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7.1-7.2
Basics of Hypothesis Testing
*** Read the Chapter Problem on pg. 367.
Hypothesis: In statistics, is a claim, statement or assumption about a property of
a population.
Example: "The proportion of all adult American drivers who admit running red
lights is greater than 0.5."
Rare Event Rule For Inferential Statistics. If, under a given assumption, the
probability of a particular observed event is exceptionally small, we conclude that
the assumption is probably not correct. [From section 3.1]
Method of Reasoning: Analyze a sample in an attempt to distinguish
between results that can easily occur and results that are highly unlikely.
*** Read together Example pg. 369 (Gender selection)
Stating conclusions about claims under the rare event rule:
***#1, pg. 385
***#2, pg. 385
***#3, pg. 385
***#4, pg. 385
Components of a Formal Hypothesis Test

Claim - The statement to be tested.

Null hypothesis - ( H o )
A statement about a value of a population parameter
Must contain condition of equality (=, ≥ or ≤ )
We test the null hypothesis by assuming it to be true and
reaching a conclusion to either reject H o or fail to reject H o

Alternate Hypothesis - ( H 1 or H a
)
It is the opposite of H o
Contains ≠, <, or > (it does not contain equality)
1
Identifying Hypotheses
Step 1: Write the claim to be tested in symbolic form.
Step 2: Write the opposite statement in symbolic form.
Step 3: Of these two statements, the one that does not contain equality
becomes the alternate hypothesis, H 1 .
Step 4: The null hypothesis, H o , is the statement that the parameter
equals the fixed value being considered.
*** Read examples page 373-374
*** Do #5-11 odd, pg. 386
Note about forming your own claims (hypotheses)
If you are conducting a study and want to use a hypothesis test to support your
claim, the claim should be worded so that it becomes the alternative
hypothesis, unless, in rare cases, there is a reason to do the opposite..
Note about testing the validity of someone else’s claim
Someone else’s claim may become the null hypothesis (because it contains
equality), and it sometimes becomes the alternative hypothesis (because it does
not contain equality).

Test Statistic – is a value computed from the sample data that is used in
making the decision about the rejection of the null hypothesis. The test
statistic is found by converting the sample statistic (such as the sample
proportion, or the sample mean) to a z or t score.
Test Statistic for Proportion:
z
Test Statistic for the Mean:
z
p p
pq
n
(x  )
(

n
)
or
t
(x  )
s
( )
n
*** Read example on page 375
2
*** Do #21, pg. 386
*** Do #23, pg. 387

Critical Region - The set of all values of the test statistic that would cause
us to reject the null hypothesis.

Critical Value - The value or values that separate the critical region
(where we reject the null hypothesis) from the values of the test statistic
that do not lead to a rejection of the null hypothesis.

Significance Level - (α) the probability that the test statistic will fall in the
critical region when the null hypothesis is actually true. It is the area in the
critical region, and the probability of being wrong if H o is rejected.
Common choices are .05, .01, .10. Use .05 if α is not given.
*** Read Example on page 376

Two-Tailed, Left-Tailed, Right-Tailed - The tails in a distribution are the
extreme regions bounded by critical values.
The tail will correspond to the critical region containing the values that
would conflict significantly with the null hypothesis.
Note: We reject the null hypothesis if the calculated test statistic falls into
the critical region because that indicates a significant discrepancy
between the null hypothesis value under test and the value indicated by
the test statistic from the sample.
*** Do #13-20 odd, pg. 386
3
Two-tailed Test
 is divided equally between
H0: =
H1:
Right-tailed Test
Slide 35
the two tails of the critical
region

H0: =
H1: >
Points Right
Means less than or greater than
Copyright © 2004 Pearson Education, Inc.
Copyright © 2004 Pearson Education, Inc.
Left-tailed Test
Slide 36
Slide 37
H0: =
H1: <
Points Left
Copyright © 2004 Pearson Education, Inc.
4

The P-value: the probability of getting a value of the test statistic that is at
least as extreme as the one representing the sample data, assuming that
the null hypothesis is true. The null hypothesis is rejected if the P-value is
very small, such as 0.05 or less. The p-value is also the probability of
being wrong if H o is rejected.
The procedure to find p-values is summarized in figure 7-6 on page 378.
*** Read example on page 379
*** Do #25-32 odd
Example: Finding P-values.
Slide 44
Figure 7-6
Copyright © 2004 Pearson Education, Inc.
5

Conclusion of Test (Must be one of the following)
Reject the Null Hypothesis or Fail to Reject the Null Hypothesis
Formulate correct wording of final conclusion based on the flowchart given
in the next page
Decision Criterion: The decision to reject or fail to reject the null
hypothesis will be explained in section 7.3 when we study the two
methods of hypothesis testing used in this chapter.
Note 1: We test the null hypothesis directly in the sense that we assume it is true
and reach a conclusion to either reject Ho or fail to reject Ho.
Note 2: Note that only one case leads to wording indicating that the sample
data actually support the conclusion. This is the case when the claim is the
alternate hypothesis, and the null hypothesis has been rejected at some level of
significance.
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.
Notes about the statement “Fail to reject the null hypothesis”
Some books use “accept the null hypothesis”
We are not proving the null hypothesis
Sample evidence is not strong enough to warrant rejection (such as not enough
evidence to convict a suspect)
*** Do #33,35, pg. 387
(Refer to the table on page 380)
6
Two Types of Errors (see the table given below)
Type I Error: The mistake of rejecting the null hypothesis when it is true.
α is used to represent a probability of a type I error. The actual p-value
would also be the probability of a Type I error.
Type II Error: The mistake of failing to reject the null hypothesis when it is
actually false. β is used to represent a probability of a type II error.
Type I and Type II Errors
Table 7-2
True State of Nature
We decide to
reject the
null hypothesis
The null
hypothesis is
true
The null
hypothesis is
false
Type I error
(rejecting a true
null hypothesis)
P
Correct
decision
Correct
decision
Type II error
(rejecting a false
null hypothesis)
P(II) = 
Decision
We fail to
reject the
null hypothesis
Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
41
*** Do #37, 39, pg. 387
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7.3
Testing a Claim About a Proportion
Much of media presentations are about proportions, percentages or probabilities
Assumptions:
1. The sample observations are a simple random sample.
2. The conditions for a binomial distribution are satisfied.
2. The conditions np  5 , and nq  5 are both satisfied, so that the binomial
distribution of sample proportions can be approximated by a normal distribution
with
  np and   npq
Notation:
n = sample size or number of trials
x = number having the characteristic we are studying
p
x
n
(sample proportion)
p = population proportion (used in the null hypothesis)
q=1–p
Test Statistic for Testing a Claim About a Proportion:
z
p p
pq
n
Note: Be sure that you do not confuse the use of the letter p for population
proportions with the p-value of a hypothesis test.
p-values: Use the standard normal distribution (Table A-2) and refer to figure7-8
Critical Values: Use the standard normal distribution (Table A-2)
Significance Level: if α is not given, use .05.
Three Methods for Testing Hypothesis:



Traditional
p-Value
Confidence Intervals (already done in chapter 6)
Refer to figure 7-8, page 384
8

Traditional Method:
Step 1:
Identify the original claim in symbolic form. (p ≤ K, p ≥ K, p ≠ K)
Step 2:
Identify the opposite of the original claim.
Step 3:
Using the preceding two symbolic expressions, identify the null
and the alternative hypothesis.
The null hypothesis is the one that contains equality.
The alternative is the one that does not contain equality.
Step 4:
Identify the critical region (Is it a left, right, or two-tailed test?)
Step 5:
Identify the significance level, α. Use 0.05 if not given.
Step 6:
Find the critical values using table A-2
Step 7:
Compute the value of the test statistic.
Step 8:
Decide whether you
Reject or Fail to Reject the Null Hypothesis
Reject Ho if the test statistic falls within the critical region.
Fail to reject Ho if the test statistic does not fall within the
critical region.
Step 9:
Write the final conclusion using the wording given in figure 7-7
*** Read example p. 389-391
*** Do # 3 page 396 (use traditional method)
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
p-Value Method
Step 1:
Identify the original claim in symbolic form.
Step 2:
Identify the opposite of the original claim.
Step 3:
Using the preceding two symbolic expressions, identify the null
and the alternative hypothesis.
The null hypothesis is the one that contains equality.
The alternative is the one that does not contain equality.
Step 4:
Identify the critical region (Is it a left, right, or two-tailed test?)
Step 5:
Identify the significance level. Use 0.05 if not given.
Step 6:
Compute the value of the test statistic.
Step 7:
Compute the p-value
Step 8:
Decide whether you
Reject or Fail to Reject the Null Hypothesis
Reject Ho if p-value ≤ α (where α is the significance level)
Fail to reject Ho if p-value > α
Step 9:
Write the final conclusion using the wording given in figure 7-7 p.
380
*** Read example p. 391-392
*** Read example p. 393-394
*** Do # 5 page 396 (Use p-value method)
10

Using the TI-83 to Test Claims About a Proportion
STAT>>TESTS 5:1PropZTest
( is , the proportion of the claim)
*** Do # 3 and 6 page 396 using the calculator
*** Do # 7 page 397 (Use p-value method)
*** Do # 13 page 396 (Use traditional method)
Decision Criterion for Rejecting or Failing to Reject Ho - Another Option
Instead of using a significance level such as 0.05, simply identify the p-value and
leave the decision to the reader.
Decision Criterion for Rejecting or Failing to Reject Ho – Confidence
Intervals:
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.
Caution: In some cases, a conclusion based on a confidence interval may be
different from a conclusion based on a hypothesis test. See the comments in the
individual sections.
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7.4
Testing Claims About a Mean: σ Known
Assumptions:
1. Sample is a simple random sample.
2. The value of the population standard deviation σ is known.
3. Either or both of these conditions is satisfied: The population is normally
distributed or n > 30.
Test Statistic for Testing a Claim About a Mean (with σ known):
z
( x  x )
(

n
where
 x is the mean of the claim.
)
P-Values and Critical Values: Use the standard normal distribution (Table A-2),
and refer to figure 7-6. It is the same method used in section 7.3.
*** Read example on p. 401-402
*** Do #5, pg. 405
(Use the traditional method. Steps are the same as in section 7.3)
12
*** Do #9, pg. 405
(Use the p-value method. Steps are the same as in section 7.3)

Using the TI-83 to Test Claims About a Mean (σ known)
STAT>>TESTS 1:Z-Test
*** Do # 11, pg. 405 (use the calculator)

Confidence interval method
*** Read the confidence interval method on p. 403
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7.5
Testing Claims About a Mean: σ Not Known
Note: The criteria for choosing between a normal distribution and a Student t
distribution are the same in this chapter as they were in Chapter 6.
(See Table 6-1, pg. 337)
Assumptions:
1. Sample is a simple random sample.
2. The value of the population standard deviation σ is not known.
3. Either or both of these conditions are satisfied: The population is normally
distributed or n > 30.
Test Statistic for Testing a Claim About a Mean (with σ not known):
z
( x  x )
, where  x is the mean of the claim.
s
( )
n
P-Values and Critical Values: Use Table A-3 and use df = n – 1 for the number
of degrees of freedom. (See Figure 7-6 for P-value procedures.)
Choosing the Appropriate Distribution
Use the Student t distribution when σ is not known and either or both of
these conditions is satisfied:
The population is normally distributed or n > 30
*** Do #1- 4, pg. 415
Finding p-Values with the Student t Distribution
Finding p-values in the Student t distribution is more complicated.
We'll use the calculator to find the p-value.

Using the TI-83 to Test Claims About a Mean (σ not known)
STAT>>TESTS 2:T-Test
14
*** Do #19, pg. 415
(Use the traditional method)
*** Do #13, pg. 416 (Use the calculator and the p-value approach)
*** Do #27, pg. 418 (Using Raw data)
We will use the calculator for this problem. Enter the data in a list. Then use
STAT>>TESTS 2: T-Test Data (instead of summary statistics)
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