1. Biostatistics Unit 5 – Samples

2. Sampling distributions Sampling distributions are important in the understanding of statistical inference. Probability distributions permit us to answer questions about sampling and they provide the foundation for statistical inference procedures.

3. Definition The sampling distribution of a statistic is the distribution of all possible values of the statistic, computed from samples of the same size randomly drawn from the same population. When sampling a discrete, finite population, a sampling distribution can be constructed. Note that this construction is difficult with a large population and impossible with an infinite population.

4. Construction of sampling distributions 1. From a population of size N, randomly draw all possible samples of size n. 2. Compute the statistic of interest for each sample. 3. Create a frequency distribution of the statistic.

5. Properties of sampling distributions We are interested in the mean, standard deviation and appearance of the graph (functional form) of a sampling distribution.

6. Types of sampling distributions We will study the following types of sampling distributions. A) Distribution of the sample mean B) Distribution of the difference between two meansDistribution of the sample meanDistribution of the difference between two means C) Distribution of the sample proportion D) Distribution of the difference between two proportionsDistribution of the sample proportionDistribution of the difference between two proportions

7. Sampling distribution of Given a finite population with mean (  ) and variance (   ). When sampling from a normally distributed population, it can be shown that the distribution of the sample mean will have the following properties.

8. Properties of the sampling distribution 1. The distribution of will be normal 2. The mean, of the distribution of the values of, will be the same as the mean of the population from which the samples were drawn; = . 3. The variance,, of the distribution of, will be equal to the variance of the population divided by the sample size; =.

9. Standard error The square root of the variance of the sampling distribution is called the standard error of the mean or the standard error.

10. Nonnormally distributed populations When the sampling is done from a nonnormally distributed population, the central limit theorem is used.

11. The central limit theorem Given a population of any nonnormal functional form with mean (  ) and variance (  2 ), the sampling distribution of, computed from samples of size n from this population will have mean, , and variance,  2 /n, and will be approximately normally distributed when the sample is large (30 or higher).

12. The central limit theorem Note that the standard deviation of the sampling distribution is used in calculations of z scores and is equal to

13. Example Given the information below, what is the probability that x is greater than 53? (1) Write the given information  = 50  = 16 n = 64 x = 53

14. Example (2) Sketch a normal curve

15. Example (3) Convert x to a z score

16. Example (4) Find the appropriate value(s) in the table A value of z = 1.5 gives an area of.9332. This is subtracted from 1 to give the probability P (z > 1.5) =.0668

17. Example (5) Complete the answer The probability that x is greater than 53 is.0668.

18. Distribution of the difference between two means It often becomes important to compare two population means. Knowledge of the sampling distribution of the difference between two means is useful in studies of this type. It is generally assumed that the two populations are normally distributed.

19. Sampling distribution of Plotting sample differences against frequency gives a normal distribution with mean equal to which is the difference between the two population means.

20. Variance The variance of the distribution of the sample differences is equal to Therefore, the standard error of the differences between two means would be equal to

21. Converting to a z score To convert to the standard normal distribution, we use the formula We find the z score by assuming that there is no difference between the population means.

22. Sampling from normal populations This procedure is valid even when Sampling from normal populations the population variances are different or when the sample sizes are different. Given two normally distributed populations with means, and, and variances, and, respectively. (continued)

23. Sampling from normal populations The sampling distribution of the difference,, between the means of independent samples of size n 1 and n 2 drawn from these populations is normally distributed with mean,, and variance,

24. Example In a study of annual family expenditures for general health care, two populations were surveyed with the following results: Population 1: n 1 = 40, = $346 Population 2: n 2 = 35, = $300

25. Example If the variances of the populations are = 2800 and = 3250, what is the probability of obtaining sample results as large as those shown if there is no difference in the means of the two populations?

26. Solution (1) Write the given information n 1 = 40, = $346, = 2800 n 2 = 35, = $300, = 3250

27. Solution (2) Sketch a normal curve

28. Solution (3) Find the z score

29. Solution (4) Find the appropriate value(s) in the table A value of z = 3.6 gives an area of.9998. This is subtracted from 1 to give the probability P (z > 3.6) =.0002

30. Solution (5) Complete the answer The probability that is as large as given is.0002.

31. Distribution of the sample proportion ( ) While statistics such as the sample mean are derived from measured variables, the sample proportion is derived from counts or frequency data.

32. Properties of the sample proportion Construction of the sampling distribution of the sample proportion is done in a manner similar to that of the mean and the difference between two means. When the sample size is large, the distribution of the sample proportion is approximately normally distributed because of the central limit theorem.

33. Mean and variance The mean of the distribution,, will be equal to the true population proportion, p, and the variance of the distribution,, will be equal to p(1-p)/n.

34. The z-score The z-score for the sample proportion is

35. Example In the mid seventies, according to a report by the National Center for Health Statistics, 19.4 percent of the adult U.S. male population was obese. What is the probability that in a simple random sample of size 150 from this population fewer than 15 percent will be obese?

36. Solution (1) Write the given information n = 150 p =.194 Find P( <.15)

37. Solution (2) Sketch a normal curve

38. Solution (3) Find the z score

39. Solution (4) Find the appropriate value(s) in the table A value of z = -1.36 gives an area of.0869 which is the probability P (z < -1.36) =.0869

40. Solution (5) Complete the answer The probability that <.15 is.0869.

41. Distribution of the difference between two proportions This is for situations with two population proportions. We assess the probability associated with a difference in proportions computed from samples drawn from each of these populations. The appropriate distribution is the distribution of the difference between two sample proportions.

42. Sampling distribution of The sampling distribution of the difference between two sample proportions is constructed in a manner similar to the difference between two means. (continued)

43. Sampling distribution of Independent random samples of size n 1 and n 2 are drawn from two populations of dichotomous variables where the proportions of observations with the character of interest in the two populations are p 1 and p 2, respectively.

44. Mean and variance The distribution of the difference between two sample proportions,, is approximately normal. The mean is The variance is These are true when n 1 and n 2 are large.

45. The z score The z score for the difference between two proportions is given by the formula

46. Example In a certain area of a large city it is hypothesized that 40 percent of the houses are in a dilapidated condition. A random sample of 75 houses from this section and 90 houses from another section yielded difference,, of.09. If there is no difference between the two areas in the proportion of dilapidated houses, what is the probability of observing a difference this large or larger?

47. Solution (1) Write the given information n 1 = 75, p 1 =.40 n 2 = 90, p 2 =.40 =.09 Find P(.09)

48. Solution (2) Sketch a normal curve

49. Solution (3) Find the z score

50. Solution (4) Find the appropriate value(s) in the table A value of z = 1.17 gives an area of.8790 which is subtracted from 1 to give the probability P (z > 1.17) =.121

51. Solution (5) Complete the answer The probability of observing of.09 or greater is.121.

52. fin