Download - Economics 105: Statistics
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Economics 105: Statistics• Any questions?• GH 11 and GH 12 due on Friday
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What is a Hypothesis?• A hypothesis is a claim
(assumption) about a population parameter:
Example: The mean monthly cell phone bill of this city is μ = $42
Example: The proportion of adults in this city with cell phones is π = 0.68
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The Null Hypothesis, H0
• States the claim or assertion to be tested
Example: The average number of TV sets in
U.S. Homes is equal to three ( )
• Is always about a population parameter, not about a sample statistic
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The Null Hypothesis, H0
• Begin with the assumption that the null hypothesis is true–Similar to the notion of innocent until
proven guilty• Refers to the status quo• Always contains “=” , “≤” or “” sign• May or may not be rejected
(continued)
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The Alternative Hypothesis, H1
• Is the opposite of the null hypothesis– e.g., The average number of TV sets in U.S. homes
is not equal to 3 ( H1: μ ≠ 3 )
• Challenges the status quo• Never contains “=” , “≤” or “” signs• May or may not be proven find evidence in
favor of H1
• Is generally the hypothesis that the researcher is trying to prove to find evidence in favor of
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Population
Claim: thepopulationmean age is 50.(Null Hypothesis:
REJECT
Supposethe samplemean age is 20: X = 20
SampleNull Hypothesis
20 likely if μ = 50?=Is
Hypothesis Testing Process
If not likely,
Now select a random sample
H0: μ = 50 )
X
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Sampling Distribution of X
μ = 50If H0 is true
If it is unlikely that we would get a sample mean of this value ...
... then we reject the null
hypothesis that μ = 50.
Reason for Rejecting H0
20
... if in fact this were the population mean…
X
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Level of Significance, • Defines the unlikely values of the sample statistic
if the null hypothesis is true
– Defines rejection region of the sampling distribution
• Is designated by , (level of significance)
– Typical values are 0.01, 0.05, or 0.10
• Is selected by the researcher at the beginning
• Provides the critical value(s) of the test
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Level of Significance and the Rejection Region
H0: μ ≥ 3
H1: μ < 30
H0: μ ≤ 3
H1: μ > 3
a
a
Represents critical value
Lower-tail test
Level of significance = a
0Upper-tail test
Two-tail test
Rejection region is shaded
/2
0
a /2aH0: μ = 3
H1: μ ≠ 3
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Hypothesis TestingStates of Nature
Decision on H0H0 is true H0 is false
Fail to reject H0
(“accept” H0)
Probability = ?
Correct decision
Probability =
Type II error
Reject H0 Probability = Significance level
Type I error
Probability = ?Power
Correct decision
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Type I & II Error Relationship
Type I and Type II errors cannot happen at the same time
Type I error can only occur if H0 is true
Type II error can only occur if H0 is false
If Type I error probability ( ) , then
Type II error probability ( β )
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Hypothesis Testing for • Pharmaceutical manufacturer is concerned about impurity concentration in pills, not wanting it to be above 3%. From past production runs, it knows that the impurity concentration in the pills is normally distributed with a standard deviation () of .4%.• A random sample of 64 pills was drawn and found to have a mean impurity level of 3.07%. • Test the following hypothesis at the 5% level on the test statistic scale.
• Perform the test on the sample statistic scale.• What is the p-value for this test? Power if true pop mean = 3.1%? • p-value is the lowest significance level at which you can reject H0.
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What are the appropriate H0 & H1?• The Federal Trade Commission wants to prosecute
General Mills for not filling its cereal boxes with the advertised weight.
• Toyota won’t accept a shipment of tires from its supplier if the tires won’t fit their cars.
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What are the appropriate H0 & H1?• A professor would like to know if having a stats lab
increases student grades relative to a class without a lab.
• Ballbearings-R-Us won’t accept a shipment of ball bearings if more than 5% of the shipment is defective.
• A firm that sends out advertising flyers wants to convince potential customers (i.e., firms) that it can increase their sales.
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Hypothesis Testing for • Pharmaceutical manufacturer is concerned about impurity concentration in pills, not wanting it to be above 3%. From past production runs, it knows that the impurity concentration in the pills is normally distributed with a standard deviation () of .4%.• A random sample of 64 pills was drawn and found to have a mean impurity level of 3.07%. • Test the following hypothesis at the 5% level on the test statistic scale.
• Perform the test on the sample statistic scale.• What is the p-value for this test? Power if true pop mean = 3.1%? • p-value is the lowest significance level at which you can reject H0.
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Hypothesis Testing for Using t• Pharmaceutical manufacturer is concerned about impurity concentration in pills, not wanting it to be different than 3%. A random sample of 16 pills was drawn and found to have a mean impurity level of 3.07% and a standard deviation (s) of .6%.• Test the following hypothesis at the 1% level on the test statistic scale.
• Perform the test on the sample statistic scale.• What is the p-value for this test? • Calculate the 99% confidence interval.