SLIDES PREPARED

1
STATISTICS for the Utterly Confused, 2nd ed.

• SLIDES PREPARED
• By
• Lloyd R. Jaisingh Ph.D.
• Morehead State University
• Morehead KY

2
Chapter 10

• Sampling Distributions and the Central Limit
  Theorem

3
Outline

• Do I Need to Read This Chapter? You should read
  the Chapter if you would like to learn about
• 10-1 Sampling Distribution of a
• Sample Proportion
• 10-2 Sampling Distribution of a Sample
  Mean

4
Outline

• Do I Need to Read This Chapter? You should read
  the Chapter if you would like to learn about
• 10-3 Sampling Distribution of a Difference
  between Two Independent Sample
  Proportions
• 10-4 Sampling Distribution of a Difference
  between Two Independent Sample Means

5
Objectives

• To investigate the sampling distribution for a
  sample proportion and sample mean.
• To investigate the sampling distribution for the
  difference between two sample proportions and
  sample means.
• To discuss the Central Limit Theorem as it
  applies to the above two situations.

6
10-1 Sampling Distribution of a Sample
Proportion

• Suppose we are interested in the true proportion
  of Americans who favor doctor-assisted suicide.
  If we let the population proportion be denoted by
  p, then p can be defined by

7
10-1 Sampling Distribution of a Sample
Proportion

• Since the population of interest is too large for
  us to observe all Americans, we can estimate the
  true proportion by observing a random sample from
  the population.

8
10-1 Sampling Distribution of a Sample
Proportion

• If we let the sample proportion be denoted by
  (read as p hat), then this single number can be
  used as an estimate for the population proportion
  p and can be defined by

9
10-1 Sampling Distribution of a Sample
Proportion

• This single number is called a point estimate
  for the population proportion p.
• Explanation of the term - point estimate A point
  estimate is a single number that is used to
  estimate a population parameter.

10
Point Estimate for the PopulationProportion p
p
n sample size x number of successes
The point estimate for the population proportion
p can be computed from (x/n)
11
10-1 Sampling Distribution of a Sample
Proportion

• Suppose we assume that the true proportion of
  Americans who favor doctor-assisted suicide is 68
  percent (source USA TODAY Snapshot).
• In general we will not know the true population
  proportion.

12
10-1 Sampling Distribution of a Sample
Proportion

• If we select a random sample of, say, 50
  Americans, we may observe that 35 of them favor
  doctor-assisted suicide.
• Thus, our sample proportion of Americans who
  favor doctor-assisted suicide will be 35/50 0.7
  or 70 percent.

13
10-1 Sampling Distribution of a Sample
Proportion

• If we were to select another random sample of
  size 50, we would most likely obtain a different
  value for the sample proportion.
• If we selected 50 different samples of the same
  sample size and compute these sample proportions,
  we should not expect these values to all be the
  same.

14
10-1 Sampling Distribution of a Sample
Proportion

• Pictorially, the situation is demonstrated in the
  following Figure.

15
10-1 Sampling Distribution of a Sample
Proportion

• These 50 sample proportions constitute a sampling
  distribution of a sample proportion.
• Explanation of the term sampling distribution
  of a sample proportion A sampling distribution
  of a sample proportion is a distribution obtained
  by using the proportions computed from random
  samples of a specific size obtained from a
  population.

16
10-1 Sampling Distribution of a Sample
Proportion

• In order to investigate properties of the
  sampling distribution of a sample proportion,
  simulations of the situation can be done.
• For example, the MINITAB statistical software can
  be used for the simulation.
• In this example, 50 samples of size 50 were
  generated.

17
10-1 Sampling Distribution of a Sample
Proportion

• The distribution used in the simulation was the
  binomial distribution with parameters n 50 and
  p 0.68.
• This assumed distribution is reasonable, since we
  are interested in the proportion (number) of
  persons in the sample of size 50 who support
  doctor-assisted suicide.

18
10-1 Sampling Distribution of a Sample
Proportion

• You may try your own simulation if you have
  access to such statistical software.
• The descriptive statistics for a simulation is
  shown below.

19
10-1 Sampling Distribution of a Sample
Proportion

• The table below shows some summary information,
  obtained for the 50 simulated sample proportions.

20
10-1 Sampling Distribution of a Sample
Proportion

• Observe from the table that the values on the
  left side are approximately equal to the
  corresponding values on the right side.
• Of course, if we do a large number of these
  simulations and take averages, we should expect
  that these values would be closer, if not equal,
  to each other.

21
10-1 Sampling Distribution of a Sample
Proportion

• The main purpose of this illustration was to help
  in understanding the stated properties given
  next.

22
10-1 Sampling Distribution of a Sample
Proportion

• Also the shape of the distribution of the
  simulated sample proportions is approximately
  bell-shaped.
• That is, the distribution of the sample
  proportions is approximately normally
  distributed.
• A histogram for the simulation is shown on the
  next slide.

23
10-1 Sampling Distribution of a Sample
Proportion
24
10-1 Sampling Distribution of a Sample
Proportion

• We can investigate with other sample sizes and
  probability p.
• However, we will generally observe the same
  properties when the sample size is large enough
  (n ? 30 or np and n(1-pgt30).
• We can generalize the observations in a very
  important theorem called the Central Limit
  Theorem for Sample Proportions.

25
10-1 Sampling Distribution of a Sample
Proportion
26
QUICK TIP
27
10-1 Sampling Distribution of a Sample
Proportion

• Example In a survey, it was reported that 33
  percent of females believe in the existence of
  aliens. If 100 females are selected at random,
  what is the probability of more than 45 percent
  of them will say that they believe in aliens?

28
10-1 Sampling Distribution of a Sample
Proportion

• Solution

This is displayed graphically on the next slide.
29
10-1 Sampling Distribution of a Sample
Proportion

• Solution (continued)

30
10-1 Sampling Distribution of a Sample
Proportion

• Example It is estimated that approximately 53
  of college students graduate in 5 years or less.
  This figure is affected by the fact that more
  students are attending college on a part-time
  basis. If 500 students on a large campus are
  selected at random, what is the probability that
  between 50 and 60 of them will graduate in 5
  years or less?

31
10-1 Sampling Distribution of a Sample
Proportion

• Solution

This is displayed graphically on the next slide.
32
10-1 Sampling Distribution of a Sample
Proportion

• Solution (continued)

33
10-2 Sampling Distribution of a Sample Mean

• Suppose we are interested in the true daily mean
  time men spend driving their motor vehicles in
  the United States. If we let the population mean
  be denoted by ? then ? can be defined by

34
10-2 Sampling Distribution of a Sample Mean

• Since the population of interest is too large for
  us to observe all American males who drive, we
  can estimate the true mean by observing a random
  sample from the population of American males who
  drive.

35
Point Estimate for the Population Mean ?
?
?
?
n sample size x-bar sample mean s sample SD
The point estimate for the population mean is
the sample mean.
36
10-2 Sampling Distribution of a Sample Mean

• If we let the sample mean be the point estimate
  for the population mean, then we can define

37
10-2 Sampling Distribution of a Sample Mean

• Suppose we assume that the true daily mean time
  American males spend driving is 81 minutes
  (source Federal Highway Administration).
• In general we will not know the true population
  mean.

38
10-2 Sampling Distribution of a Sample Mean

• If we select a random sample of 50 American males
  who drive, we may observe that the average daily
  time spent behind the wheel for this sample is 85
  minutes.
• If we were to select another random sample of
  size 50, we are most likely to obtain a different
  value for the sample mean.

39
10-2 Sampling Distribution of a Sample Mean

• If we selected 100 different samples, say, of the
  same sample size, and compute the average time
  spent behind the wheel by American males, we
  should not expect these 100 sample means to all
  be the same.
• That is, there will be some variability in these
  computed sample means.

40
10-2 Sampling Distribution of a Sample Mean

• Pictorially, the situation is demonstrated in the
  following Figure.

41
10-2 Sampling Distribution of a Sample Mean

• These 100 sample means constitute a sampling
  distribution of a sample mean.
• Explanation of the term sampling distribution
  of a sample mean A sampling distribution of a
  sample mean is a distribution obtained by using
  the means computed from random samples of a
  specific size obtained from a population.

42
10-2 Sampling Distribution of a Sample Mean

• In order to investigate properties of the
  sampling distribution of a sample mean,
  simulations of the situation can be done.
• For example, the MINITAB statistical software can
  be used for the simulation.
• In this example, 100 samples of size 50 were
  generated.

43
10-2 Sampling Distribution of a Sample Mean

• Here we will assume that the time spent driving
  is normally distributed with a mean of 81 and a
  standard deviation of 1, for the sake of the
  simulation.

44
10-2 Sampling Distribution of a Sample Mean

• You may try your own simulation if you have
  access to such a statistical software.
• The descriptive statistics for a simulation is
  shown below.

45
10-2 Sampling Distribution of a Sample Mean

• The table below shows some summary information,
  obtained for the 100 simulated sample means.

46
10-2 Sampling Distribution of a Sample Mean

• Observe from the table that the values on the
  left side are approximately equal to the
  corresponding values on the right side.
• Of course, if we do a large number of these
  simulations and take averages, we should expect
  that these values would be closer, if not equal,
  to each other.

47
10-2 Sampling Distribution of a Sample Mean

• The main purpose of this illustration was to help
  in understanding the stated properties given
  next.

48
10-2 Sampling Distribution of a Sample Mean

• Also the shape of the distribution of the
  simulated sample means is approximately
  bell-shaped.
• That is, the distribution of the sample means is
  approximately normally distributed.
• A histogram for the simulation is shown on the
  next slide.

49
10-2 Sampling Distribution of a Sample Mean
50
10-2 Sampling Distribution of a Sample Mean

• We can investigate with other sample sizes.
• However, we will generally observe the same
  properties when the sample size is large enough
  (n ? 30).
• We can generalize the observations in a very
  important theorem called the Central Limit
  Theorem for Sample Means.

51
10-2 Sampling Distribution of a Sample Mean
52
QUICK TIP
53
10-2 Sampling Distribution of a Sample Mean

• Example A tire manufacturer claims that its
  tires will last an average of 60,000 miles with a
  standard deviation of 3,000 miles. Sixty-four
  tires were placed on test and the average failure
  miles, for these tires, was recorded. What is
  the probability that the average failure miles
  will be more than 59,500 miles?

54
10-2 Sampling Distribution of a Sample Mean

• Solution

This is displayed graphically on the next slide.
55
10-2 Sampling Distribution of a Sample Mean

• Solution (continued)

56
10-2 Sampling Distribution of a Sample Mean

• Example A supervisor has determined that the
  average salary of the employees in his department
  is 40,000 with a standard deviation of 15,000.
  A sample of 25 of the employees salaries was
  selected at random. Assuming that the
  distribution of the salaries is normal, what is
  the probability that the average for this sample
  is between 36,000 and 42,000?

57
10-2 Sampling Distribution of a Sample Mean

• Solution

This is displayed graphically on the next slide.
58
10-2 Sampling Distribution of a Sample Mean

• Solution (continued)

59
10-3 Sampling Distribution of a Difference
between Two Independent Sample
Proportions

• We may be interested in comparing the proportions
  of two populations.
• For example, we may have to compare the
  effectiveness of two different drugs, drug 1 and
  drug 2, say, on a certain medical condition.
• One way of doing this, is to select a homogeneous
  group of people with the given medical condition
  and randomly divide into two groups.

60
10-3 Sampling Distribution of a Difference
between Two Independent Sample
Proportions

• These groups can then be treated with the
  different medications over a period of time, and
  then the effectiveness of the medication for
  these two groups can be determined.
• A general sampling situation is shown on the next
  slide.

61
10-3 Sampling Distribution of a Difference
between Two Independent Sample Proportions
p1
p2
n1 sample size x1 number of successes
n2 sample size x2 number of successes
Population 1
Population 2
62
10-3 Sampling Distribution of a Difference
between Two Independent Sample
Proportions
63
10-3 Sampling Distribution of a Difference
between Two Independent Sample
Proportions

• Explanation of the term - sampling distribution
  of the difference between two independent sample
  proportions A sampling distribution of the
  difference between two independent sample
  proportions is a distribution obtained by using
  the difference of the proportions computed from
  random samples obtained from the two populations.

64
10-3 Properties of the Sampling Distribution of
a Difference between Two Independent Sample
Proportions
65
10-3 Properties of the Sampling Distribution of
a Difference between Two Independent Sample
Proportions

• Also, the distribution for these
• differences will be approximately
• normally distributed.
• We will generally observe these same properties
  when n1p1 gt 5, n1(1-p1) gt 5, n2p2 gt 5, and
  n2(1-p2) gt 5.

66
10-3 Central Limit Theorem for a Difference
between Two Independent Sample Proportions

• We can summarize the properties
• of the Central Limit Theorem for
• the difference of Sample Proportions
• with the following

67
10-3 Central Limit Theorem for a Difference
between Two Independent Sample Proportions
68
10-3 Z-score for a Difference between Two
Independent Sample Proportions
69
Quick Tip

• In order for us to use the equation to compute
  probabilities with respect to two population
  proportions, we would need to estimate the true
  proportions with corresponding sample
  proportions.

70
10-3 Example

• Example A study was conducted to determine
  whether remediation in developmental mathematics
  enabled students to be more successful in an
  elementary statistics course. Success here means
  that a student received a grade of C or better
  and remediation was for one year. The following
  table shows the summary results of the study.

71
10-3 Example (Continued)
72
10-3 Example (Continued)

• Example (continued) Based on past history, it is
  known that 75 of students who enroll in remedial
  mathematics are successful while only 50 are
  successful for nonremedial students.
• What is the probability that the difference in
  proportion of success for the remedial and
  nonremedial students is at least 10 percent?

73
10-3 Example (Solution)
This is displayed graphically on the next slide.
74
10-3 Example (Solution)
75
10-4 Sampling Distribution of a Difference
between Two Independent Sample Means

• We may be interested in comparing the means of
  two populations.
• For example, we may have to compare the
  effectiveness of two different diets, diet 1 and
  diet 2, say, for weight loss.
• One way of doing this, is to select a homogeneous
  group of people who are classified as overweight,
  and randomly divide into two groups.

76
10-4 Sampling Distribution of a Difference
between Two Independent Sample Means

• These groups can then be treated with the
  different diets over a period of time, and the
  effectiveness of the diets for these two groups
  can be determined.
• A general sampling situation is shown on the next
  slide.

77
10-4 Sampling Distribution of a Difference
between Two Independent Sample Means
78
10-4 Sampling Distribution of a Difference
between Two Independent Sample Means
79
10-4 Sampling Distribution of a Difference
between Two Independent Sample Means

• Explanation of the term - sampling distribution
  of the difference between two independent sample
  means A sampling distribution of the difference
  between two independent sample means is a
  distribution obtained by using the difference of
  the sample means computed from random samples
  obtained from the two populations.

80
10-4 Properties of the Sampling Distribution of
a Difference between Two Independent Sample Means
81
10-4 Properties of the Sampling Distribution of
a Difference between Two Independent Sample Means

• Also, the distribution for these
• differences will be approximately
• normally distributed.

82
10-4 Central Limit Theorem for a Difference
between Two Independent Sample Means

• We can summarize the properties
• of the Central Limit Theorem for
• the difference of Sample Means
• with the following

83
10-4 Central Limit Theorem for a Difference
between Two Independent Sample Means
84
10-4 Z-score for a Difference between Two
Independent Sample Means
85
Quick Tip

• If the population standard deviations are unknown
  but the sample sizes are large ( n1 and n2 ? 30),
  then we can approximate the population variances
  by the corresponding sample variances.

86
10-4 Example

• Example Based on extensive use of two methods
  (Method 1 and Method 2) of teaching a high school
  advance placement (AP) statistics course, the
  following summary information, given on the next
  slide, for a random sample of final scores for
  each teaching method were obtained.

87
10-4 Example (Continued)
88
10-4 Example (Continued)

• Example (continued) Find the probability that
  Method 1, on average, was more successful than
  Method 2.
• That is, we need to find P(
  ) ? P( ? 0).

89
10-4 Example (Solution)
This is displayed graphically on the next slide.
90
10-3 Example (Solution)
91
Quick Tip