ECON 2300 LEC

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ECON 2300 LEC11

• 10/16/06

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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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• Type I error (Example Automobile product
  research group) Researchers claim that the new
  system improves the miles-per-gallon rating
  (µgt24) when in fact the new system is not any
  better than the current system.
• Type II error Researchers conclude that the new
  system is not any better than the current system
  (µlt24) when in fact the new system improves the
  miles-per-gallon performance.

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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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Tests about a Population Mean Large-Sample s
known

• Hypotheses
• Upper Tailed Test Lower Tailed
  Test
• H0 ?????? ?
  H0 ?????? Ha???????? ?
  Ha????????
• Test Statistic
• Rejection Rule
• Reject H0 if z gt z???
  Reject H0 if z lt -z?
• Reject H0 if p lt a
  Reject H0 if p lt a
• 
  

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• Example Federal Trade Commission periodically
  conducts statistical studies designed to test the
  claims that manufacturers make about their
  products. For example, the label on a large can
  of Hilltop Coffee states that the can contains 3
  pounds of coffee. FTC wants to check Hilltops
  claim by conducting a hypothesis test.
• Null hypothesis H0µgt3
• Alternate hypothesis
  Haµlt3

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• Alternate hypothesis indicates that it is lower
  tailed test.
• A sample of 36 cans of coffee is selected and the
  sample mean 2.92 is obtained.
• Director of FTC selected a significance level of
  .01
• Test Statistic Historical data tells that the
  population standard deviation can be assumed to
  be .18 and the population of filling weights
  follows a normal distribution.

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s Known

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

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Tests about a Population Mean Large-Sample s
known

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

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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 lt a Reject H0 if p
  lt a Reject H0 if p lt a
• 
  

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Example

• US Golf Association establishes rules that
  manufacturers of golf equipment must meet if
  their products are to be acceptable for use in
  USGA events. MaxFlight uses a high-technology
  manufacturing process to produce golf balls with
  a mean driving distance of 295 yards. If the
  balls produced give mean driving distance lower
  than 295 yards their sales will drop and if it
  gives more than 295 yards then golf balls may be
  rejected by the USGA for exceeding the overall
  distance standard concerning carry and roll.
• Hypotheses
• H0 ???295
• H0 ????295

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• Historical data showed that the population
  standard deviation can be assumed known with a
  value of s12.
• A sample of size 50 was collected to calculate
  the sample mean, x 297.6
• Level of significance taken as 0.05

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Example

• The mean length of a work week for the population
  of
• workers was reported to be 39.2 hours. Suppose
  that we
• would like to take a current sample of workers to
  see
• whether the mean length of a work week has
  changed from
• the previously reported 39.2 hours.
• State the hypothesis that will help us determine
  whether a change occurred in the mean length of a
  work week.
• Suppose a current sample of 112 workers provided
  a sample mean of 38.5 hours. Use a population
  standard deviation equal to 4.8 hours. What is
  the p-value?
• At alpha .05, can the null hypothesis be
  rejected? What is your conclusion?
• Repeat the preceding hypothesis test using the
  critical value approach.

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Example

• In 2001, the U.S. Department of Labor
• reported the average hourly earnings of U.S.
• production workers to be 14.32 per hour. A
• sample of 75 production workers during 2003
• showed a sample mean of 14.68 per hour.
• Assuming the population standard deviation
• equal to 1.45, can we conclude that an increase
• occurred in the mean hourly earnings since
• 2001? Use alpha .05

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Relationship between Interval Estimation and
Hypothesis Testing

• Suppose the Hypothesis is of the following
• form
• H0µ µ0
• Haµ ? µ0
• Steps
• Select a simple random sample from the population
  and use the value of the sample mean to develop
  the confidence interval for the population

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• 2. If the confidence interval contains the
  hypothesized value, do not reject Ho. Otherwise,
  reject Ho
• Confidence interval for MaxFlight Example
• H0µ 295
• Haµ ? 295

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s unknown

• Standard deviation unknown sample standard
  deviation used to estimate population standard
  deviation
• One-Tailed Test

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Tests about a Population Mean Large-Sample s
unknown

• Hypotheses
• Upper Tailed Test Lower Tailed
  Test
• H0 ?????? ?
  H0 ?????? Ha???????? ?
  Ha????????
• Test Statistic
• Rejection Rule
• Reject H0 if t gt t???
  Reject H0 if t lt -t?
• Reject H0 if p lt a
  Reject H0 if p lt a
• 
  

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s unknown

• Standard deviation unknown Sample standard
  deviation used to estimate population standard
  deviation
• Two-Tailed Test

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Tests about a Population Mean Large-Sample s
unknown

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

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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
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 lt a Reject H0 if p
  lt a Reject H0 if p lt a
• 
  

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Example

• Holiday Toys manufactures and distributes
  its products through more than 1000 retail
  outlets. In planning production levels for the
  coming winter season. Holiday must decide how
  many units of each product to produce prior to
  knowing the actual demand at the retail level.
  For this years most important new toy, Holidays
  marketing director is expecting demand to average
  40 units per outlet. Prior to making the final
  production decision based upon the estimate,
  Holiday decided to survey a sample of 25
  retailers in order to develop more information
  about the new product.
• Hypotheses
• H0 ???40
• H0 ????40
• Sample of 25 retailers provided a sample mean of
  37.4 and sample standard deviation of 11.79 units.

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Example

• The National Association of Professional
  Baseball Leagues, Inc., reported that attendance
  for 176 minor league baseball teams reached an
  all-time high during the 2001 season. On a
  per-game basis, the mean attendance for minor
  league baseball was 3530 people per game. Midway
  through 2002 season, the president of the
  association asked for an attendance report that
  would hopefully show that the mean attendance for
  2002 was exceeding the 2001 level.
• Formulate hypotheses that could be used to
  determine whether the mean attendance per game in
  2002 was greater than the previous years level.
• Assume that a sample of 92 minor league baseball
  games played during the first half of the 2002
  season showed a mean attendance of 3740 people
  per game with a sample standard deviation of 810.
  What is the p-value?
• At alpha .01, what is your conclusion?

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Summary of Test Statistics to be Used in
aHypothesis Test about a Population Mean
Yes
s known
No
Use s to estimate s
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Tests about a Population Mean Large-Sample s
unknown

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

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• Example Over the past year, 20 of the players
  at Pine Creek were women. In an effort to
  increase the proportion of women players, Pine
  Creek implemented a special promotion designed to
  attract women golfers. One month after the
  promotion was implemented, the course manager
  requested a statistical study to determine
  whether the proportion of women players at Pine
  Creek had increased.
• ? Hypothesis
• Ho plt0.20
• Ha pgt0.20

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• A sample of 400 players was selected and 100 of
  the players were women. The proportion of women
  golfers is .25
• Since npgt5 and n(1-p)gt5, the sampling
  distribution is normal
• Level of significance chosen for the test is 0.05

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Example

• An article about driving practices in
  Strathcona County, Alberta, Canada, claimed that
  48 of drivers did not stop at stop sign
  intersections on county roads. Two months later,
  a follow-up study collected data in order to see
  whether this percentage had changed.
• Formulate the hypothesis to determine whether the
  proportion of drivers who did not stop at stop
  sign intersection had changed.
• Assume the study found 360 of 800 drivers did not
  stop at stop sign intersections. What is the
  sample proportion? What is the p-value?
• At alpha.05, what is your conclusion?