Sampling Distributions. Chapter 9

Transcription

1 Sampling Distributions Chapter 9

2 First, a word from our textbook A statistic is a numerical value computed from a sample. EX. Mean, median, mode, etc. A parameter is a numerical value determined by the entire population and is assumed that the value is fixed,unchanging and unknown.

3 Introduction to Statistics and Sampling Variability Consider a small population consisting of the board of directors of a day care center. Board member and number of children: Jay Carol Allison Teresa Anselmo Bob Roxy Vishal Find the average number of children for the entire group of eight: m = 2 children

4 Discovery question ONE: How is the parameter of the population related to a sampling distribution based on the population?

5 Introduction to Statistics and Sampling Variability Board member and number of children: Jay Carol Allison Teresa Anselmo Bob Roxy Vishal List all possible samples of size 2. Calculate the average number of children represented by the group. Samples: Jay Carol 5 2 x 3.5 Jay Allison 5 1 x 3

6 Answer question ONE: the average of all possible values for a sampling distribution will equal the population parameter mx m

7 Variability of a statistic What is the relationship between the population parameter and each sample statistic? The observed value of a statistic will vary from sample to sample. This fact is called sampling variability.

8 Sampling distributions If we calculated using only the first 3 columns of values, would we get the same results? Explain. How did the spread change from the population to the sampling distribution?

9 Definition In summary, a sampling distribution is the distribution of all possible values for a given sample size for a fixed population. Sampling distribution applet

10 Discovery question TWO: For a normal population, how will the shape and spread of a sampling distribution change as we increase the sample size? Population distribution: m = 16 s = 5

11 Discovery question TWO: m 16 m 16 X X s X s X m X mx s X s X

12 Answer question TWO: For a normal population, the shape of the sampling distribution remains mound shaped and symmetrical (taller/thinner)for all sample sizes. We can conclude the sampling distribution remains approximately normal. The standard deviation for the sampling distribution is equal to the population standard deviation divided by the square root of the sample size. s X s n

13 Sample means parameter statistic mean m x standard deviation s s sampling distribution m x s X Formulas: mx m s X s n

14 Example ONE The average sales price of a single-family house in the United States is $243,756. Assume that the sales prices are normally distributed with a standard deviation of $44,000.

15 Draw the normal distribution. Within what range would the middle 68% of the houses fall? m $243,756 s $44,000 $199,756 $243,756 $287,756

16 s x Draw the sampling distribution for a sample size of 4 houses. Within what range would the middle 68% of the samples of size 4 houses fall? m $243,756 x $44,000 s $44,000 4 s $22,000 x $221,756 $243,756 $265,756

17 Draw the sampling distribution for a sample size of 16 houses. Within what range would the middle 68% of the samples of size 16 houses fall? m x s x s x $243,756 $44, $11,000 $232,756 $243,756 $254,756

18 Draw the sampling distribution for a sample size of 25 houses. Within what range would the middle 68% of the samples of size 25 houses fall? m x s x s x $243,756 $44, $8,800 $234,956 $243,756 $252,556

19 Example TWO Suppose the mean room and board expense per year at a certain four-year college is $7,850. You randomly select 9 dorms offering room and board near the college. Assume that the room and board expenses are normally distributed with a standard deviation of $1125.

20 Draw the population distribution. m $7,850 s 1125 $4,475 $5,600 $6,725 $7,850 $8,975 $10,100 $11,225

21 What is the probability that a randomly dorm has room and board of less than $8,180? m $7,850 $8,180 s $1125 Px ( 8180) $4,475 $5,600 $6,725 $7,850 $8,975 $10,100 $11,225

22 What is the probability that a randomly dorm has room and board of less than $8,180? Given normal distribution m $7,850 s $1125 Px ( 8180).6141 z x m s

23 Draw the sampling distribution for a sample size of 9 dorms. m $7,850 x 1125 s x 9 $375 $6,725 $7,100 $7,475 $7,850 $8,225 $8,600 $8,975

24 What is the probability that the mean room and board of the nine dorms is less than $8,180? m $7,850 x 1125 s x 9 $375 $8,180 Px ( 8180) $6,725 $7,100 $7,475 $7,850 $8,225 $8,600 $8,975

25 What is the probability that the mean room and board of the nine dorms is less than $8,180? Given normal distribution m x $7,850 Px ( 8180) 1125 s x $375 z x s m n

26 What is the probability that the mean cost of a sample of four dorms is more than $7,250? Given normal distribution m x $7,850 Px ( 7250) 1125 s $ x z x s m n

27 Central Limit Theorem Take a random sample of size n from any population with mean m and standard deviation s. When n is large, the sampling distribution of the sample mean is close to the normal distribution. How large a sample size is needed depends on the shape of the population distribution.

28 Uniform distribution Sample size 1

29 Uniform distribution Sample size 2

30 Uniform distribution Sample size 3

31 Uniform distribution Sample size 4

32 Uniform distribution Sample size 8

33 Uniform distribution Sample size 16

34 Uniform distribution Sample size 32

35 Triangle distribution Sample size 1

36 Triangle distribution Sample size 2

37 Triangle distribution Sample size 3

38 Triangle distribution Sample size 4

39 Triangle distribution Sample size 8

40 Triangle distribution Sample size 16

41 Triangle distribution Sample size 32

42 Inverse distribution Sample size 1

43 Inverse distribution Sample size 2

44 Inverse distribution Sample size 3

45 Inverse distribution Sample size 4

46 Inverse distribution Sample size 8

47 Inverse distribution Sample size 16

48 Inverse distribution Sample size 32

49 Parabolic distribution Sample size 1

50 Parabolic distribution Sample size 2

51 Parabolic distribution Sample size 3

52 Parabolic distribution Sample size 4

53 Parabolic distribution Sample size 8

54 Parabolic distribution Sample size 16

55 Parabolic distribution Sample size 32

56 Loose ends An unbiased statistic falls sometimes above and sometimes below the actual mean, it shows no tendency to over or underestimate. As long as the population is much larger than the sample (rule of thumb, 10 times larger), the spread of the sampling distribution is approximately the same for any size population.

57 Loose ends As the sampling standard deviation continually decreases, what conclusion can we make regarding each individual sample mean with respect to the population mean m? As the sample size increases, the mean of the observed sample gets closer and closer to m. (law of large numbers)