Interval Estimation of the Population Mean for a Normal Population with s Unknown

1
Interval Estimation of the Population Mean for a
Normal Population with s Unknown
2
Students t-distribution

• If s is not known and n 30, the derivation of
  the confidence interval must be changed slightly.
• Provided the population from which the sample is
  drawn is normally distributed, the distribution
  of the quantity
• has a Students t-distribution, which was
  discovered by W. S. Gossett in 1908.

3
Degrees of Freedom

• The t-distribution has one parameter, degrees of
  freedom.
• Degrees of freedom for any t-distribution is
  computed in the following manner
• d.f. n - 1,
• where n is the number of sample observations.

4
Shape of the t-distribution

• The t-distribution is very much like the normal
  it is a symmetrical, bell-shaped distribution
  with slightly larger tails than a normal.
• In fact, as the degrees of freedom become larger,
  the t-distribution approaches the normal
  distribution.

normal
t
5
Confidence Interval for the Population Mean

• Definition
• If n 30, s is unknown, and the sample is drawn
  from a normal population, a (1 - a) confidence
  interval for the population mean is given by
• where is the critical value
• for a t-distribution with n - 1 degrees of
  freedom which captures an area of a/2 in the
  right tail of the distribution.

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Example 5

• Construct an 80 confidence interval for the mean
  of a normal population assuming that the values
  listed below comprise a random sample taken from
  the population.
• 83.9 87.4 65.2 86.0 73.1
• 80.3 92.7 87.5 69.3 77.5
• 91.9 71.1 79.1 72.4 88.2
• The population standard deviation is unknown.

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Example 5 - Solution

• We are given
• X has a normal distribution,
• the variance is unknown,
• n 15,
• and the confidence level .80.

8
Example 5 - Solution
.40
.40
.80
.10
.10
9
Example 5 - Solution

• We must calculate 80.37 and s 8.68.
  We want to determine an 80 confidence interval
  for the mean.
• Since X is normal, s is unknown and n 30, the
  confidence interval is given by

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Precision and Sample Size
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Precision

• The best interval estimates a decision maker
  could hope for would be those which are very
  small in size and possess a large amount of
  confidence.
• The width of the confidence interval defines the
  precision with which the population mean is
  estimated the smaller the interval the greater
  the precision.

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Components Which Affect the Width

• Note The entire confidence interval
• width 2 .
• represents the distance the confidence
  interval boundary is from the mean in standard
  deviation units. The distance is related to the
  specified level of confidence.
• s represents the population standard deviation.
• n represents the sample size.

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Which component can change?

• Since the population standard deviation (s) is a
  constant, it does not change.
• However, the sample size, n, is selected by the
  decision maker.
• Since the sample size can enlarge or reduce the
  width of the confidence interval, how large
  should the sample be?

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Error

• The sample size should be selected in relation to
  the size of the maximum positive or negative
  error the decision maker is willing to accept.
• This can be achieved by setting the error equal
  to one half the confidence interval width,

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Determining the Sample Size

• The equation for error can be solved for the
  sample size,
• By selecting a level of confidence and the
  maximum error, the relationship can be used to
  determine the sample size necessary to estimate
  the sample mean with the desired accuracy.
• In order to assure the desired level of
  confidence, always round the value obtained for
  the sample size up to the next whole integer.

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Example 6

• A computer software company would like to
  estimate how long it will take a beginner to
  become proficient at creating a graph using their
  new spreadsheet package.

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Example 6

• Past experience has indicated that the time
  required for a beginner to become proficient with
  a particular function of new software products
  has an approximately a normal distribution with a
  standard deviation of 15 minutes.
• Find the sample size necessary to estimate the
  average time required for a beginner to become
  proficient at creating a graph with the new
  spreadsheet package to within 5 minutes with 95
  confidence.

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Example 6 - Solution

• We are given
• s 15,
• error E 5,
• and the confidence level .95.

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Example 6 - Solution
1 - a .95
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Example 6 - Solution

• We want to determine the sample size necessary to
  estimate the mean time required for a beginner to
  become proficient at creating a graph with the
  new spreadsheet package.
• The sample size is given by
• To get the desired accuracy, we must round up to
  35.

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Sample Size for s Unknown

• The most obvious method for obtaining an estimate
  of s is to take a small sample and use the sample
  standard deviation as an estimate of the
  population standard deviation.
• Replacing s with s in the sample size
  determination relationship will provide an
  initial estimate of the required sample size.

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Estimating Population Attributes

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Attribute

• An attribute is a characteristic that members of
  a population either do or do not possess.
• Attributes are almost always measured as the
  proportion of the population that possess the
  characteristic.
• Example
• An attribute of a person would be whether they
  smoke cigarettes or not.
• p proportion of the population which smoke
  cigarettes

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Estimating the Proportion

• Estimating the proportion of the population that
  possesses an attribute is straightforward.
• A random sample is selected and the sample
  proportion is computed as follows
• X number in the sample that possess the
  attribute,
• n sample size, and

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Example 7

• The Richland Gazette, a local newspaper,
  conducted a poll of 1,000 randomly selected
  readers to determine their views concerning the
  city's handling of snow removal. The paper found
  that 650 people in the sample felt the city did a
  good job.

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Example 7

• Compute the best point estimate for the
  percentage of readers who believe the city is
  doing a good job in snow removal.

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Example 7 - Solution

• X number of people who believe the city is
  doing a good job
• X 650
• n number of randomly selected readers
• n 1000

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Interval Estimation of a Population Attribute

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Sampling Distribution of the Point Estimate

• In order to develop the confidence interval for a
  population proportion the sampling distribution
  of the point estimate must be developed.
• The random variable, , has a binomial
  distribution which is approximated with a normal
  random variable.

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The Sample Proportion

• The sample proportion, , is distributed
  normally with mean, p, and variance, .

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The Sample Proportion

• The standard deviation of the sample proportion (
  ) is denoted symbolically as and is given
  by
• where is used as an estimate of p if the
  population proportion is unknown.

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Confidence Interval for the Population Proportion

• Definition
• If the sample size is sufficiently large, np 5
  and n(1-p) 5, the 1 - a confidence interval
  for the population proportion is given by the
  expression
• where is the distance from the
• point estimate to the end of the
• interval in standard deviation units,
• and is the standard deviation of .

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Example 8

• The Peacock Cable Television Company thinks that
  40 of their customers have more outlets wired
  than they are paying for.
• A random sample of 400 houses reveals that 110 of
  the houses have excessive outlets.

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Example 8

• Construct a 99 confidence interval for the true
  proportion of houses having too many outlets.
• Do you feel the company is accurate in its belief
  about the proportion of customers who have more
  outlets wired than they are paying for?

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Example 8 - Solution

• We are given
• X number of houses that have excessive outlets
• 110,
• n 400,
• and the confidence level .99.
• Thus,

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Example 8 - Solution
1 - a .99
a .01
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Example 8 - Solution

• A 99 confidence interval for the true proportion
  of houses having too many outlets is given by

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Example 8 - Solution

• Interpretation We are 99 confident that the
  true proportion of houses having too many outlets
  is between .2175 and .3325.
• Note We do not believe that 40 of customers
  have more outlets wired than they are paying for
  because we are 99 confident that the true
  proportion is in the interval 21.75 to 33.25
  and 40 is not in that interval.

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Precision and Sample Size for Population
Attributes

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Accuracy in Estimating a Population Proportion

• Just as for the population mean, a specific level
  of accuracy in estimating a population proportion
  is desirable.
• When we estimate extremely small quantities,
  highly precise estimates are necessary.

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Deriving the Sample Size

• The technique for deriving the sample size
  parallels the discussion of the sample mean.
• Setting one half the entire width of the
  confidence interval equal to the maximum
  allowable error yields
• error
• Solving for n yields
• n

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Sample Size when is Known

• Generally the population proportion is unknown
  and is estimated from a pilot study.
• In this case, the sample size necessary to
  estimate the population proportion to within a
  particular error with a certain level of
  confidence is given by
• n
• where is the estimate obtained from the pilot
  study.

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Sample Size when is Unknown

• If an estimate of the population proportion is
  not available, then the population proportion is
  set equal to .5 and the sample size is given by
• n
• The value .5 maximizes the quantity p(1 - p) and
  thus provides the most conservative estimate of
  the sample size.

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Example 9

• Researchers working in a remote area of Africa
  feel that 40 of the families in the area are
  without adequate drinking
  water either
  through
  contamination or
  unavailability.
• What sample size will be
  necessary to estimate the
  percent without adequate water to within 5 with
  95 confidence?

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Example 9 - Solution

• We are given
• ,
• error E .05,
• and the confidence level .95.

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Example 9 - Solution

1 - a .95
47
Example 9 - Solution

• We want to determine the sample size necessary to
  estimate the proportion of families in the area
  without adequate drinking water. Since we have an
  estimate of , the sample size is given by
• To get the desired accuracy, we must round up to
  369 families.