1. Biostatistics Dr. Chenqi Lu Telephone: 021-55665269 E-mail: luchenqi@fudan.edu.cn@fudan.edu.cn Office: 2309 GuangHua East Main Building

2. Population Sample Parameter Statistic Random Sampling Sampling Distribution Estimate parameter Statistical inference

3. Chapter 4. Sampling Distributions § 4.1. Concept of random samples A sample from any given population is recognized as a random sample if every individual in the population is equally likely to be present in that sample. i.e. a random sample is a “fair” presentation of the population. An example of wing shape inheritance in Drosophila Vv x Vv ¼ VV ½ Vv ¼ vv (vestigial wing) randomly sampling 20 flies Estimated value of p: p = 0.25 of vv genotype Only those samples containing 5 vv individuals give, the probability is probability frequency

4. B(n,p)EVarInterval Binomialxnpnpq1 Binomial Proportion x/nppq/n1/n

5. 0.2023 0.1444 0.1025 0.0727 x0123456x0123456

6. Larger sample size tend to improve the probability that is close to p but decreases as n increases

7. § 4.2. Distribution of the sample mean Let X = {X 1, X 2, …, X n } be a random sample of size n from a population with mean  and variance  2, and {X i } are i.i.d. It is clear that is a r.v. Its mean and variance are given by Mean of a random sample equals mean of the population from which the sample is collected Variance of a random sample is proportional to its population variance to a factor 1/n Independent identical distribution

8. Revisit of the example of two rolls of a die (sampling with replacement )

9. the example of two from six balls (sampling without replacement ) 0 1 2 3 2 1 0 15

10. If a random sample is from a population exhibiting normal distribution mean  and variance  2, then the sample mean as a r.v. follows normal distribution with Standard deviation of mean is also referred as to standard error In fact, if the sample size (n) is large enough, the sample mean approaches to a normal distribution no matter what population distribution the sample is from! (Central Limit Theorem)

11. § 4.2. Confidence Intervals for population mean In many cases where population size is very large or even infinity, it is not practical to calculate the exact value of population mean. In many practices, it is useful to have ideas about the range of value of the population mean with certain confidence. Sampling distribution of a sample mean allows this to be done. In fact For a given probability confidence 1-  0.05 -1.96 0.95 1.96

12. is called the 1-  confidence interval of population mean  When  = 0.05

13. In practice, calculation of C.I. for population mean needs , the population standard deviation which is unknown but can be estimated from sample s.d., s. However is no longer a normal distributed r.v. Its distribution follows a t-distribution with d.f. = n-1. Thus, estimated C.I. of  from a sample is

14. § 4.3. Confidence Intervals for population variance Let s 2 be variance of a random sample from a population with mean normal standard deviation

15. In a similar way but need to consider asymmetry in  2 distribution, the C.I. for population variance  2 can be constructed for a given probability confidence that is, Calculation of using Minitab MiniTab => Calculate => Probability Distributions => Chi-square => Inverse CP (DF, input constant)