi  og dreifni 2, með X sem summu og X sem meðaltal þessara hendinga. Eftir því sem n stækkar mun dreifingin sem hér er leidd út, nálgast hina stöðluðu normaldreifðu hendingu.

Sample Proportions Úrtakshlutfall Let X be the number of successes in a binomial sample of n observations, with parameter . The parameter  is the proportion of the population members that have a characteristic of interest. We define the sample proportion as The sum X is the sum of a set of n independent Bernoulli random variables each with a probability of success . As a result p is the mean of a set of independent random variables and the results developed in the previous sections for sample means apply. In addition the central limit theorem can be used to argue that the probability distribution for p can be modeled as a normal. Látum X vera fjölda heppnaðra atburða í úrtaki tvíliðunardreifingar með n athugunum og með stikanum . Stikinn  er hlutfall þýðismeðlima sem hafa ákveðin einkenni. Við skilgreinum úrtakshlutfall Summan X er summa af n óháðum bernoulli hendingum. Þar sem hver hending hefur líkur  á að atburður heppnist. Þannig er p meðalfjöldi þeirra atburða óháðra hendinga sem heppnaðist. Niðurstöður um úrtaksmeðaltal gilda því. Ennfremur gefur miðleitnisetningin það til kynna að dreifing p nálgist Normal í stórum úrtökum.

Sampling Distribution of the Sample Proportion Dreifing úrtakshlutfallsins Let p denote the sample proportion of successes in a random sample from a population with proportion of success . Then The sampling distribution of p has mean  The sampling distribution of p has standard deviation If the sample size is large, the random variable is approximately distributed as a standard normal. The approximation is good if

Sample Variance Úrtaksdreifni Let X1, X2, . . . , Xn be a random sample from a population. The quantity Látum X1, X2, . . . , Xn vera slembið úrtak úr þýði. Þá er stærðin Is called the sample variance and its square root s is called the sample standard deviation. Given a specific random sample we would compute the sample variance and the sample variance would be different for each random sample, because of differences in sample observations. Kölluð úrtaksdreifni og kvaðratrótin úrtaksstaðalfrávik. Fyrir hvert úrtak sem tekið er tekur þessi stærð mismunandi gildi því er þessi stærð hending með ákveðna líkindadreifingu.

Sampling Distribution of the Sample Variances Líkindadreifing úrtaksdreifni Let s2X denote the sample variance for a random sample of n observations from a population with variance 2. Then Látum s2X tákna úrtaksdreifni slembins úrtaks n athugana úr þýði með dreifni 2. Þá gildir The sampling distribution of s2 has mean 2 Hendingin s2 hefur vongildi 2 The variance of the sampling distribution of s2X depends on the underlying population distribution. If that distribution is normal, then Dreifni s2X er háður dreifingu gagna þýðisins ef hún er normal þá If the population distribution is normal then (n-1)s2/ 2 is distributed as 2(n-1) Ef dreifing þýðisins er normaldreifð þá mun (n-1)s2/ 2 vera dreifð sem 2(n-1)

Key Words Central Limit Theorem Chi-Square Distribution Sample Mean Sample Proportion Sample Variance Sampling Distribution of the Sample Mean Sampling Distribution of the Sample Proportion Sampling Distribution of the Sample Variance Simple Random Sample Standard Normal for Sample Mean Statistics and Sampling Distribution