Transcript CHAPTER EIGHT

Chapter Seven
Hypothesis Testing with ONE Sample
Section 7.1
Introduction to Hypothesis
Testing
Hypothesis Tests
… A process that uses sample statistics
to test a claim about a population
parameter.
 Test includes:

◦ Stating a NULL and an ALTERNATIVE
Hypothesis.
◦ Determining whether to REJECT or to NOT
REJECT the Null Hypothesis. (If the Null is
rejected, that means the Alternative must be
true.)
Stating a Hypothesis
The Null Hypothesis (H0) is a
statistical hypothesis that contains
some statement of equality, such as
=, <, or >
 The Alternative Hypothesis (Ha) is
the complement of the null
hypothesis. It contains a statement
of inequality, such as ≠, <, or >

Left, Right, or Two-Tailed Tests
If the Alternative Hypotheses, Ha ,
includes <, it is considered a LEFT
TAILED test.
 If the Alternative Hypotheses, Ha ,
includes >, it is considered a RIGHT
TAILED test.
 If the Alternative Hypotheses, Ha ,
includes ≠, it is considered a TWO
TAILED test.

EX: State the Null and
Alternative Hypotheses.
26. As stated by a company’s shipping
department, the number of
shipping errors per mission
shipments has a standard deviation
that is less than 3.
28. A state park claims that the mean
height of oak trees in the park is at
least 85 feet.
Types of Errors
When doing a test, you will decide whether
to reject or not reject the null
hypothesis. Since the decision is based
on SAMPLE data, there is a possibility
the decision will be wrong.
Type I error: the null hypothesis is rejected
when it is true.
Type II error: the null hypothesis is not
rejected when it is false.
4 possible outcomes…
Do not reject
H0
TRUTH OF H0
H0 is TRUE
H0 is FALSE
Correct
Type II Error
Decision
Reject H0
Type I Error
DECISION
Correct
Decision
Level of Significance

The level of significance is the
maximum allowed probability of
making a Type I error. It is denoted
by the lowercase Greek letter alpha.

The probability of making a Type II
error is denoted by the lowercase
Greek letter beta.
p-Values
If the null hypothesis is true, a pValue of a hypothesis test is the
probability of obtaining a sample
statistic with a value as extreme or
more extreme than the one
determined from the sample data.
 The p-Value is connected to the
area under the curve to the left
and/or right on the normal curve.

Making and Interpreting your
Decision

Decision Rule based on the p-Value
Compare the p-Value with alpha.
◦ If p < alpha, reject H0
◦ If p > alpha, do not reject H0
General Steps for Hypothesis
Testing
1.
2.
3.
4.
5.
6.
7.
State the null and alternative hypotheses.
Specify the level of significance.
Sketch the curve.
Find the standardized statistic add to
sketch and shade. (usually z or t-score)
Find the p-Value
Compare p-Value to alpha to make the
decision.
Write a statement to interpret the
decision in context of the original claim.
Section 7.2
Hypothesis Testing for the
MEAN (Large Samples)
Using p-Value to Make
Decisions

Decision Rule based on the p-Value
Compare the p-Value with alpha.
◦ If p < alpha, reject H0
◦ If p > alpha, do not reject H0
Finding the p-Value for a
Hypothesis Test – using the table

To find p-Value
◦ Left tailed: p = area in the left tail
◦ Right tailed: p = area in the right tail
◦ Two Tailed: p = 2(area in one of the tails)
This section we’ll be finding the z-values
and using the standard normal table.
Find the p-value. Decide whether
to reject or not reject the null
hypothesis

4. Left tailed test, z = -1.55, alpha =
0.05

8. Two tailed test, z = 1.23, alpha =
0.10
Using p-Values for a z-Test

Z-Test used when the population is
normal, δ is known, and n is at least
30. If n is more than 30, we can use s
for δ.
Guidelines – using the p-value
1. find H0 and Ha
 2. identify alpha
 3. find z
 4. find area that corresponds to z
(the p-value)
 5. compare p-value to alpha
 6. make decision
 7. interpret decision


30. A manufacturer of sprinkler
systems designed for fire protection
claims the average activating
temperature is at least 135oF. To
test this claim, you randomly select
a sample of 32 systems and find
mean = 133, and s = 3.3. At alpha =
0.10, do you have enough evidence
to reject the manufacturer’s claim?
Rejection Regions & Critical
Values
The Critical value (z0) is the z-score
that corresponds to the level of
significance (alpha)
 Z0 separates the rejection region
from the non-rejection region
 Sketch a normal curve and shade
the rejection region. (Left, right, or
two tailed)

Find z0 and shade rejection
region

18. Right tailed test, alpha = 0.08

22. Two tailed test, alpha = 0.10
Guidelines – using rejection
regions

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
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1. find H0 and Ha
2. identify alpha
3. find z0 – the critical value(s)
4. shade the rejection region(s)
5. find z
6. make decision (Is z in the rejection
region?)
7. interpret decision

38. A fast food restaurant estimates
that the mean sodium content in
one of its breakfast sandwiches is no
more than 920 milligrams. A
random sample of 44 sandwiches
has a mean sodium content of 925
with s = 18. At alpha = 0.10, do you
have enough evidence to reject the
restaurant’s claim?