1. Where Are You?
Children Adults

2. Generating Sampling Distributions
Section 7.1
Generating Sampling Distributions

3. Why Statistics?

4. Why Statistics?
Statistics allow us to make inferences (or draw conclusions) about a population.

5. Population
How do we describe populations?

6. Population How do we describe populations?
If we can graph data about the population, we can use shape, center, and spread.

7. Population How do we describe populations?
If we can graph data about the population, we can use shape, center, and spread.
We may use summary numbers such as the mean ( ), standard deviation ( ), median, quartiles, etc.

8. Population
What do we call a summary number that describes a population or a probability distribution?

9. Population
What do we call a summary number that describes a population or a probability distribution?
parameter

10. Summary Statistic
What is a summary statistic?

11. Summary Statistic
Summary statistic: a summary number calculated from a sample taken from a population

12. Summary Statistic Examples: sample mean x, standard deviation ,
5-number summary

13. Sample statistic Population parameter

14. Sampling Distribution
Suppose we take a random sample of a fixed size n from our population and compute a summary statistic.

15. Sampling Distribution
Suppose we take a random sample of a fixed size n from our population and compute a summary statistic.
Then, suppose we repeat this process many times.

16. Sampling Distribution
Sampling distribution: distribution of summary statistics you get from taking repeated random samples

17. Sampling Distribution
Sampling distribution: distribution of summary statistics you get from taking repeated random samples
Answers the question “How does my summary statistic behave when I repeat the process many times?”
(Think Law of Large Numbers).

18. Sampling Distribution
Two types of sampling distributions:

19. Sampling Distribution
Two types of sampling distributions:
Exact Sampling Distribution

20. Sampling Distribution
Two types of sampling distributions:
Exact Sampling Distribution
Approximate Sampling Distribution

21. Exact Sampling Distributions
When the population size is very small,

22. Exact Sampling Distributions
When the population size is very small, you can construct sampling distributions exactly by listing distribution of summary statistic for all possible samples for a fixed size, n.

23. Exact Sampling Distributions
Utah has five national parks. Your company has been hired to make maps of two of these parks, which will be selected at random.
1. Construct the sampling distribution for the total number of square miles you would map.
2. Find P(map > 600 mi2)

24. Exact Sampling Distributions
Utah has five national parks. Your company has been hired to make maps of two of these parks, which will be selected at random.
How many possible samples of size n = 2 are there?

25. Exact Sampling Distributions
Utah has five national parks. Your company has been hired to make maps of two of these parks, which will be selected at random.
How many possible samples of size n = 2 are there? 5C2 = 10 samples

26. Exact Sampling Distributions
Construct the sampling distribution for the total number of square miles you would map. Find P(map > 600 mi2).

27. Exact Sampling Distributions

28. Find P(map > 600 mi2).

29. Find P(map > 600 mi2) = 4/10

30. Exact Sampling Distributions

31. Should we always use exact sampling distributions?

32. Always use Exact Sampling Distribution?
Suppose you have a population of 100 rectangles with varying dimensions and you had to construct a sampling distribution for a sample of size 5.
How many ways can you choose 5 rectangles at a time from the population of 100?

33. Always use Exact Sampling Distribution?
Suppose you have a population of 100 rectangles with varying dimensions and you had to construct a sampling distribution for a sample of size 5.
How many ways can you choose 5 rectangles at a time from the population of 100? 100C5 = 75,287,520
Would you construct an exact sampling distribution here?

34. Approximate Sampling Distribution
Approximate sampling distribution is AKA simulated sampling distribution.

35. Approximate Sampling Distribution
Approximate sampling distribution is AKA simulated sampling distribution.
4-step process:

36. Approximate Sampling Distribution
Approximate sampling distribution is AKA simulated sampling distribution.
4-step process:
1. Take random sample of fixed size n from population

37. Approximate Sampling Distribution
Approximate sampling distribution is AKA simulated sampling distribution.
4-step process:
1. Take random sample of fixed size n from population
2. Compute summary statistic of interest
For example: mean, median, min, or max

38. Approximate Sampling Distribution
Approximate sampling distribution is AKA simulated sampling distribution.
4-step process:
1. Take random sample of fixed size n from population
2. Compute summary statistic of interest
3. Repeat steps 1 and 2 many times

39. Approximate Sampling Distribution
Approximate sampling distribution is AKA simulated sampling distribution.
4-step process:
1. Take random sample of fixed size n from population
2. Compute summary statistic of interest
3. Repeat steps 1 and 2 many times
4. Display distribution of the summary statistic

41. Each dot represents 1 rectangle.
Display 7.2, p. 411
Each dot represents 1 rectangle.

42. Display 7.2, p. 411
Shape: skewed right Center: μx = Spread: σx = 5.2

43. Now,
5 rectangles were selected at random

44. Now,
5 rectangles were selected at random
Mean area of these five was calculated

45. Now,
5 rectangles were selected at random
Mean area of these five was calculated
This was repeated 1000 times

46. Display 7.3, p. 411

47. Display 7.3, p. 411
Shape: approx. normal Center: μx = 7.4 Spread: σx = 2.3

48. Population Sampling Dist.
Shape: skewed right approx. normal
Center: μx = μx = 7.4
Spread: σx = σx = 2.3

49. Reasonably Likely vs Rare Events
What are these?

50. Reasonably Likely vs Rare Events
Reasonably likely events: values that lie in the middle 95% of sampling distribution

51. Reasonably Likely vs Rare Events
Reasonably likely events: values that lie in the middle 95% of sampling distribution
Rare events: values that lie in outer 5% of sampling distribution

52. Display 7.3, p. 411

53. Reasonably Likely vs Rare Events
In normal distribution, rare events lie more than approximately 2 standard deviations from the mean

54. Vocabulary
Population standard deviation is the standard deviation of the population,

55. Vocabulary

56. Point Estimators
Point estimator: a statistic from a sample that provides a single point (number) as a plausible value of a population parameter

57. Sampling Bias Recall, when we discussed sampling bias.
What happens if you have biased results?

58. Sampling Bias Recall, when we discussed sampling bias.
What happens if you have biased results?
The estimate from the sample is larger or smaller, on average, than the population parameter being estimated

59. Point Estimators
A summary statistic is a biased estimator of a population parameter if it gives results that are too large or too small on average

60. Biased or Unbiased Estimators?
For a sampling distribution, are these biased or unbiased estimators?
1) Sample mean
Sample median
Sample maximum
Sample minimum
Sample range
6) Sample standard deviation

61. Estimators
Sample mean is unbiased estimator of the population mean because the mean of the sampling distribution of the sample mean is equal to the population mean
Sample median?

62. Estimators
Sample median is nearly unbiased estimator of population median for large samples
Sample maximum?

63. Estimators
Sample maximum is biased estimator of population maximum and is biased in direction of being too small
Sample minimum?

64. Estimators
Sample minimum is biased estimator of population minimum and tends to be too large
Sample range?

65. Estimators
Sample range is biased estimator of population range and tends to be too small
Sample standard deviation?

66. Estimators
Sample standard deviation is biased estimator of population standard deviation and tends to be too small

67. Biased or Unbiased Estimators?
For a sampling distribution, are these biased or unbiased estimators?
1) Sample mean : unbiased
Sample median: nearly unbiased
Sample maximum
Sample minimum
Sample range
6) Sample standard deviation

68. Questions?