Chapter 8 Confidence Intervals

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Chapter 8Confidence Intervals

• 8.1
• Confidence Intervals about a Population Mean, ?
  Known

2

• Recall
• A population parameter is a characteristic of a
  population.

3

• Recall
• A population parameter is a characteristic of a
  population.
• population mean ?

4

• Recall
• A population parameter is a characteristic of a
  population.
• population mean ?
• population s.d. ?

5

• Recall
• A population parameter is a characteristic of a
  population.
• population mean ?
• population s.d. ?
• It is a fixed but unknown quantity.

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• A function of the data sample is called a
  statistic.

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• A function of the data sample is called a
  statistic.
• A statistic is a random quantity because the
  data sample is random.

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• A function of the data sample is called a
  statistic.
• A statistic is a random quantity because the
  data sample is random.
• Statistics which are used to estimate
  population parameters are called estimators.

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• The population mean ? is a parameter.

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• The population mean ? is a parameter.
• The random variable X is an estimator of ?.

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• The population mean ? is a parameter.
• The random variable X is an estimator of ?.
• The actual value of the sample mean x is a
  point estimate of ?.

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• The population mean ? is a parameter.
• The random variable X is an estimator of ?.
• The actual value of the sample mean x is a
  point estimate of ?.
• Sampling error is the error in using x to
  estimate ? due to the random sample.

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• The population mean ? is a parameter.
• The random variable X is an estimator of ?.
• The actual value of the sample mean x is a
  point estimate of ?.
• Sampling error is the error in using x to
  estimate ? due to the random sample.
• Because of the unavoidable sampling error, we
  would like to attach a level of confidence to the
  estimate x

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• For continuous random variables,
• P(X ?) 0

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• For continuous random variables,
• P(X ?) 0
• Probabilities for continuous random variables
  are expressed in terms of intervals of numbers.

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P(x1lt X lt x2)
x1
x2
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• For continuous random variables,
• P(X ?) 0
• Probabilities for continuous random variables
  are expressed in terms of intervals of numbers.
• Instead of attaching a level of confidence to a
  single point estimate, we attach a level of
  confidence to an interval of numbers containing
  the point estimate.

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A confidence interval estimate of a parameter
consists of
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A confidence interval estimate of a parameter
consists of (1) An interval of numbers
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A confidence interval estimate of a parameter
consists of (1) An interval of numbers (2)
The probability that the interval contains the
unknown parameter.
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• The level of confidence in a confidence
  interval is the probability that the parameter is
  contained in the interval.

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• The level of confidence in a confidence
  interval is the probability that the parameter is
  contained in the interval.
• It represents the percentage of intervals that
  would contain the population parameter if samples
  were taken repeatedly.

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• For example
• Construct a 95 confidence interval for the
  mean.

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• For example
• Construct a 95 confidence interval for the
  mean.
• Of 100 such confidence intervals, about 95 of
  the intervals should contain the population mean.

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The confidence interval for ? depends on three
things
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• The confidence interval for ? depends on three
  things
• The sample mean x.

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• The confidence interval for ? depends on three
  things
• The sample mean x.
• The level of confidence

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• The confidence interval for ? depends on three
  things
• The sample mean x.
• The level of confidence (1 - ?)100

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• The confidence interval for ? depends on three
  things
• The sample mean x.
• The level of confidence (1 - ?)100
• The standard deviation of the sample mean

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• The confidence interval for ? depends on three
  things
• The sample mean x.
• The level of confidence (1 - ?)100
• The standard deviation of the sample mean

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• Suppose a random variable X N(?,?).

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• Suppose a random variable X N(?,?).
• We want to estimate the mean ?.

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• Suppose a random variable X N(?,?).
• We want to estimate the mean ?.
• We obtain a random sample of size n
• X1, X2 ,. . . ,Xn

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• Suppose a random variable X N(?,?).
• We want to estimate the mean ?.
• We obtain a random sample of size n
• X1, X2 ,. . . ,Xn
• X (X1 X2 . . . Xn) / n

35

• Suppose a random variable X N(?,?).
• We want to estimate the mean ?.
• We obtain a random sample of size n
• X1, X2 ,. . . ,Xn
• X (X1 X2 . . . Xn) / n
• How precise is our estimate?

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• Suppose a random variable X N(?,?).
• We want to estimate the mean ?.
• We obtain a random sample of size n
• X1, X2 ,. . . ,Xn
• X (X1 X2 . . . Xn) / n
• How precise is our estimate?
• In other words, how confident are we that the
  sample mean X is close to ??

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• Since X N(?,?)

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• Since X N(?,?) X N(?X, ?X)

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• Since X N(?,?) X N(?X, ?X)
• ?X ?

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• Since X N(?,?) X N(?X, ?X)
• ?X ? ?X ? / sqrt(n)

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• Since X N(?,?) X N(?X, ?X)
• ?X ? ?X ? / sqrt(n)
• Suppose we want to construct a 95 confidence
  interval for the mean.

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• Since X N(?,?) X N(?X, ?X)
• ?X ? ?X ? / sqrt(n)
• Suppose we want to construct a 95 confidence
  interval for the mean.
• In other words we want an interval (x1, x2)

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• Since X N(?,?) X N(?X, ?X)
• ?X ? ?X ? / sqrt(n)
• Suppose we want to construct a 95 confidence
  interval for the mean.
• In other words we want an interval (x1, x2)
• P(x1 lt ? lt x2) 0.95

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• Z (X - ?x) / ?x

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• Z (X - ?x) / ?x
• Z (X - ?) / ?/ sqrt(n)

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• Z (X - ?x) / ?x
• Z (X - ?) / ?/ sqrt(n)
• P( -1.96 lt Z lt 1.96) 0.95

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• Z (X - ?x) / ?x
• Z (X - ?) / ?/ sqrt(n)
• P( -1.96 lt Z lt 1.96) 0.95
• -1.96 lt Z lt1.96

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• Z (X - ?x) / ?x
• Z (X - ?) / ?/ sqrt(n)
• P( -1.96 lt Z lt 1.96) 0.95
• -1.96?/sqrt(n)lt Z?/sqrt(n) lt1.96?/sqrt(n)

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• Z (X - ?x) / ?x
• Z (X - ?) / ?/ sqrt(n)
• P( -1.96 lt Z lt 1.96) 0.95
• ?-1.96?/sqrt(n)lt ?Z?/sqrt(n)lt ?1.96?/sqrt(n)
  

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• Z (X - ?x) / ?x
• Z (X - ?) / ?/ sqrt(n)
• P( -1.96 lt Z lt 1.96) 0.95
• ?-1.96?/sqrt(n)lt ?Z?/sqrt(n)lt ?1.96?/sqrt(n)
  
• X ? Z?/sqrt(n)

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• Z (X - ?x) / ?x
• Z (X - ?) / ?/ sqrt(n)
• P( -1.96 lt Z lt 1.96) 0.95
• ?-1.96?/sqrt(n)lt X lt ?1.96?/sqrt(n)
• X ? Z?/sqrt(n)

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• Z (X - ?x) / ?x
• Z (X - ?) / ?/ sqrt(n)
• P( -1.96 lt Z lt 1.96) 0.95
• P(?-1.96?/sqrt(n)lt X lt ?1.96?/sqrt(n)) 0.95
• X ? Z?/sqrt(n)

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95 of all sample means are in the interval
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• P(?-1.96?/sqrt(n)lt X lt ?1.96?/sqrt(n)) 0.95

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• ?-1.96?/sqrt(n)lt X lt ?1.96?/sqrt(n)

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• ?-1.96?/sqrt(n)lt X lt ?1.96?/sqrt(n)
• subtract ? from all terms

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• -1.96?/sqrt(n)lt X - ? lt 1.96?/sqrt(n)

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• 1.96?/sqrt(n)lt X - ? lt 1.96?/sqrt(n)
• subtract X from all terms

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• -1.96?/sqrt(n) - X lt - ? lt 1.96?/sqrt(n) -
  X

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• 1.96?/sqrt(n) - X lt - ? lt 1.96?/sqrt(n) -
  X
• multiply all terms by -1

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• X -1.96?/sqrt(n) lt ? lt X 1.96?/sqrt(n)

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• P(X -1.96?/sqrt(n) lt ? lt X 1.96?/sqrt(n))
• 0.95

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EXAMPLE Computing a Point Estimate for the
Population Mean

• Specifications call for the penny to weigh 2.50
  grams.

67
EXAMPLE Computing a Point Estimate for the
Population Mean

• Specifications call for the penny to weigh 2.50
  grams.
• The following data represent the weights (in
  grams) of 17 randomly selected pennies minted
  after 1982
• 2.46 2.47 2.49 2.48 2.50 2.44
  2.46 2.45 2.49
• 2.47 2.45 2.46 2.45 2.46 2.47
  2.44 2.45

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EXAMPLE Computing a Point Estimate for the
Population Mean

• Specifications call for the penny to weigh 2.50
  grams.
• The following data represent the weights (in
  grams) of 17 randomly selected pennies minted
  after 1982
• 2.46 2.47 2.49 2.48 2.50 2.44
  2.46 2.45 2.49
• 2.47 2.45 2.46 2.45 2.46 2.47
  2.44 2.45
• Estimate the population mean weight of pennies

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EXAMPLE Computing a Point Estimate for the
Population Mean

• Specifications call for the penny to weigh 2.50
  grams.
• The following data represent the weights (in
  grams) of 17 randomly selected pennies minted
  after 1982
• 2.46 2.47 2.49 2.48 2.50 2.44
  2.46 2.45 2.49
• 2.47 2.45 2.46 2.45 2.46 2.47
  2.44 2.45
• Estimate the population mean weight of pennies
• x 2.464

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EXAMPLE Computing a Point Estimate for the
Population Mean

• The population mean is supposed to be 2.50
  grams.

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EXAMPLE Computing a Point Estimate for the
Population Mean

• The population mean is supposed to be 2.50
  grams.
• The sample mean x 2.46.

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EXAMPLE Computing a Point Estimate for the
Population Mean

• The population mean is supposed to be 2.50
  grams.
• The sample mean x 2.46.
• Is there evidence that the pennies are not being
  manufactured according to specification?

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EXAMPLE Computing a Point Estimate for the
Population Mean

• The population mean is supposed to be 2.50
  grams.
• The sample mean x 2.46.
• Is there evidence that the pennies are not being
  manufactured according to specification?
• The weights of the pennies are normally
  distributed with standard deviation ? 0.02.

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EXAMPLE Computing a Point Estimate for the
Population Mean

• The population mean is supposed to be 2.50
  grams.
• The sample mean x 2.46.
• Is there evidence that the pennies are not being
  manufactured according to specification?
• The weights of the pennies are normally
  distributed with standard deviation ? 0.02.
• Additionally, if the pennies are being
  manufactured according to specification, ?
  2.50.

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• X weight of penny

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• X weight of penny
• X N(2.50 g, 0.02 g)

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• X weight of penny
• X N(2.50 g, 0.02 g)
• n 17

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• X weight of penny
• X N(2.50 g, 0.02 g)
• n 17
• X 2.46 g

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• X weight of penny
• X N(2.50 g, 0.02 g)
• n 17
• X 2.46 g
• ? 0.02

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• X weight of penny
• X N(2.50 g, 0.02 g)
• n 17
• X 2.46 g
• ? 0.02
• P(X -1.96?/sqrt(n) lt ? lt X 1.96?/sqrt(n))
• 0.95

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• X weight of penny
• X N(2.50 g, 0.02 g)
• n 17
• X 2.46 g
• ? 0.02
• P(2.46-1.96?/sqrt(n) lt ? lt 2.461.96?/sqrt(n))
  
• 0.95

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• X weight of penny
• X N(2.50 g, 0.02 g)
• n 17
• X 2.46 g
• ? 0.02
• P(2.46-1.96(.02)/sqrt(n)lt?lt2.461.96(.02)/sqrt(n)
  )
• 0.95

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• X weight of penny
• X N(2.50 g, 0.02 g)
• n 17
• X 2.46 g
• ? 0.02
• P(2.46-1.96(.005)lt?lt2.461.96(.005))
• 0.95

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• X weight of penny
• X N(2.50 g, 0.02 g)
• n 17
• X 2.46 g
• ? 0.02
• P(2.46- .010lt?lt2.46.010)
• 0.95

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• X weight of penny
• X N(2.50 g, 0.02 g)
• n 17
• X 2.46 g
• ? 0.02
• P(2.45 lt ? lt 2.47)
• 0.95

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• A 95 confidence interval for the mean is
  (2.45g, 2.47g)

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• A 95 confidence interval for the mean is
  (2.45g, 2.47g)
• In other words, 95 of the time the true mean ?
  should lie between 2.45 grams and 2.47 grams.
  

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• A 95 confidence interval for the mean is
  (2.45g, 2.47g)
• In other words, 95 of the time the true mean ?
  should lie between 2.45 grams and 2.47 grams.
• Since 2.50 grams is not in this interval, there
  is evidence that the pennies are not being
  manufactured to specification.

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• P(X -1.96?/sqrt(n) lt ? lt X 1.96?/sqrt(n))
• 0.95

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• P(X -1.96?/sqrt(n) lt ? lt X 1.96?/sqrt(n))
• 0.95
• Recall that 1.96 z0.025

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P( Z gt z0.025) 0.025
0.975
0.025

z0.025
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P( Z gt 1.96) 0.025
0.975
0.025

1.96
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• P(X -1.96?/sqrt(n) lt ? lt X 1.96?/sqrt(n))
• 0.95
• Recall that 1.96 z0.025
• In general z?/2 is the number such that
• P(Z gt z?/2 ) ?/2

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P( Z gt z?/2 ) ?/2
1- ?/2
?/2

z?/2
95

• P(X -1.96?/sqrt(n) lt ? lt X 1.96?/sqrt(n))
• 0.95
• Recall that 1.96 z0.025
• In general z?/2 is the number such that
• P(Z gt z?/2 ) ?/2
• P(Z lt z?/2 ) 1 - ?/2

96
P( Z lt -z?/2 ) ?/2
?/2
?/2

z?/2
-z?/2
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P(-z?/2 lt Z lt z?/2 ) 1 - ?
1- ?
?/2
?/2

z?/2
-z?/2
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• P(X -1.96?/sqrt(n) lt ? lt X 1.96?/sqrt(n))
• 0.95
• Recall that 1.96 z0.025
• In general z?/2 is the number such that
• P(Z gt z?/2 ) ?/2
• P(Z lt z?/2 ) 1 - ?/2
• P(- z?/2 lt Z lt z?/2 ) 1 - ?

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P(X - z?/2 ?/sqrt(n) lt ? lt X z?/2 ?/sqrt(n))
1 - ?
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P(X - z?/2 ?/sqrt(n) lt ? lt X z?/2 ?/sqrt(n))
1 - ? A (1 - ?)100 confidence interval
for the mean is given by (X - z?/2 ?/sqrt(n) ,
X z?/2 ?/sqrt(n) )
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EXAMPLE Constructing a Confidence
Interval 2.46 2.47 2.49 2.48 2.50 2.44 2.46 2.
45 2.49 2.47 2.45 2.46 2.45 2.46 2.47 2.44 2.45
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Weight (in grams) of Pennies
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EXAMPLE Role of Margin of Error Construct a 90
confidence interval for the mean weight of
pennies minted after 1982. Comment on the effect
decreasing the level of confidence has an the
margin of error.
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EXAMPLE The Role of Sample Size on the Margin
of Error
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Determining the Sample Size n
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EXAMPLE Determining Sample Size