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Economics 105: Statistics
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Transcript of Economics 105: Statistics
Economics 105: Statistics• Any questions?• GH 11 and GH 12 due on Friday
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
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
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)
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
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
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
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
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
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
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 )
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.
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.
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.
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.
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.