1. Sampling Distribution of the Sample Mean
Section 7.2
Sampling Distribution of the Sample Mean

2. For a finite population, the sampling distribution of the sample mean is the

3. For a finite population, the sampling distribution of the sample mean is the set of means from all possible samples of a specific size, n.

4. Simulated Sampling Distribution
4 steps to construct:

5. Simulated Sampling Distribution
4 steps to construct:
1) Take random sample from population

6. Simulated Sampling Distribution
4 steps to construct:
1) Take random sample from population
2) Compute summary statistic for sample

7. Simulated Sampling Distribution
4 steps to construct:
1) Take random sample from population
2) Compute summary statistic for sample
3) Repeat steps 1 and 2 many times

8. Simulated Sampling Distribution
4 steps to construct:
1) Take random sample from population
2) Compute summary statistic for sample
3) Repeat steps 1 and 2 many times
4) Display distribution of summary statistic

9. Find mean and standard deviation using Freq
Find mean and standard deviation using Freq. Table of Population (not a sample distribution)
Page 428

10. Find mean and standard deviation using Freq
Find mean and standard deviation using Freq. Table of Population (not a sample distribution) L2 L1

11. Find mean and standard deviation using Freq
Find mean and standard deviation using Freq. Table of Population (not a sample distribution) L1 L2
STAT
CALC
1-Var Stats L1, L2

12. Find mean and standard deviation using Freq
Find mean and standard deviation using Freq. Table of Population (not a sample distribution)

13. Find mean and standard deviation using a sample distribution (not of population)
Suppose we make 5 sampling distributions of the sample mean for samples of size:
n = 1
n = 4
n = 10
n = 20
n = 40

14. Find mean and standard deviation using a sample distribution, n=1 (from a population)
Note: this column is a Sample
Distribution using a sample size n=1
(looks just like the Population Frequency Graph)
Note: this column is of a Population

15. Note: this graph is of a Population
Using Population Freq. Table, generate Sampling Distributions of different sample sizes (next slides)
n = 4
n = 10
n = 20
n = 40
4 Steps not shown…see Page 428 & 429 in textbook for calculations when n = 4
Note: this graph is of a Population

16. To clarify Step 1 in textbook, page 428, we can use a random table of digits OR a calculator to assign digits to possible outcomes
Using random
table of digits:
975 – 999, 1000
Or using Calculator:
MATH
PRB
randInt (1, 1000)

17. Results
After repeating Steps 1-4 many times (i.e. Fathom) using different size samples, the results are:

18. As n increases, what happens to:
Shape
Center
Spread

19. As n increases, what happens to:
Shape: more normal
Center
Spread

20. As n increases, what happens to:
Shape: more normal
Center: stays same
Spread

21. As n increases, what happens to:
Shape: more normal
Center: stays same
Spread: decreases

22. Common Symbols
Note: a calculator cannot tell if a list (L1 or L2) is from a population or sample distribution…you have to know which is which.
Page 430

23. Properties of Sampling Distribution of the Sample Mean
These properties depend on using ________ samples.

24. Properties of Sampling Distribution of the Sample Mean
These properties depend on using random samples.

25. Properties of Sampling Distribution of the Sample Mean
These properties depend on using random samples.
These properties will cover center, spread, and shape.

26. Properties of Sampling Distribution of the Sample Mean
If a random sample of size n is selected from a population with mean and standard deviation , then:

27. Properties of Sampling Distribution of the Sample Mean
Center
The mean, x, of the sampling distribution of x equals the mean of the population, :
x =

28. Properties of Sampling Distribution of the Sample Mean
Center
The mean, x, of the sampling distribution of x equals the mean of the population, :
x =
In other words, the means of random samples are centered at the population mean.

29. Properties of Sampling Distribution of the Sample Mean
Spread
The standard deviation, x, of the sampling distribution, sometimes called the standard error of the mean, equals the standard deviation of the population, , divided by the square root of the sample size n.
x =

30. Properties of Sampling Distribution of the Sample Mean
Spread
The standard deviation, x, of the sampling distribution, sometimes called the standard error of the mean, equals the standard deviation of the population, , divided by the square root of the sample size n.
X =
When sample size increases, spread ________.

31. Properties of Sampling Distribution of the Sample Mean
Spread
The standard deviation, x, of the sampling distribution, sometimes called the standard error of the mean, equals the standard deviation of the population, , divided by the square root of the sample size n.
X =
When sample size increases, spread decreases

32. Properties of Sampling Distribution of the Sample Mean
Shape
The shape of the sampling distribution will be approximately normal if the population is approximately normal.

33. Properties of Sampling Distribution of the Sample Mean
Shape
The shape of the sampling distribution will be approximately normal if the population is approximately normal.
For other populations, the sampling distribution becomes more normal as n increases (Central Limit Theorem).

34. Using These Properties
When can you use property that mean of
sampling distribution of the mean is equal to
the mean of the population, x = ?

35. Using These Properties
When can you use property that mean of
sampling distribution of the mean is equal to
the mean of the population, x = ?
Anytime you use random sampling. Shape of pop., size of sample, how large pop. is, or whether sample with or without replacement do not matter.

36. Using These Properties
When can we use property that standard
error of sampling distribution of the mean,
x, is equal to ?

37. Using These Properties
With a population of any shape and with any
sample size as long as you randomly
sample with replacement or

38. Using These Properties
With a population of any shape and with any
sample size as long as you randomly
sample with replacement or
you randomly sample without replacement
and the sample size is less than 10% of
population size.

39. Using These Properties
3. When can we assume the sampling distribution is approximately normal?

40. Using These Properties
3. When can we assume the sampling distribution is approximately normal?
If we are told the population is approximately normally distributed (or bell-shaped or mound-shaped), you can assume sampling distribution is approximately normal regardless of sample size or _______.

41. Using These Properties
3. When can we assume the sampling distribution is approximately normal?
If we are told the population is approximately normally distributed (or bell-shaped or mound-shaped), you can assume sampling distribution is approximately normal regardless of sample size or if we are told sample size is very large.

42. Using These Properties
3. When can we assume the sampling distribution is approximately normal?
If we are told the population is approximately normally distributed (or bell-shaped or mound-shaped), you can assume sampling distribution is approximately normal regardless of sample size or if we are told sample size is very large (think Central Limit Theorem).

43. Using These Properties
4. Isn’t the size of the population really important?

44. Using These Properties
4. Isn’t the size of the population really important?
As long as the sample was randomly selected and as long as the population is much larger than the sample, it does not matter how large the population is.

45. Example: Average Number of Children
What is the probability that a random sample of 20 families in the United States will have an average of 1.5 children or fewer?

46. Example: Average Number of Children
What is the probability that a random sample of 20 families in the United States will have an average of 1.5 children or fewer?
Could add up heights of all bars ≤ 1.5

47. Example: Average Number of Children
What is the probability that a random sample of 20 families in the United States will have an average of 1.5 children or fewer?
Could add up heights of all bars ≤ 1.5
What shape is this?

48. Example: Average Number of Children
What is the probability that a random sample of 20 families in the United States will have an average of 1.5 children or fewer?
How do we find the area under the normal curve?

49. Average of 1.5 children or fewer?
normalcdf(lower bound, upper bound, mean, standard deviation)

50. Example: Average Number of Children
What is the probability that a random sample of 20 families in the United States will have an average of 1.5 children or fewer?
Recall, for the population μ = 0.9 and σ = 1.1
What are the sampling distribution mean and standard deviation (standard error)?

51. Example: Average Number of Children
What is the probability that a random sample of 20 families in the United States will have an average of 1.5 children or fewer?
Recall, = 0.9 and = 1.1

52. What is the probability that a random sample of 20 families in the United States will have an average of 1.5 children or fewer?
normalcdf (-1E99, 1.5, .9, .25) =

53. Reasonably Likely Averages
What average numbers of children are reasonably likely in a random sample of 20 families?

54. Reasonably Likely Averages
What average numbers of children are reasonably likely in a random sample of 20 families?
Reasonably likely values are those in the middle 95% of the sampling distribution.

55. Reasonably Likely Averages
What average numbers of children are reasonably likely in a random sample of 20 families?

56. Reasonably Likely Averages
What average numbers of children are reasonably likely in a random sample of 20 families?
For approx. normal distributions, reasonably likely outcomes are those within 1.96 standard errors of the mean.

57. Reasonably Likely Averages
What average numbers of children are reasonably likely in a random sample of 20 families?
(0.25)
which gives an average number of children between 0.41 and 1.39

58. Page 437, P7

59. Page 437, P7

60. Page 437, P7
population
n = 4
n = 10

61. Page 437, P7
population
n = 4
n = 10
mean ≈ 1.7 for each

62. Page 437, P7
population
n = 4
n = 10
mean ≈ 1.7 for each
SD ≈ 1

63. Page 437, P7 population n = 4 n = 10 mean ≈ 1.7 for each SE ≈ 0.3
SD ≈ 1

64. Page 437, P7 population n = 4 n = 10 mean ≈ 1.7 for each SE ≈ 0.3
SD ≈ 1
SE ≈ 0.5

65. Page 437, P7 a. Plot I is the population, starting out with only
five different values and a slight skew.
Plot III is for a sample size of 4, having more
possible values, a smaller spread, and a more
nearly normal shape.
Plot II is for a sample size of 10, which has even
more possible values, an even smaller spread, and
a shape that is closest to normal of all the
distributions.

66. Page 437, P7
b. Theoretically, the mean will be the same 1.7 for each distribution, which is consistent with the estimates.

67. Page 437, P7

68. Page 437, P7 d. The population distribution is roughly
mound-shaped with a slight skew. The two
sampling distributions; however, are
approximately normal with the distributions
becoming more nearly normal as the sample
size increases. This is consistent with the
Central Limit Theorem.

69. Page 438, P8

70. Page 438, P8

71. Page 438, P10

72. Page 438, P10
Decrease in price means < 0

73. Page 438, P10

74. Page 438, P10

75. Page 438, P10

76. Page 443, E21

77. Page 443, E21

78. Page 443, E21
normalcdf (510, 1E99, 500, 50)

79. Page 443, E21
normalcdf (510, 1E99, 500, 20)