Estimating the Value of a Parameter

1
Chapter 9

• Estimating the Value of a Parameter
• Using Confidence Intervals

2
Overview

• We apply the results about the sample mean to the
  problem of estimation
• Estimation is the process of using sample data to
  estimate the value of a population parameter
• We will quantify the accuracy of our estimation
  process

3
Chapter 9 Sections

• Sections in Chapter 9
• The Logic in Constructing Confidence Intervals
  about a Population Mean where the Population
  Standard Deviation is Known
• Confidence Intervals about a Population Mean
  in Practice where the Population Standard
  Deviation is Unknown
• Confidence Intervals about a Population
  Proportion
• Confidence Intervals about a Population
  Standard Deviation
• Putting It All Together Which Procedure Do I
  Use?

4
Chapter 9Section 1

• The Logic in ConstructingConfidence Intervals
  about aPopulation Mean where thePopulation
  Standard Deviation is Known

5
Confidence Intervals

• Learning objectives
• Compute a point estimate of the population mean
• Construct and interpret a confidence interval
  about the population mean (assuming the
  population standard deviation is known)
• Understand the role of margin of error in
  constructing a confidence interval
• Determine the sample size necessary for
  estimating the population mean within a specified
  margin of error

6
Confidence Intervals

• Learning objectives
• Compute a point estimate of the population mean
• Construct and interpret a confidence interval
  about the population mean (assuming the
  population standard deviation is known)
• Understand the role of margin of error in
  constructing a confidence interval
• Determine the sample size necessary for
  estimating the population mean within a specified
  margin of error

7
Chapter 9 Section 1

• The environment of our problem is that we want to
  estimate the value of an unknown population mean
• The process that we use is called estimation
• This is one of the most common goals of statistics

8
Chapter 9 Section 1

• Estimation involves two steps
• Step 1 to obtain a specific numeric estimate,
  this is called the point estimate

• Estimation involves two steps
• Step 1 to obtain a specific numeric estimate,
  this is called the point estimate
• Step 2 to quantify the accuracy and precision
  of the point estimate

• Estimation involves two steps
• Step 1 to obtain a specific numeric estimate,
  this is called the point estimate
• Step 2 to quantify the accuracy and precision
  of the point estimate
• The first step is relatively easy
• The second step is why we need statistics

9
Chapter 9 Section 1

• Some examples of point estimates are
• The sample mean to estimate the population mean
• The sample standard deviation to estimate the
  population standard deviation
• The sample proportion to estimate the population
  proportion
• The sample median to estimate the population
  median

10
Confidence Intervals

• Learning objectives
• Compute a point estimate of the population mean
• Construct and interpret a confidence interval
  about the population mean (assuming the
  population standard deviation is known)
• Understand the role of margin of error in
  constructing a confidence interval
• Determine the sample size necessary for
  estimating the population mean within a specified
  margin of error

11
Chapter 9 Section 1

• The most obvious point estimate for the
  population mean is the sample mean
• Now we will use the material in Chapter 8 on the
  sample mean to quantify the accuracy and
  precision of this point estimate

12
Chapter 9 Section 1

• An example of what we want to quantify
• We want to estimate the miles per gallon for a
  certain car
• We test some number of cars
• We calculate the sample mean it is 27
• 27 miles per gallon would be our best guess

13
Chapter 9 Section 1

• How sure are we that the gas economy is 27 and
  not 28.1, or 25.2?
• We would like to make a statement such as
• We think that the mileage is 27 mpg
• and were pretty sure that were
• not too far off

14
Chapter 9 Section 1

• A confidence interval for an unknown parameter is
  an interval of numbers
• Compare this to a point estimate which is just
  one number, not an interval of numbers

• A confidence interval for an unknown parameter is
  an interval of numbers
• Compare this to a point estimate which is just
  one number, not an interval of numbers
• The level of confidence represents the expected
  proportion of intervals that will contain the
  parameter if a large number of different samples
  is obtained

15
Chapter 9 Section 1

• What does the level of confidence represent?

• What does the level of confidence represent?
• If we have a process for calculating confidence
  intervals with a 90 level of confidence
• Assume that we know the population mean
• We then obtain a series of 50 random samples
• We apply our process to the data from each random
  sample to obtain a confidence interval for each

• What does the level of confidence represent?
• If we have a process for calculating confidence
  intervals with a 90 level of confidence
• Assume that we know the population mean
• We then obtain a series of 50 random samples
• We apply our process to the data from each random
  sample to obtain a confidence interval for each
• Then, we would expect that 90 of those 50
  confidence intervals (or about 45) would contain
  our population mean

16
Chapter 9 Section 1

• If we expect that a method would create intervals
  that contain the population mean 90 of the time,
  we call those intervals
• 90 confidence intervals

• If we expect that a method would create intervals
  that contain the population mean 90 of the time,
  we call those intervals
• 90 confidence intervals
• If we have a method for intervals that contain
  the population mean 95 of the time, those are
• 95 confidence intervals

• If we expect that a method would create intervals
  that contain the population mean 90 of the time,
  we call those intervals
• 90 confidence intervals
• If we have a method for intervals that contain
  the population mean 95 of the time, those are
• 95 confidence intervals
• And so forth

17
Chapter 9 Section 1

• The level of confidence is always expressed as a
  percent

• The level of confidence is always expressed as a
  percent
• The level of confidence is described by a
  parameter a

• The level of confidence is always expressed as a
  percent
• The level of confidence is described by a
  parameter a
• The level of confidence is (1 a) 100
• When a .05, then (1 a) .95, and we have a
  95 level of confidence
• When a .01, then (1 a) .99, and we have a
  99 level of confidence

18
Chapter 9 Section 1

• To tie the definitions together (in English)
• We are using the sample mean to estimate the
  population mean

• To tie the definitions together (in English)
• We are using the sample mean to estimate the
  population mean
• With each specific sample, we can construct a 95
  confidence interval

• To tie the definitions together (in English)
• We are using the sample mean to estimate the
  population mean
• With each specific sample, we can construct a 95
  confidence interval
• As we take repeated samples, we expect that 95
  of these intervals would contain the population
  mean

19
Chapter 9 Section 1

• To tie all the definitions together (using
  statistical terms)

• To tie all the definitions together (using
  statistical terms)
• We are using a point estimator to estimate the
  population mean

• To tie all the definitions together (using
  statistical terms)
• We are using a point estimator to estimate the
  population mean
• We wish to construct a confidence interval with
  parameter a, the level of confidence is (1 a)
  100

• To tie all the definitions together (using
  statistical terms)
• We are using a point estimator to estimate the
  population mean
• We wish to construct a confidence interval with
  parameter a, the level of confidence is (1 a)
  100
• As we take repeated samples, we expect that(1
  a) 100 of the resulting intervals will contain
  the population mean

20
Chapter 9 Section 1

• Back to our 27 miles per gallon car
• We think that the mileage is 27 mpgand were
  pretty sure thatwere not too far off

• Back to our 27 miles per gallon car
• We think that the mileage is 27 mpgand were
  pretty sure thatwere not too far off
• Putting in numbers,
• We estimate the gas mileage is 27 mpgand we are
  90 confident thatthe real mileage of this model
  of caris between 25 and 29 miles per gallon

21
Chapter 9 Section 1

• We estimate the gas mileage is 27 mpg
• This is our point estimate

• We estimate the gas mileage is 27 mpg
• This is our point estimate
• and we are 90 confident that
• Our confidence level is 90 (i.e. a 0.10)

• We estimate the gas mileage is 27 mpg
• This is our point estimate
• and we are 90 confident that
• Our confidence level is 90 (i.e. a 0.10)
• the real mileage of this model of car
• The population mean

• We estimate the gas mileage is 27 mpg
• This is our point estimate
• and we are 90 confident that
• Our confidence level is 90 (i.e. a 0.10)
• the real mileage of this model of car
• The population mean
• is between 25 and 29 miles per gallon
• Our confidence interval is (25, 29)

22
Chapter 9 Section 1

• In this section, we assume that we know the
  standard deviation of the population (s)

• In this section, we assume that we know the
  standard deviation of the population (s)
• This is not very realistic but we need it for
  right now

• In this section, we assume that we know the
  standard deviation of the population (s)
• This is not very realistic but we need it for
  right now
• Well solve this problem in a better way (where
  we dont know what s is) in the next section
  but first well do this one

23
Chapter 9 Section 1

• If n is large enough, i.e. n 30, we can assume
  that the sample means have a normal distribution
  with standard deviation s / v n

• If n is large enough, i.e. n 30, we can assume
  that the sample means have a normal distribution
  with standard deviation s / v n
• We look up a standard normal calculation
• 95 of the values in a standard normal are
  between 1.96 and 1.96 in other words between
  1.96
• We now use this applied to a general normal
  calculation

24
Chapter 9 Section 1

• The values of a general normal random variable
  are less than 1.96 times its standard deviation
  away from its mean 95 of the time
• Thus the sample mean is within
• 1.96 s / v n
• of the population mean 95 of the time

25
Chapter 9 Section 1

• If the sample mean is within
• 1.96 s / v n
• of the population mean 95 of the time, then we
  can flip this around to say that the population
  mean is within
• 1.96 s / v n
• of the population mean 95 of the time

26
Chapter 9 Section 1

• Thus a 95 confidence interval for the mean is
• This is in the form
• Point estimate margin of error
• The margin of error here is 1.96 s / v n

27
Chapter 9 Section 1

• For our car mileage example
• Assume that the sample mean was 27 mpg
• Assume that we tested 40 cars
• Assume that we knew that the population standard
  deviation was 6 mpg

• For our car mileage example
• Assume that the sample mean was 27 mpg
• Assume that we tested 40 cars
• Assume that we knew that the population standard
  deviation was 6 mpg
• Then our 95 confidence interval would be
• or 27 1.9

28
Chapter 9 Section 1

• If we wanted to compute a 90 confidence
  interval, or a 99 confidence interval, etc., we
  would just need to find the right standard normal
  value

• If we wanted to compute a 90 confidence
  interval, or a 99 confidence interval, etc., we
  would just need to find the right standard normal
  value
• Frequently used confidence levels, and their
  critical values, are
• 90 corresponds to 1.645
• 95 corresponds to 1.960
• 99 corresponds to 2.575

29
Chapter 9 Section 1

• The numbers 1.645, 1.960, and 2.575 are written
  as Z-values
• z0.05 1.645 P(Z 1.645) .05
• z0.025 1.960 P(Z 1.960) .025
• z0.005 2.575 P(Z 2.575) .005
• where Z is a standard normal random variable

30
Chapter 9 Section 1

• Why do we use a 0.025 for 95 confidence?

• Why do we use a 0.025 for 95 confidence?
• To be within something 95 of the time
• We can be too low 2.5 of the time
• We can be too high 2.5 of the time

• Why do we use a 0.025 for 95 confidence?
• To be within something 95 of the time
• We can be too low 2.5 of the time
• We can be too high 2.5 of the time
• Thus the 5 confidence that we dont have is
  split as 2.5 being too high and 2.5 being too
  low needing a 0.025 (or 2.5)

31
Chapter 9 Section 1

• In general, for a (1 a) 100 confidence
  interval, we need to find za/2, the critical
  Z-value
• za/2 is the value such that
• P(Z za/2) a/2

32
Chapter 9 Section 1

• Once we know these critical values for the normal
  distribution, then we can construct confidence
  intervals for the sample mean
• to

33
Chapter 9 Section 1

• Learning objectives
• Compute a point estimate of the population mean
• Construct and interpret a confidence interval
  about the population mean (assuming the
  population standard deviation is known)
• Understand the role of margin of error in
  constructing a confidence interval
• Determine the sample size necessary for
  estimating the population mean within a specified
  margin of error

34
Chapter 9 Section 1

• If we write the confidence interval as
• 27 2
• then we would call the number 2 (after the )
  the margin of error

• If we write the confidence interval as
• 27 2
• then we would call the number 2 (after the )
  the margin of error
• So we have three ways of writing confidence
  intervals
• (25, 29)
• 27 2
• 27 with a margin of error of 2

35
Chapter 9 Section 1

• The margin of error depends on three factors
• The level of confidence (a)
• The sample size (n)
• The standard deviation of the population (s)

• The margin of error depends on three factors
• The level of confidence (a)
• The sample size (n)
• The standard deviation of the population (s)
• Well now calculate the margin of error
• Once we know the margin of error, we can state
  the confidence interval

36
Chapter 9 Section 1

• The margin of errors would be
• 1.645 s / v n for 90 confidence intervals
• 1.960 s / v n for 95 confidence intervals
• 2.575 s / v n for 99 confidence intervals

37
Chapter 9 Section 1

• Learning objectives
• Compute a point estimate of the population mean
• Construct and interpret a confidence interval
  about the population mean (assuming the
  population standard deviation is known)
• Understand the role of margin of error in
  constructing a confidence interval
• Determine the sample size necessary for
  estimating the population mean within a specified
  margin of error

38
Chapter 9 Section 1

• Often we have the reverse problem where we want
  an experiment to result in an answer with a
  particular accuracy
• We have a target margin of error
• We need to find the sample size (n) needed to
  achieve this goal

39
Chapter 9 Section 1

• For our car miles per gallon, we had s 6
• If we wanted our margin of error to be 1 for a
  95 confidence interval, then we would need
• Solving for n would get us n (1.96 6)2 or
  that n 138 cars would be needed

40
Chapter 9 Section 1

• We can write this as a formula
• The sample size n needed to result in a margin of
  error E for (1 a) 100 confidence is
• Usually we dont get an integer for n, so we
  would need to take the next higher number (the
  one lower wouldnt be large enough)

41
Summary Chapter 9 Section 1

• We can construct a confidence interval around a
  point estimator if we know the population
  standard deviation s
• The margin of error is calculated using s, the
  sample size n, and the appropriate Z-value
• We can also calculate the sample size needed to
  obtain a target margin of error

42
Chapter 9Section 2

• Confidence Intervals about aPopulation Mean in
  Practice wherethe Population Standard Deviation
• is Unknown

43
Chapter 9 Section 2

• Learning objectives
• Know the properties of t-distribution
• Determine t-values
• Construct and interpret a confidence interval
  about a population mean

44
Chapter 9 Section 2

• Learning objectives
• Know the properties of t-distribution
• Determine t-values
• Construct and interpret a confidence interval
  about a population mean

45
Chapter 9 Section 2

• In Section 1, we assumed that we knew the
  population standard deviation s
• Since we did not know the population mean µ, this
  seems to be unrealistic
• In this section, we construct confidence
  intervals in the case where we do not know the
  population standard deviation
• This is much more realistic

46
Chapter 9 Section 2

• If we dont know the population standard
  deviation s, we obviously cant use the formula
• Margin of error 1.96 s / v n
• because we have no number to use for s

• If we dont know the population standard
  deviation s, we obviously cant use the formula
• Margin of error 1.96 s / v n
• because we have no number to use for s
• However, just as we can use the sample mean to
  approximate the population mean, we can also use
  the sample standard deviation to approximate the
  population standard deviation

47
Chapter 9 Section 2

• Because weve changed our formula (by using s
  instead of s), we cant use the normal
  distribution any more

• Because weve changed our formula (by using s
  instead of s), we cant use the normal
  distribution any more
• Instead of the normal distribution, we use the
  Students t-distribution

• Because weve changed our formula (by using s
  instead of s), we cant use the normal
  distribution any more
• Instead of the normal distribution, we use the
  Students t-distribution
• This distribution was developed specifically for
  the situation when s is not known

48
Chapter 9 Section 2

• Properties of the t-distribution
• Several properties are familiar about the
  Students t distribution

• Properties of the t-distribution
• Several properties are familiar about the
  Students t distribution
• Just like the normal distribution, it is centered
  at 0 and symmetric about 0

• Properties of the t-distribution
• Several properties are familiar about the
  Students t distribution
• Just like the normal distribution, it is centered
  at 0 and symmetric about 0
• Just like the normal curve, the total area under
  the Students t curve is 1, the area to left of 0
  is ½, and the area to the right of 0 is also ½

• Properties of the t-distribution
• Several properties are familiar about the
  Students t distribution
• Just like the normal distribution, it is centered
  at 0 and symmetric about 0
• Just like the normal curve, the total area under
  the Students t curve is 1, the area to left of 0
  is ½, and the area to the right of 0 is also ½
• Just like the normal curve, as t increases, the
  Students t curve gets close to, but never
  reaches, 0

49
Chapter 9 Section 2

• So whats different?

• So whats different?
• Unlike the normal, there are many different
  standard t-distributions
• There is a standard one with 1 degree of
  freedom
• There is a standard one with 2 degrees of
  freedom
• There is a standard one with 3 degrees of
  freedom
• Etc.

• So whats different?
• Unlike the normal, there are many different
  standard t-distributions
• There is a standard one with 1 degree of
  freedom
• There is a standard one with 2 degrees of
  freedom
• There is a standard one with 3 degrees of
  freedom
• Etc.
• The number of degrees of freedom is crucial for
  the t-distributions

50
Chapter 9 Section 2

• When s is known, the Z-score
• follows a standard normal distribution

• When s is known, the Z-score
• follows a standard normal distribution
• When s is not known, the t-statistic
• follows a t-distribution with n 1 degrees of
  freedom

51
Chapter 9 Section 2

• Comparing three curves
• The standard normal curve
• The t curve with 14 degrees of freedom
• The t curve with 4 degrees of freedom

52
Chapter 9 Section 2

• Learning objectives
• Know the properties of t-distribution
• Determine t-values
• Construct and interpret a confidence interval
  about a population mean

53
Chapter 9 Section 2

• The calculation of t-distribution values can be
  done in similar ways as the calculation of normal
  values
• Using tables (such as Table V on the inside back
  cover)
• Using technology (such as Excel, MINITAB,
  calculators, StatCrunch, etc.)

• The calculation of t-distribution values can be
  done in similar ways as the calculation of normal
  values
• Using tables (such as Table V on the inside back
  cover)
• Using technology (such as Excel, MINITAB,
  calculators, StatCrunch, etc.)
• Because t-distribution tables are not complete,
  it is suggested that the calculations be done
  with one of the technology methods

54
Chapter 9 Section 2

• Critical values for various degrees of freedom
  for the t-distribution are (compared to the
  normal)

55
Chapter 9 Section 2

• Learning objectives
• Know the properties of t-distribution
• Determine t-values
• Construct and interpret a confidence interval
  about a population mean

56
Chapter 9 Section 2

• The difference between the two formulas
• is that the sample standard deviation s is used
  to approximate the population standard deviation s

• The difference between the two formulas
• is that the sample standard deviation s is used
  to approximate the population standard deviation
  s
• The Z-score has a normal distribution,
  thet-statistic (or the t-score) has
  at-distribution

57
Chapter 9 Section 2

• A 95 confidence interval, with s unknown, is
• to
• where t0.025 is the critical value for the
  t-distribution with (n 1) degrees of freedom

58
Chapter 9 Section 2

• The different confidence intervals with t0.025
  would be
• For n 6, the sample mean 2.571 s / v 6
• For n 16, the sample mean 2.131 s / v 16
• For n 31, the sample mean 2.042 s / v 31
• For n 101, the sample mean 1.984 s / v 101
• For n 1001, the sample mean 1.962 s / v
  1001
• When s is known, the sample mean 1.960 s / v n

59
Chapter 9 Section 2

• In general, the (1 a) 100 confidence
  interval, when s is unknown, is
• to
• where ta/2 is the critical value for the
  t-distribution with (n 1) degrees of freedom

60
Chapter 9 Section 2

• As the sample size n gets large, there is less
  and less of a difference between the critical
  values for the normal and the critical values for
  the t-distribution

• As the sample size n gets large, there is less
  and less of a difference between the critical
  values for the normal and the critical values for
  the t-distribution
• It is correct to use the t-distribution when s is
  not known
• Technology should always use t-distribution
• When doing rough assessment by hand, the normal
  critical values can be used, particularly when n
  is large, for example if n is 30 or more

61
Chapter 9 Section 2

• When does the t-distribution and normal differ by
  a lot?

• When does the t-distribution and normal differ by
  a lot?
• In either of two situations
• When the sample size n is small (particularly if
  n is 10 or less), or
• When the confidence level needs to be high
  (particularly if a is 0.005 or lower)

• When does the t-distribution and normal differ by
  a lot?
• In either of two situations
• When the sample size n is small (particularly if
  n is 10 or less), or
• When the confidence level needs to be high
  (particularly if a is 0.005 or lower)
• For n 5 and a .001, when n and a both are
  small, the t-distribution critical value is 5.893
  compared to the normal critical value of 3.091

62
Chapter 9 Section 2

• Assume that we want to estimate the average
  weight of a particular type of very rare fish
• We are only able to borrow 7 specimens of this
  fish
• The average weight of these was 1.38 kg (the
  sample mean)
• The standard deviation of these was 0.29 kg (the
  sample standard deviation)
• What is a 95 confidence interval for the true
  mean weight?

63
Chapter 9 Section 2

• n 7, the critical value t0.025 for 6 degrees of
  freedom is 2.447
• Our confidence interval thus is
• to
• or (1.11, 1.65)

64
Chapter 9 Section 2

• Outliers are always a concern, but they are even
  more of a concern for confidence intervals using
  the t-distribution
• The number of values n is small, so each outlier
  has a major affect on the data set
• The sample mean is sensitive to outliers
• The sample standard deviation is sensitive to
  outliers
• This is a problem!

65
Chapter 9 Section 2

• So what can we do?
• We always must check to see that the outlier is a
  legitimate data value (and not just a typo)
• We can collect more data, for example to increase
  n to be over 30

• So what can we do?
• We always must check to see that the outlier is a
  legitimate data value (and not just a typo)
• We can collect more data, for example to increase
  n to be over 30
• If neither method above will work, i.e. if the
  data value is a legitimate value and we are not
  able to collect more data, then there are other
  methods (nonparametric methods) that could
  apply these are in Chapter 15

66
Summary Chapter 9 Section 2

• We used values from the normal distribution when
  we knew the value of the population standard
  deviation s
• When we do not know s, we estimate s using the
  sample standard deviation s
• We use values from the t-distribution when we use
  s instead of s, i.e. when we dont know the
  population standard deviation

67
Chapter 9Section 3

• Confidence Intervals about aPopulation Proportion

68
Chapter 9 Section 3

• Learning objectives
• Obtain a point estimate for the population
  proportion
• Construct and interpret a confidence interval for
  the population proportion
• Determine the sample size necessary for
  estimating a population proportion within a
  specified margin of error

69
Chapter 9 Section 3

• Learning objectives
• Obtain a point estimate for the population
  proportion
• Construct and interpret a confidence interval for
  the population proportion
• Determine the sample size necessary for
  estimating a population proportion within a
  specified margin of error

70
Chapter 9 Section 3

• In Section 1, we calculated confidence intervals
  for the mean, assuming that we knew s
• In Section 2, we calculated confidence intervals
  for the mean, assuming that we did not know s
• In this section, we construct confidence
  intervals for situations when we are analyzing a
  population proportion
• The issues and methods are quite similar

71
Chapter 9 Section 3

• When we analyze the population mean, we use the
  sample mean as the point estimate
• The sample mean is our best guess for the
  population mean

• When we analyze the population mean, we use the
  sample mean as the point estimate
• The sample mean is our best guess for the
  population mean
• When we analyze the population proportion, we use
  the sample proportion as the point estimate
• The sample proportion is our best guess for the
  population proportion

72
Proportions Point Estimate

• Using the sample proportion is the natural choice
  for the point estimate
• If we are doing a poll, and 68 of the
  respondents said yes to our question, then we
  would estimate that 68 of the population would
  say yes to our question also
• The sample proportion is written

73
Chapter 9 Section 3

• Learning objectives
• Obtain a point estimate for the population
  proportion
• Construct and interpret a confidence interval for
  the population proportion
• Determine the sample size necessary for
  estimating a population proportion within a
  specified margin of error

74
Chapter 9 Section 3

• Confidence intervals for the population mean are
• Centered at the sample mean
• Plus and minus za/2 times the standard deviation
  of the sample mean (the standard error from the
  sampling distribution)

• Confidence intervals for the population mean are
• Centered at the sample mean
• Plus and minus za/2 times the standard deviation
  of the sample mean (the standard error from the
  sampling distribution)
• Similarly, confidence intervals for the
  population proportion will be
• Centered at the sample proportion
• Plus and minus za/2 times the standard deviation
  of the sample proportion

75
Chapter 9 Section 3

• We have already studied the distribution of the
  sample proportion is approximately normal with
• under most conditions
• We use this to construct confidence intervals for
  the population proportion

76
Chapter 9 Section 3

• The (1 a) 100 confidence interval for the
  population proportion is from
• to
• where za/2 is the critical value for the
  normaldistribution

77
Chapter 9 Section 3

• Like for confidence intervals for population
  means, the quantity
• is called the margin of error

78
Chapter 9 Section 3

• Example
• We polled n 500 voters
• When asked about a ballot question, 47 of
  them were in favor
• Obtain a 99 confidence interval for the
  population proportion in favor of this ballot
  question (a 0.005)

79
Chapter 9 Section 3

• The critical value z0.005 2.575, so
• to
• or (0.41, 0.53) is a 99 confidence interval for
  the population proportion

80
Chapter 9 Section 3

• Learning objectives
• Obtain a point estimate for the population
  proportion
• Construct and interpret a confidence interval for
  the population proportion
• Determine the sample size necessary for
  estimating a population proportion within a
  specified margin of error

81
Chapter 9 Section 3

• We often want to know the minimum sample size to
  obtain a target margin of error for the
  population proportion

• We often want to know the minimum sample size to
  obtain a target margin of error for the
  population proportion
• A common use of this calculation is in polling
  how many people need to be polled for the result
  to have a certain margin of error
• News stories often say the latest polls show
  that so-and-so will receive X of the votes with
  a E margin of error

82
Chapter 9 Section 3

• For our polling example, how many people need to
  be polled so that we are within 1 percentage
  point with 99 confidence?
• The margin of error is
• which must be 0.01
• We have a problem, though what is ?

83
Chapter 9 Section 3

• The way around this is that using
  will always yield a sample size that is large
  enough
• In our case, if we do this, then we have
• so
• and n 16,577

84
Chapter 9 Section 3

• We understand now why political polls often have
  a 3 or 4 percentage points margin of error

• We understand now why political polls often have
  a 3 or 4 percentage points margin of error
• Since it takes a large sample (n 16,577) to get
  to be 99 confident to within 1 percentage point,
  the 3 or 4 percentage points margin of error
  targets are good compromises between accuracy and
  cost effectiveness

85
Summary Chapter 9 Section 3

• We can construct confidence intervals for
  population proportions in much the same way as
  for population means
• We need to use the formula for the standard
  deviation of the sample proportion
• We can also compute the minimum sample size for a
  desired level of accuracy

86
Chapter 9Section 4

• Confidence Intervals about aPopulation Standard
  Deviation

87
Chapter 9 Section 4

• Learning objectives
• Find critical values for the chi-square
  distribution
• Construct and interpret confidence intervals
  about the population variance and standard
  deviation

88
Chapter 9 Section 4

• Learning objectives
• Find critical values for the chi-square
  distribution
• Construct and interpret confidence intervals
  about the population variance and standard
  deviation

89
Chapter 9 Section 4

• The previous sections have shown techniques for
  constructing confidence intervals for population
  means and population proportions
• Both of these are mean type problems these
  use the normal and the t-distributions
• Analyzing the standard deviation and the variance
  are different

90
Chapter 9 Section 4

• To analyze the variance and the standard
  deviation, we follow the same general approach
• We use the probability distribution that models
  the point estimator
• We find critical values for that distribution
• We construct confidence intervals based on the
  critical values
• However, the details are different

91
Chapter 9 Section 4

• The distribution of the sample mean is
  approximately normal (by the Central Limit
  Theorem)

• The distribution of the sample mean is
  approximately normal (by the Central Limit
  Theorem)
• The distribution of the sample variance is not
  normal

• The distribution of the sample mean is
  approximately normal (by the Central Limit
  Theorem)
• The distribution of the sample variance is not
  normal
• If a population has a normal distribution, then
  the sample variance s2 has a chi-square
  distribution

92
Chapter 9 Section 4

• The chi-square distribution is defined by its
  degrees of freedom
• A chi-square curve for 6 degrees of freedom

93
Chapter 9 Section 4

• Properties of the chi-square distribution

• Properties of the chi-square distribution
• The values are always greater than or equal to 0

• Properties of the chi-square distribution
• The values are always greater than or equal to 0
• It is not symmetric

• Properties of the chi-square distribution
• The values are always greater than or equal to 0
• It is not symmetric
• The shape of the distribution depends on the
  degrees of freedom (just like for the
  t-distribution)

• Properties of the chi-square distribution
• The values are always greater than or equal to 0
• It is not symmetric
• The shape of the distribution depends on the
  degrees of freedom (just like for the
  t-distribution)
• As the number of degrees of freedom increases,
  the distribution becomes more nearly symmetric

94
Chapter 9 Section 4

• The distribution of the sample variance s2 is not
  described precisely as a chi-squared
  distribution, but as
• having a chi-square distribution with n 1
  degrees of freedom (where the sample size is n,
  as usual)

95
Chapter 9 Section 4

• The calculation of chi-square distribution values
  can be done in similar ways as the calculation of
  normal values
• Using tables (such as Table VI in the textbook)
• Using technology (such as Excel, MINITAB,
  calculators, StatCrunch, etc.)

• The calculation of chi-square distribution values
  can be done in similar ways as the calculation of
  normal values
• Using tables (such as Table VI in the textbook)
• Using technology (such as Excel, MINITAB,
  calculators, StatCrunch, etc.)
• Because chi-square tables are not complete, it is
  suggested that the calculations be done with one
  of the technology methods

96
Chapter 9 Section 4

• The critical values for the chi-square
  distribution are those values that correspond to
  specific areas under the density curve
• This diagram shows the critical values ?20.05
  ?20.95 (the curve is for 15 degrees of freedom)

97
Chapter 9 Section 4

• The concept of finding critical values for the
  chi-square distribution is similar to the
  concepts of finding critical values for the
  normal and thet-distributions
• Specify a distribution (normal, t, chi-square)
• Specify a probability a
• Find the value (za, ta, ?2a) such that the area
  to the right of these values is equal to a

98
Chapter 9 Section 4

• A comparison of the three processes (for the
  normal

• A comparison of the three processes (for the
  normal, the t

• A comparison of the three processes (for the
  normal, the t, and the chi-square)

99
Chapter 9 Section 4

• The critical values for the chi-square
  distribution are those values that correspond to
  specific areas under the density curve

100
Chapter 9 Section 4

• Learning objectives
• Find critical values for the chi-square
  distribution
• Construct and interpret confidence intervals
  about the population variance and standard
  deviation

101
Chapter 9 Section 4

• Once we know the critical values for
  thechi-square distribution, then we can
  construct confidence intervals for the sample
  variance
• to

102
Chapter 9 Section 4

• The confidence intervals for the standard
  deviation are just the square roots
• to

103
Chapter 9 Section 4

• Example
• We have measured a sample standard deviation of s
  8.3 from a sample of size n 12

• Example
• We have measured a sample standard deviation of s
  8.3 from a sample of size n 12
• A 90 confidence interval for the standard
  deviation would be computed as follows
• n 12, so there are 11 degrees of freedom
• 90 confidence means that a 0.05
• ?20.05 19.68 and ?20.95 4.57

104
Chapter 9 Section 4

• The 90 confidence interval for the sample
  variance would be
• to
• And the 90 confidence interval for the sample
  standard deviation would be (6.2, 12.9) (where
  6.2 and 12.9 are the square roots of 38.5 and
  165.8 respectively)

105
Chapter 9 Section 4

• The process for the population mean and the
  process for the population variance are similar,
  but different

106
Summary Chapter 9 Section 4

• We can construct confidence intervals for
  population variances and standard deviations in
  much the same way as for population means and
  proportions
• We use the chi-square distribution to obtain
  critical values
• We divide the sample variances and standard
  deviations by the critical values to obtain the
  confidence intervals

107
Chapter 9Section 5

• Putting It All Together
• Which Procedure Do I Use?

108
Chapter 9 Section 5

• Learning objectives
• Determine the appropriate confidence interval to
  construct

109
Chapter 9 Section 5

• There are four different confidence interval
  calculations covered in Chapter 9
• It can be confusing which one is appropriate for
  which situation
• I should use the normal no, the chi-square no
  the ???

110
Chapter 9 Section 5

• The one main question right at the beginning
• Which parameter are we trying to estimate?
• A mean?
• A proportion?
• A variance or a standard deviation?
• This the single most important question

111
Chapter 9 Section 5

• The more complicated case
• Analyzing population means
• The two easy cases
• Analyzing population proportions
• Analyzing population variances and standard
  deviations

112
Chapter 9 Section 5

• For the analysis of a population mean
• If
• The data is OK (reasonably normal)
• The variance is known
• then we can use the normal distribution (section
  9.1) with a confidence interval of
• to

113
Chapter 9 Section 5

• For the analysis of a population mean
• If
• The data is OK (reasonably normal)
• The variance is NOT known
• then we can use the t-distribution (section 9.2)
  with a confidence interval of
• to

114
Chapter 9 Section 5

• For the analysis of a population mean
• If
• The data is strange
• (i.e. not normal)
• then we should use nonparametric methods

115
Chapter 9 Section 5

• For the analysis of a population proportion
• If
• n p (1 p) 10
• n .05 N
• then we can use the proportions method (section
  9.3) with a confidence interval of
• to

116
Chapter 9 Section 5

• For the analysis of a population variance or of a
  population standard deviation
• As long as
• The data is from a normal distribution
• There are no outliers
• then we can use the chi-square method (section
  9.4) with a confidence interval of
• to

117
Summary Chapter 9 Section 5

• The main questions that determine the method
• Is it a
• Population mean?
• Population proportion?
• Population variance / population standard
  deviation?
• In the case of a population mean, we need to
  determine
• Is the population variance known?
• Does the data look reasonably normal?

118
Chapter 9Section 5

• Putting It All Together
• Which Procedure Do I Use?

119
Chapter 9 Section 5

• Learning objectives
• Determine the appropriate confidence interval to
  construct

120
Chapter 9 Section 5

• There are four different confidence interval
  calculations covered in Chapter 9
• It can be confusing which one is appropriate for
  which situation
• I should use the normal no, the chi-square no
  the ???

121
Chapter 9 Section 5

• The one main question right at the beginning
• Which parameter are we trying to estimate?
• A mean?
• A proportion?
• A variance or a standard deviation?
• This the single most important question

122
Chapter 9 Section 5

• The more complicated case
• Analyzing population means
• The two easy cases
• Analyzing population proportions
• Analyzing population variances and standard
  deviations

123
Chapter 9 Section 5

• For the analysis of a population mean
• If
• The data is OK (reasonably normal)
• The variance is known
• then we can use the normal distribution (section
  9.1) with a confidence interval of
• to

124
Chapter 9 Section 5

• For the analysis of a population mean
• If
• The data is OK (reasonably normal)
• The variance is NOT known
• then we can use the t-distribution (section 9.2)
  with a confidence interval of
• to

125
Chapter 9 Section 5

• For the analysis of a population mean
• If
• The data is strange
• (i.e. not normal)
• then we should use nonparametric methods

126
Chapter 9 Section 5

• For the analysis of a population proportion
• If
• n p (1 p) 10
• n .05 N
• then we can use the proportions method (section
  9.3) with a confidence interval of
• to

127
Chapter 9 Section 5

• For the analysis of a population variance or of a
  population standard deviation
• As long as
• The data is from a normal distribution
• There are no outliers
• then we can use the chi-square method (section
  9.4) with a confidence interval of
• to

128
Summary Chapter 9 Section 5

• The main questions that determine the method
• Is it a
• Population mean?
• Population proportion?
• Population variance / population standard
  deviation?
• In the case of a population mean, we need to
  determine
• Is the population variance known?
• Does the data look reasonably normal?

129
Chapter 9

• Estimating the Value of a Parameter
• Using Confidence Intervals
• Summary

130
Chapter 9 Summary

• We can use a sample mean, proportion, variance,
  standard deviation to estimate the population
  mean, proportion, variance, standard deviation
• In each case, we can use the appropriate model to
  construct a confidence interval around our
  estimate
• The confidence interval expresses the confidence
  we have that our calculated interval contains the
  true parameter