ECON 3300 LEC

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ECON 3300 LEC 9

• 10/09/06

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Mean (Average)

• Most important measure of central location
• Arithmetic average sum of the observations
  divided by the number of observations
• Mean (sample) - and Mean (population)

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Mean
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Variance

• Indicates closeness of scores in the distribution
  to the middle of the distribution.
• Variance-Average squared difference of the scores
  from the mean.
• Measure of variation for interval-ratio values
• Procedure
• Calculate the mean of the data set
• Obtain the deviation of the scores from the mean
• Square and sum all the deviations
• Note Mean deviation from the mean is zero

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Population variance

• Variance of data for a population population
  variance
• It is denoted by s2

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Sample variance

• It is used to estimate population variance.
• Sum of squared deviations about the sample mean
  divided by n-1 Unbiased estimate of the
  population variance
• It is denoted by s2

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Sample variance

• Alternative formula

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Standard Deviation

• Positive square root of variance
• Measure of variation for interval-ratio values
• It is especially useful measure normal or
  approximately normal
• Proportion of distribution within a given number
  of standard deviations from the mean
• 68 of the distribution within one standard
  deviations of the mean
• 95 of the distribution within two standard
  deviations of the mean

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Standard deviation

• Sample standard deviation
• Measured in the same units as the original data

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Sampling

• Population Set of all elements of interest in a
  study
• Sample subset of the population
• Statistical Inference Develop estimates and
  test hypothesis about population parameters with
  help of samples.
• Example 1 A tire manufacturer using a sample of
  120 tires to obtain the mean mileage expected
  from the new tires
• Example 2 Members of party trying to decide on
  supporting a candidate use a sample of 400
  registered voters to estimate the proportion
  supporting the candidate.
• Appropriate sampling methods necessary to obtain
  the correct population parameter estimates and
  make the correct decision

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Point Estimation

• In point estimation we use the data from the
  sample to compute a value of a sample statistic
  that serves as an estimate of a population
  parameter.
• We refer to as the point estimator of the
  population mean ?.
• s is the point estimator of the population
  standard deviation ?.
• is the point estimator of the population
  proportion p.

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Example St. Andrews

• St. Andrews College receives 900 applications
• annually from prospective students. The
  application
• forms contain a variety of information including
  the
• individuals scholastic aptitude test (SAT) score
  and
• whether or not the individual desires on-campus
• housing.

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Example St. Andrews

• The director of admissions would like to know
• the following information
• the average SAT score for the applicants, and
• the proportion of applicants that want to live on
  campus.

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Example St. Andrews

• We will now look at three alternatives for
  obtaining
• the desired information.
• Conducting a census of the entire 900 applicants
• Selecting a sample of 30 applicants, using a
  random number table
• Selecting a sample of 30 applicants, using
  computer-generated random numbers

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Example St. Andrews

• Taking a Census of the 900 Applicants
• SAT Scores
• Population Mean
• Population Standard Deviation

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Example St. Andrews

• Taking a Census of the 900 Applicants
• Applicants Wanting On-Campus Housing
• Population Proportion

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Example St. Andrews

• Point Estimates
• as Point Estimator of ?
• s as Point Estimator of ?
• as Point Estimator of p

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Interval Estimate

• Point estimator cannot be expected to provide the
  exact value of the population parameter
• Interval Estimate Provides information on how
  close the point estimate provided by the sample,
  is to the value of the population parameter.
• General Form Point estimate Margin of error.
• Population mean

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Summary of Interval Estimation Procedures for a
Population Mean
Yes
No
s known
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Hypothesis Testing

• Determines whether the statement about the value
  of a population parameter should or should not be
  rejected.
• Steps involved
• A tentative assumption is made about a population
  parameter Null hypothesis (H0)
• Another hypothesis called Alternate hypothesis is
  defined (Ha) which is exactly opposite of null
  hypothesis
• Hypothesis testing involves using data from
  sample to test the two competing statements.

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Developing Null and Alternate Hypothesis

• Not always obvious how null and alternate
  hypothesis should be formulated.
• Hypothesis needs to be structured appropriately
  Conclusion provides information the researcher or
  decision maker wants.
• Example Testing Research Hypothesis
• Consider a particular automobile company that
  currently attains an average fuel efficiency of
  24 miles per gallon. A product research group
  developed a new fuel injection system
  specifically designed to increase the
  miles-per-gallon rating. To evaluate the new
  system, several will be manufactured, installed
  in automobiles, and subjected to
  research-controlled driving tests. The product
  research group is looking for evidence to
  conclude that

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• the new system will increase the mean
  miles-per-gallon rating.
• Null hypothesis H0µlt24
• Alternate hypothesis
  Haµgt24
• Example 2 Consider the situation of a
• manufacturer of soft drinks who states that two
• liter containers of its products contain an
  average
• of at least 67.6 fluid ounces. A sample of
  two-liter
• containers will be selected, and the contents
  will
• be measured to test the manufacturers claim.
• Null hypothesis
  H0µgt67.6
• Alternate hypothesis
  Haµlt67.6

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• Example 3 On the basis of a sample of parts from
  a shipment just received, a quality control
  inspector must decide whether to accept a
  shipment or to return the shipment to the
  supplier because it does not meet specifications.
  Assume that specifications for a particular part
  require mean length of 2 inches per part. If mean
  length is greater or less than 2-inch standard,
  the parts will cause quality problems in the
  assembly operation.
• Null hypothesis H0µ2
• Alternate hypothesis
  Haµ?2

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A Summary of Forms for Null and Alternative
Hypotheses about a Population Mean

• The equality part of the hypotheses always
  appears in the null hypothesis.
• In general, a hypothesis test about the value of
  a population mean ?? must take one of the
  following three forms (where ?0 is the
  hypothesized value of the population mean).
• H0 ? gt ?0 H0 ? lt ?0 H0
  ? ?0
• Ha ? lt ?0 Ha ? gt ?0
  Ha ? ? ?0

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Type I and II errors

• Null and alternate hypotheses are competing
  statements about the population.
• Either the Null hypothesis is true or the
  alternate hypothesis is true.
• Ideally the hypothesis testing procedure should
  lead to acceptance of H0 when H0 is true and
  rejection of H0 when Ha is true
• Errors are possible because conclusions made on
  basis of samples.

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Type I and II errors
Population Condition
Conclusion
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Level of Significance

• The level of significance is the probability of
  making a Type I error when the null hypothesis is
  true as an equality
• a is used to represent the level of significance
• Common choices of value for a 0.05 and 0.01
• By selecting a person is controlling the
  probability of making a Type I error
• Selection of a depends on the cost incurred if
  Type I error occurs.
• Hypothesis testing applications control for the
  Type I error and not always for Type II error
  deciding to accept Ho does not indicate any
  confidence in the decision.

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Level of Significance

• Thereby preferred statement Do not reject Ho
  instead of Accept Ho

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s Known

• Standard deviation known when large amount of
  historical data available
• One-Tailed Test

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Steps of Hypothesis Testing

• Determine the null and alternative hypotheses.
• Specify the level of significance ?.
• Select the test statistic that will be used to
  test the hypothesis.
• p-Value Approach
• Use the value of the test statistic to compute
  the p-
• value.
• 5. Reject H0 if p-value lt a.
• Critical Value Approach
• 4. Use level of significance to determine the
  critical value and the rejection rule for H0.
• 5. Use the value of the test statistic and the
  rejection rule to determine whether to reject H0.

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Tests about a Population Mean Large-Sample s
known

• Hypotheses
• H0 ?????? ? H0 ?????? H0 ?????
• Ha???????? ?Ha???????? H0
  ??????
• Test Statistic
• Rejection Rule
• Reject H0 if z gt z???Reject H0 if z lt -z?
  Reject H0 if z lt -z?/2
• 
  Reject H0 if z gt z?/2
• Reject H0 if p gt a Reject H0 if p lt
  a Reject H0 if p lt a
• 
  

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Tests about a Population Mean Large-Sample s
unknown

• Hypotheses
• H0 ?????? ? H0 ?????? H0 ?????
• Ha???????? ?Ha???????? H0
  ??????
• Test Statistic
• Rejection Rule
• Reject H0 if t gt t???Reject H0 if t lt -t?
  Reject H0 if t lt -t?/2
• 
  Reject H0 if t gt t?/2
• Reject H0 if p gt a Reject H0 if p lt
  a Reject H0 if p lt a
• 
  

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Summary of Test Statistics to be Used in
aHypothesis Test about a Population Mean
Yes
No
n gt 30 ?
No
Popul. approx. normal ?
s known ?
Yes
Yes
Use s to estimate s
No
s known ?
No
Use s to estimate s
Yes
Increase n to gt 30