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.