Section 8 Chapter 8 Hypothesis Testing Scientific Process Analysis Paper How would you describe Hypothesis Testing to a friend outside of this statistics c

Section 8 Chapter 8 Hypothesis Testing Scientific Process Analysis Paper How would you describe Hypothesis Testing to a friend outside of this statistics class?You may use the information in the textbook or create your own problem. Formulas are not required.Please remember to use APA formatting if you use help from the textbook or a website. Try to use your own words! Elementary Statistics
Thirteenth Edition
Chapter 8
Hypothesis
Testing
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Hypothesis Testing
8-1 Basics of Hypothesis Testing
8-2 Testing a Claim about a Proportion
8-3 Testing a Claim About a Mean
8-4 Testing a Claim About a Standard Deviation or
Variance
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Key Concept
In this section we present key components of a
formal hypothesis test. The concepts in this section
are general and apply to hypothesis tests involving
proportions, means, or standard deviations or
variances.
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Hypothesis and Hypothesis Test
• Hypothesis
– In statistics, a hypothesis is a claim or statement
about a property of a population.
• Hypothesis Test
– A hypothesis test (or test of significance) is a
procedure for testing a claim about a property of a
population.
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Example: Majority of Consumers are not
Comfortable with Drone Deliveries (1 of 6)
1009 consumers were asked if they are comfortable
with having drones deliver their purchases, and 54% (or
545) of them responded with “no.” Using p to denote the
proportion of consumers not comfortable with drone
deliveries, the “majority” claim is equivalent to the claim
that the proportion is greater than half, or p > 0.5. The
expression p > 0.5 is the symbolic form of the original
claim.
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Example: Majority of Consumers are not
Comfortable with Drone Deliveries (2 of 6)
The Big Picture We have the claim that the population
proportion p is such that p > 0.5. Among 1009
consumers, how many do we need to get a
significantly high number who are not comfortable
with drone delivery?
• A result of 506 (or 50.1%) is just barely more than half, so
506 is clearly not significantly high.
• A result of 1006 (or 99.7%) is clearly significantly high.
But what about the result of 545 (or 54.0%) that was
actually obtained in the Pitney Bowes survey?
• Is 545 (or 54.0%) significantly high? The method of
hypothesis testing allows us to answer that key question.
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Example: Majority of Consumers are not
Comfortable with Drone Deliveries (3 of 6)
Using Technology It is easy to obtain hypothesistesting results using technology. The accompanying
screen displays show results from four different
technologies, so we can use computers or
calculators to do all of the computational heavy
lifting.
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Example: Majority of Consumers are not
Comfortable with Drone Deliveries (4 of 6)
Using Technology
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Example: Majority of Consumers are not
Comfortable with Drone Deliveries (5 of 6)
Using Technology
Examining the four screen displays, we see some
common elements. They all display a “test statistic” of
z = 2.55 (rounded), and they all include a “P-value” of
0.005 (rounded).
Focus on understanding how the hypothesis-testing
procedure works and learn the associated terminology.
Only then will results from technology make sense.
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Example: Majority of Consumers are not
Comfortable with Drone Deliveries (6 of 6)
Significance
Hypothesis tests are also called tests of significance.
In Section 4-1 we used probabilities to determine when
sample results are significantly low or significantly
high. This chapter formalizes those concepts in a
unified procedure that is used often throughout many
different fields of application.
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Null Hypothesis
• Null Hypothesis
– The null hypothesis (denoted by H0) is a
statement that the value of a population parameter
(such as proportion, mean, or standard deviation)
is equal to some claimed value.
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Alternative Hypothesis
• Alternative Hypothesis
– The alternative hypothesis (denoted by H1 or Ha
or HA) is a statement that the parameter has a
value that somehow differs from the null
hypothesis. For the methods of this chapter, the
symbolic form of the alternative hypothesis must
use one of these symbols: , ≠.
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Procedure for Hypothesis Tests
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Confidence Interval Method
Confidence Interval Method
Construct a confidence interval with a confidence level selected as in
Table 8-1.
Table 8-1 Confidence Level for Confidence Interval
Significance Level for Hypothesis Test
Two-Tailed Test
One-Tailed Test
0.01
99%
98%
0.05
95%
90%
0.10
90%
80%
Because a confidence interval estimate of a population parameter
contains the likely values of that parameter, reject a claim that the
population parameter has a value that is not included in the confidence
interval.
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Use the Original Claim to Create a Null
Hypothesis H0 and an Alternative Hypothesis H1
Step 1. Identify the claim to be tested and express it in
symbolic form.
Step 2. Give the symbolic form that must be true when
the original claim is false.
Step 3. Consider the two symbolic expressions
obtained so far:
– Alternative hypothesis H1 is the one NOT containing
equality, so H1 uses the symbol or ≠.
– Null hypothesis H0 is the symbolic expression that the
parameter equals the fixed value being considered.
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Example: Drone Delivery (1 of 8)
Given the claim that “the majority of consumers are
uncomfortable with drone delivery,” we can apply Steps
1, 2, and 3 as follows.
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Example: Drone Delivery (2 of 8)
Step 1: Identify the claim to be tested and express it in
symbolic form. Using p to denote the probability of
selecting a consumer uncomfortable with drone delivery,
the claim that “the majority is uncomfortable with drone
delivery” can be expressed in symbolic form as p > 0.5.
Step 2: Give the symbolic form that must be true when
the original claim is false. If the original claim of p > 0.5
is false, then p ≤ 0.5 must be true.
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Example: Drone Delivery (3 of 8)
Step 3: This step is in two parts: Identify the alternative
hypothesis H1 and identify the null hypothesis H0.
• Identify H1: Using the two symbolic expressions p > 0.5
and p ≤ 0.5, the alternative hypothesis H1 is the one
that does not contain equality. Of those two
expressions, p > 0.5 does not contain equality, so we
get H1: p > 0.5
• Identify H0: The null hypothesis H0 is the symbolic
expression that the parameter equals the fixed value
being considered, so we get H0: p = 0.5
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Step 4: Significance Level α
• Significance Level
– The significance level α for a hypothesis test is the
probability value used as the cutoff for determining
when the sample evidence constitutes significant
evidence against the null hypothesis. By its nature,
the significance level α is the probability of mistakenly
rejecting the null hypothesis when it is true:
Significance level α = P (rejecting H0 when H0 is true)
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Select the Significance Level α
Step 4. The significance level α is the same α
introduced in sections 7-1, where we defined “critical
value”. Common choices for α are 0.05, 0.01, and 0.10;
0.05 is most common.
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Identify the test statistic that is relevant to the
test and determine its sampling distribution
(such as normal, t, χ²) (1 of 2)
Step 5. Identify the test statistic that is relevant to the
test and determine its sampling distribution (such as
normal, t, χ²). The table on the following slide lists
parameters along with the corresponding sampling
distributions.
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Identify the test statistic that is relevant to the
test and determine its sampling distribution
(such as normal, t, χ²) (2 of 2)
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Example: Drone Delivery (4 of 8)
The claim p > 0.5 is a claim about the population
proportion p, so use the normal distribution, provided that
the requirements are satisfied. (With n = 1009, p = 0.5,
and q = 0.5 from the ongoing example, np ≥ 5 and nq ≥ 5
are both true.)
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Find the Value of the Test Statistic, Then Find
Either the P-Value or the Critical Values(s)
Step 6. Find the value of the test statistic and the
P-value or critical value(s).
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Test Statistic
• Test Statistic
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Example: Drone Delivery (5 of 8)
With the null hypothesis of H0: p = 0.5, we are working with
the assumption that p = 0.5, and it follows that q = 1 − p = 0.5.
We can evaluate the test statistic as shown below. The test
statistic of z = 2.55 from each of the previous technology
displays is more accurate than the result of z = 2.54 shown
below.
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Critical Region
• Critical Region
– The critical region (or rejection region) is the
area corresponding to all values of the test statistic
that cause us to reject the null hypothesis.
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Two-Tailed, Left-Tailed, Right-Tailed
• Two-tailed test: The critical
region is in the two extreme
regions (tails) under the
curve.
• Left-tailed test: The critical
region is in the extreme left
region (tail) under the curve.
• Right-tailed test: The critical
region is in the extreme right
region (tail) under the curve.
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P-Value Method
• P-Value Method
– In a hypothesis test, the P-value is the probability of
getting a value of the test statistic that is at least as
extreme as the test statistic obtained from the sample
data, assuming that the null hypothesis is true.
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Example: Drone Delivery (6 of 8)
Using the data from the previous problem, the test
statistic is z = 2.55, and it has a normal distribution
area of 0.0054 to its right, so a right-tailed test with
test statistic z = 2.55 has a P-value of 0.0054. See the
different technology displays given earlier, and note
that each of them provides the same P-value of 0.005
after rounding.
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Finding P-Values
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Caution
• P-value = probability of a test statistic at least as
extreme as the one obtained
• p = population proportion
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Critical Value Method
• Critical Values
– In a hypothesis test, the critical value(s)
separates the critical region (where we reject the
null hypothesis) from the values of the test statistic
that do not lead to rejection of the null hypothesis.
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Example: Drone Delivery (7 of 8)
The critical region is shaded in green. The figure shows
that with a significance level of α = 0.05, the critical
value is z = 1.645.
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Make a decision to Either Reject H0 or
Fail to Reject H0
Step 7. Make a decision to either reject H0 or fail to
reject H0.
Decision Criteria for the P-Value Method:
• If P-value ≤ α, reject H0 (“If the P is low, the null must
go.”)
• If P-value > α, fail to reject H0.
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Restate the Decision Using Simple
and Nontechnical Terms
Step 8. Restate the decision using simple and
nontechnical terms.
Without using technical terms not understood by most
people, state a final conclusion that addresses the
original claim with wording that can be understood by
those without knowledge of statistical procedures.
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Example: Drone Delivery (8 of 8)
There is sufficient evidence to support the claim that
the majority of consumers are uncomfortable with
drone deliveries.
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Restate the Decision Using Simple
and Nontechnical Terms (1 of 4)
Wording the Final Conclusion
For help in wording the final conclusion, refer to the
table on the next slide, which lists the four possible
circumstances and their corresponding conclusions.
Note that only the first case leads to wording
indicating support for the original conclusion. If you
want to support some claim, state it in such a way that
it becomes the alternative hypothesis, and then hope
that the null hypothesis gets rejected.
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Restate the Decision Using Simple
and Nontechnical Terms (2 of 4)
Wording the Final Conclusion
Condition
Conclusion
Original claim does not include
equality, and you reject H0.
“There is sufficient evidence to
support the claim that …
(original claim)”
Original claim does not include
equality, and you fail to reject H0.
“There is not sufficient evidence
to support the claim that …
(original claim)”
Original claim includes equality,
and you reject H0.
“There is sufficient evidence to
warrant rejection of the claim
that … (original claim).”
Original claim includes equality,
and you fail to reject H0.
“There is not sufficient evidence
to warrant rejection of the claim
that … (original claim).”
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Restate the Decision Using Simple
and Nontechnical Terms (3 of 4)
Accept or Fail to Reject?
We should say that we “fail to reject the null hypothesis”
instead of saying that we “accept the null hypothesis.”
The term accept is misleading, because it implies
incorrectly that the null hypothesis has been proved, but
we can never prove a null hypothesis. The phrase fail
to reject says more correctly that the available
evidence isn’t strong enough to warrant rejection of the
null hypothesis.
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Restate the Decision Using Simple
and Nontechnical Terms (4 of 4)
Multiple Negatives
Final conclusions can include as many as three
negative terms. For such confusing conclusions, it is
better to restate them to be understandable. Instead of
saying that “there is not sufficient evidence to warrant
rejection of the claim of no difference between 0.5 and
the population proportion,” a better statement would be
this: “Until stronger evidence is obtained, continue to
assume that the population proportion is equal to 0.5.”
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Confidence Intervals for Hypothesis
Tests (1 of 2)
A confidence interval estimate of a population parameter
contains the likely values of that parameter. We should
therefore reject a claim that the population parameter has
a value that is not included in the confidence interval.
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Confidence Intervals for Hypothesis
Tests (2 of 2)
Equivalent Methods
A confidence interval estimate of a proportion might lead
to a conclusion different from that of a hypothesis test.
Parameter
Is a confidence interval equivalent to a
hypothesis test in the sense that they
always lead to the same conclusion?
Proportion
No
Mean
Yes
Standard deviation or variance
Yes
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Type I and Type II Errors (1 of 2)
• Type I error: The mistake of rejecting the null hypothesis
when it is actually true. The symbol α (alpha) is used to
represent the probability of a type I error.
α = P(type I error) = P(rejecting H0 when H0 is true)
• Type II error: The mistake of failing to reject the null
hypothesis when it is actually false. The symbol β(beta) is
used to represent the probability of a type II error.
β = P(type II error) = P(failing to reject H0 when H0 is false)
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Type I and Type II Errors (2 of 2)
Preliminary Conclusion
True State of Nature
Null hypothesis is true
True State of Nature
Null hypothesis is false
Reject H0
Type I error:
Reject a true H0.
P(type I error) = α
Correct decision
Fail to reject H0
Correct decision
Type II error:
Fail to reject a false H0.
P(type II error) = β
HINT FOR DESCRIBING TYPE I AND TYPE II ERRORS
Descriptions of a type I error and a type II error refer to the null
hypothesis being true or false, but when wording a statement
representing a type I error or a type II error, be sure that the
conclusion addresses the original claim (which may or may not
be the null hypothesis).
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Example: Describing Type I and Type
II Errors (1 of 3)
Consider the claim that a medical procedure designed
to increase the likelihood of a baby girl is effective, so
that the probability of a baby girl is p > 0.5. Given the
following null and alternative hypotheses, write
statements describing (a) a type I error, and (b) a type II
error.
H0: p = 0.5
H1: p > 0.5 (original claim that will be addressed in the
final conclusion)
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Example: Describing Type I and Type
II Errors (2 of 3)
Solution
a. Type I Error: A type I error is the mistake of rejecting
a true null hypothesis, so the following is a type I
error: In reality p = 0.5, but sample evidence leads us
to conclude that p > 0.5. (In this case, a type I error
is to conclude that the medical procedure is effective
when in reality it has no effect.)
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Example: Describing Type I and Type
II Errors (3 of 3)
Solution
b. Type II Error: A type II error is the mistake of failing
to reject the null hypothesis when it is false, so the
following is a type II error: In reality p > 0.5, but we
fail to support that conclusion. (In this case, a type II
error is to conclude that the medical procedure has
no effect, when it really is effective in increasing the
likelihood of a baby girl.)
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Power of a Hypothesis
• Power
– The power of a hypothesis test is the probability 1 − β
of rejecting a false null hypothesis. The value of the
power is computed by using a particular significance
level α and a particular value of the population
parameter that is an alternative to the value assumed
true in the null hypothesis.
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Example: Power of a Hypothesis (1 of 5)
Consider these preliminary results from the XSORT
method of gender selection: There were 13 girls among
the 14 ba…
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