Transcript MATH 1410/7.1 and 7.2 pp

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.
*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.
*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
(α).

The probability of making a Type II
error is denoted by the lowercase
Greek letter beta (β).
Section 7.2
Hypothesis Testing for the
MEAN (Large Samples)
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.

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.
Using p-Values for a z-Test

Z-Test used when the population is
normal, σ is known, and n is at least
30. If n is more than 30, we can use s
for σ.
Making and Interpreting your
Decision

Decision Rule based on the p-Value
Compare the p-Value with alpha.
◦ If p < α, reject H0
◦ If p > α, do not reject H0
Find the p-value. Decide whether
to reject or not reject the null
hypothesis

Left tailed test, z = -1.55, α = 0.05

Two tailed test, z = 1.23, α = 0.10
General Steps for Hypothesis
Testing – P Value METHOD
1.
2.
3.
4.
5.
6.
7.
State the null and alternative hypotheses.
Specify the level of significance, α
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.

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 α = 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

Right tailed test, alpha = 0.08

Two tailed test, alpha = 0.10
Guidelines – using rejection
regions

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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

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?