Download Document

• Have HW out to be checked.
Introduction to Hypothesis Testing
Section 7.1 Objectives
• State a null hypothesis and an alternative hypothesis
• Identify type I and type II errors and interpret the
level of significance
• Determine whether to use a one-tailed or two-tailed
statistical test and find a p-value
• Make and interpret a decision based on the results of
a statistical test
• Write a conclusion for a hypothesis test
Larson/Farber 4th ed.
3
Hypothesis Tests
Hypothesis test
• A process that uses sample statistics to test a claim
about the value of a population parameter.
• For example: An automobile manufacturer
advertises that its new hybrid car has a mean mileage
of 50 miles per gallon. To test this claim, a sample
would be taken. If the sample mean differs enough
from the advertised mean, you can decide the
advertisement is wrong.
Larson/Farber 4th ed.
4
USES
• Marketing strategies
• New medicines and treatments
Larson/Farber 4th ed.
5
REMEMBER
• Central limit theorem
• 1) if n ≥ 30 then sample means will approximate a
normal distribution.
• 2) if population is normally distributed then sample
means will be normally distributed for any n.
2 𝜎²
• Variance: 𝜎𝑥 =
𝑛
• Standard deviation : 𝜎𝑥 =
𝜎
𝑛
6
Normal curve
7
Hypothesis Tests
Statistical hypothesis
• A statement, or claim, about a population parameter.
• Need a pair of hypotheses
• one that represents the claim
• the other, its complement
• When one of these hypotheses is false, the other must
be true.
Larson/Farber 4th ed.
8
Stating a Hypothesis
Null hypothesis
• A statistical hypothesis
that contains a statement
of equality such as , =,
or .
• Denoted H0 read “H
subzero” or “H naught.”
Alternative hypothesis
• A statement of
inequality such as >, ,
or <.
• Must be true if H0 is
false.
• Denoted Ha read “H
sub-a.”
complementary
statements
Larson/Farber 4th ed.
9
To test a population parameter
• 1) state 2 hypotheses: 𝐻0 & 𝐻𝑎
• 2) prove one of them is true or false
• 3) if one is true other must be false
Larson/Farber 4th ed.
10
To write a hypothesis
• 1) write claim in math form
• 2) write its complement
• 3) 𝐻0 always one with = or ≤ 𝑜𝑟 ≥
• Ex: 𝐻0 = 5 then 𝐻𝑎 ≠ 5
•
𝐻0 ≥70 then 𝐻𝑎 < 70
•
𝐻0 ≤8 then 𝐻𝑎 > 8
•
Larson/Farber 4th ed.
11
EXAMPLE
• In prob and stats the average age of a student is 17.
• 𝐻0 is µ = 17
• 𝐻𝑎 is µ ≠ 17
Larson/Farber 4th ed.
12
Stating a Hypothesis
• To write the null and alternative hypotheses, translate
the claim made about the population parameter from
a verbal statement to a mathematical statement.
• Then write its complement.
H0: μ ≤ k
Ha: μ > k
H0: μ ≥ k
Ha: μ < k
H0: μ = k
Ha: μ ≠ k
• Regardless of which pair of hypotheses you use, you
always assume μ = k and examine the sampling
distribution on the basis of this assumption.
Larson/Farber 4th ed.
13
Larson/Farber 4th ed.
14
Example: Stating the Null and Alternative
Hypotheses
Write the claim as a mathematical sentence. State the null
and alternative hypotheses and identify which represents
the claim.
1. A university publicizes that the proportion of its
students who graduate in 4 years is 82%.
Solution:
H0: p = 0.82
Equality condition (Claim)
Ha: p ≠ 0.82
Complement of H0
Larson/Farber 4th ed.
15
Example: Stating the Null and Alternative
Hypotheses
Write the claim as a mathematical sentence. State the null
and alternative hypotheses and identify which represents
the claim.
2. A water faucet manufacturer announces that the mean
flow rate of a certain type of faucet is less than 2.5
gallons per minute.
Solution:
H0: μ ≥ 2.5 gallons per minute
Ha: μ < 2.5 gallons per minute
Larson/Farber 4th ed.
Complement of Ha
Inequality
(Claim)
condition
16
Example: Stating the Null and Alternative
Hypotheses
Write the claim as a mathematical sentence. State the null
and alternative hypotheses and identify which represents
the claim.
3. A cereal company advertises that the mean weight of
the contents of its 20-ounce size cereal boxes is more
than 20 ounces.
Solution:
H0: μ ≤ 20 ounces
Ha: μ > 20 ounces
Larson/Farber 4th ed.
Complement of Ha
Inequality
(Claim)
condition
17
Types of Errors
• No matter which hypothesis represents the claim,
always begin the hypothesis test assuming that the
equality condition in the null hypothesis is true.
• At the end of the test, one of two decisions will be
made:
 reject the null hypothesis
 fail to reject the null hypothesis
• Because your decision is based on a sample, there is
the possibility of making the wrong decision.
Larson/Farber 4th ed.
18
Types of Errors
Actual Truth of H0
Decision
Do not reject H0
Reject H0
H0 is true
Correct Decision
Type I Error
H0 is false
Type II Error
Correct Decision
• A type I error occurs if the null hypothesis is rejected
when it is true.
• A type II error occurs if the null hypothesis is not
rejected when it is false.
Larson/Farber 4th ed.
19
Example: Identifying Type I and Type II
Errors
The USDA limit for salmonella contamination for
chicken is 20%. A meat inspector reports that the
chicken produced by a company exceeds the USDA
limit. You perform a hypothesis test to determine
whether the meat inspector’s claim is true. When will a
type I or type II error occur? Which is more serious?
(Source: United States Department of Agriculture)
Larson/Farber 4th ed.
20
Solution: Identifying Type I and Type II
Errors
Let p represent the proportion of chicken that is
contaminated.
Hypotheses: H0: p ≤ 0.2
Ha: p > 0.2 (Claim)
Chicken meets
USDA limits.
H0: p ≤ 0.20
Chicken exceeds
USDA limits.
H0: p > 0.20
p
0.16
Larson/Farber 4th ed.
0.18
0.20
0.22
0.24
21
Solution: Identifying Type I and Type II
Errors
Hypotheses: H0: p ≤ 0.2
Ha: p > 0.2 (Claim)
A type I error is rejecting H0 when it is true.
The actual proportion of contaminated chicken is less
than or equal to 0.2, but you decide to reject H0.
A type II error is failing to reject H0 when it is false.
The actual proportion of contaminated chicken is
greater than 0.2, but you do not reject H0.
Larson/Farber 4th ed.
22
Solution: Identifying Type I and Type II
Errors
Hypotheses: H0: p ≤ 0.2
Ha: p > 0.2 (Claim)
• With a type I error, you might create a health scare
and hurt the sales of chicken producers who were
actually meeting the USDA limits.
• With a type II error, you could be allowing chicken
that exceeded the USDA contamination limit to be
sold to consumers.
• A type II error could result in sickness or even death.
Larson/Farber 4th ed.
23
Level of Significance
Level of significance
• Your maximum allowable probability of making a
type I error.
 Denoted by , the lowercase Greek letter alpha.
• By setting the level of significance at a small value,
you are saying that you want the probability of
rejecting a true null hypothesis to be small.
• Commonly used levels of significance:
  = 0.10
 = 0.05
 = 0.01
• P(type II error) = β (beta)
Larson/Farber 4th ed.
24
Statistical Tests
• After stating the null and alternative hypotheses and
specifying the level of significance, a random sample
is taken from the population and sample statistics are
calculated.
• The statistic that is compared with the parameter in
the null hypothesis is called the test statistic.
Population
parameter
Test statistic
μ
x
p
σ2
p̂
Larson/Farber 4th ed.
s2
Standardized test
statistic
z (Section 7.2 n  30)
t (Section 7.3 n < 30)
z (Section 7.4)
χ2 (Section 7.5)
25
Assignment
• Page 367 1-36
Larson/Farber 4th ed.
26