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1、CHAPTER-6Sampling error and confidence intervalspopulationpopulationsamplesamplestatisticstatisticParameterParametererrorSection 1 sampling error of meanSection 2 t distribution Section 3 confidence intervals for the population meanSection 1 sampling error of mean A simple random sample is a sample
2、of size n drawn from a population of size N in such a way that every possible random samples n has the same probability of being selected. Variability among the simple random samples drawn from the same population is called sampling variability, and the probability distribution that characterizes so
3、me aspect of the sampling variability, usually the mean but not always, is called a sampling distribution. These sampling distributions allow us to make objective statements about population parameters without measuring every object in the population.Example 1 The population mean of DBP in the Chine
4、se adult men is 72mmHg with standard deviation 5mmHg. 10 adult participants was chosen randomly from the Chinese adult men, here we can calculate the sample mean and sample standard deviation. Supposing sampling 100 times, whats the result? linkageN If random samples are repeatedly drawn from a popu
5、lation with a mean and standard deviation , we can find:1 the sample means are different from the others 2 The sample mean are not necessary equal to population mean 3 The distribution of sample mean is symmetric about HOW TO EXPLORE THE SAMPLING DISTRIBUTION FOR THE MEAN? The difference between sam
6、ple statistics and population parameter or the difference among sample statistics are called sampling error.vIn real life we sample only once, but we realize that our sample comes from a theoretical sampling distribution of all possible samples of a particular size. The sampling distribution concept
7、 provides a link between sampling variability and probability. Choosing a random sample is a chance operation and generating the sampling distribution consists of many repetitions of this chance operation.When sampling from a normally distributed population with mean , the distribution of the sample
8、 mean will be normal with mean Central limit Theorem = 50= 50= 50 =10=10=10X X XPopulation Population Population distributiondistributiondistributionn n = 4 = 4Sampling Sampling Sampling distributiondistributiondistributionXn n =16 =16When sampling from a nonnormally distributed population with mean
9、 , the distribution of the sample mean will be approximately normal with mean as long as n is larger enough (n50).Central limit TheoremX X Standard error (SE) can be used to assess sampling error of mean. Although sampling error is inevitable, it can be calculated accurately.theoretical value of SEe
10、stimation of SECalculation of standard error (SE)sSEnSElinkagevExample 5.2 One analyst chose randomly a sample (n=100) and measured their weights with a mean of 72kg and standard deviation of 15kg. Question: what is the standard error?Solution: Exercise 5.1 Consider a sample of measurement 100 with
11、mean 121cm and standard deviation 7cm drawn from a normal population. Try to compute its standard error.Solution:Section 2 t distribution1. Definition N(, 2)N(0, 1)Random samplingUsually standard deviation is unknown, so we can only get s, then we can calculate This sampling distribution was develop
12、ed by W.S Gossett and published under the pseudonym “student” in 1908. it is, therefore, sometimes called the “students t distribution and is really a family of distributions dependent on the n-1. n-1Z distributiont distribution2. the characteristics of t distribution graphFIG 4 the graph of t distr
13、ibution with different degrees of freedomv1 symmetric about 0;v2 the shape of t curve is determined by degree of freedom, df=n-1.v3 t-distribution is approximated to standard normal distribution when n is infinite. t critical value with one-sided probability t(, )t critical value with two-sided prob
14、ability t t( (/2, ) )Example 5.2 With n=15, find t0 such that P(-t0t t0 )=0.90solution From t value table, df=15-1=14, the two-tailed shaded area equals 0.10, so -t0=-1.761 and t0 =1.761 Section 3 confidence intervals for the population meandescriptive statisticsdescriptive statistics1. Basic concep
15、ts Parameter estimation: Deduce the population parameter basing on the sample statisticsPoint Estimate A single-valued estimate.A single element chosen from a sampling distribution.Conveys little information about the actual value of the population parameterabout the accuracy of the estimate.Confide
16、nce Interval or Interval Estimation An interval or range of values believed to include the unknown population parameter.Point Point estimationestimationLower limitLower limitUpper limitUpper limitIntervals estimation1 - 1 - a a a aa a a a/ /2 2a a a a/ /2 2 2. MethodsZ distribution1. is known2. is u
17、nknown ，n50 t distribution is unknown，n5050CICICICI Example 5.3 A horticultural scientist is developing a new variety of apple. One of the important traits, in addition to taste, color, and storability, is the uniformity of the fruit size. To estimate the weight she samples 100 mature fruit and calc
18、ulates a sample mean of 220g and standard deviation 5g Develop 95% confidence intervals for the population mean from her samplesolution 95% confidence intervals for the population mean is between 219.02 and 220.98gExercise A forester is interested in estimating the average number of count trees per
19、acre. A random sample of n=64 one acre is selected and examined. The average (mean) number of count trees per acre is found to be 27.3, with a standard deviation of 12.1. Use this information to construct 95% confidence interval for .solution 95% confidence intervals for the population mean is betwe
20、en 24.36 and 30.24 The forester is 95% confident that the population mean for “count trees” per acre is between 24.36 and 30.24Example 5.4 The ecologist samples 25 plants and measures their heights. He finds that the sample has a mean of 15cm and a sample deviation of 4cm . what is the 95% confidenc
21、e interval for the population mean solutiondf=25-1=24 The plant ecologist is 95% confident that the population mean for heights of these plants is between 13.349 and 16.651cmExercise 1 one doctor samples 25 men and measures their heights. He finds that the sample has a mean of 172.12cm and a sample
22、deviation of 4.50cm . what is the 95% confidence interval for the population mean solution 95% confidence intervals for the population mean is between 170.26 and 173.98 Exercise 2 Random samples of size 9 are repeatedly drawn from a normal distribution with a mean of 65 and a standard deviation of 18. Describe the sampling distribution of mean.65Lower limitLower limitUpper limitUpper limitPROBLEM1.What are the difference of SD and SE?2.What is the medical reference range? What is the confidence intervals for population mean?
