Chapter 8 Hypothesis Testing
8.1
Review & Preview
Inferential statistics involve using sample data to…
1.
2.
8.2
Basics of Hypothesis Testing
Part I: Basic Concepts of Hypothesis Testing
Hypothesis:
Hypothesis test (or test of significance):
Examples of typical hypotheses:
 the mean body temperature of humans is less than 98.6°F
µ < 98.6°F

the XSORT method of gender selection increases the probability that a
baby born will be a girl, so the probability of a girl is greater than 0.5
p > 0.5

the population of college students has IQ scores with a standard deviation
equal to 15
σ = 15
Rare Event Rule for Inferential Statistics
 To analyze sample data to test a claim we choose between the following 2
explanations:
1. The sample results COULD easily occur by chance
1
Ex: In testing the XSORT gender selection method that is supposed to
make babies more likely to be girls, the result of 52 girls in 100 births is
greater than 50% but 52 girls could easily occur by chance, so there is not
sufficient evidence to conclude that the XSORT method is effective.
2. The sample results are NOT LIKELY to occur by chance
Ex: In testing the XSORT gender-selection method, the result of 95 girls
in 100 births is greater than 50% and 95 girls is so extreme that it could
not easily occur by chance so there is sufficient evidence to conclude that
the XSORT method is effective.
Ex: Assume that 100 babies are born to 100 couples treated with XSORT method of
gender selection that is claimed to make girls more likely. If 58 of the 100 babies born
are girls, test the claim that “with the XSORT method, the proportion of girls is
greater than the proportion of 0.5 that occurs without any treatment.”

Using p to denote the proportion of girls born with XSORT method, test
the claim that p > 0.5.

Given the probability of getting 58 or more girls = 0.0548
2
Components of a Formal Hypothesis Test
 Null Hypothesis:
 denoted by H0
 We test the null hypothesis directly in the sense that we assume it is true
and reach a conclusion to either reject H0 or fail to reject H0.
 Ex: H0 : p = 0.5
H0 :  = 98.6
H0 :  = 15
 Alternative Hypothesis:
 denoted by H1, Ha, or HA
 must use one of these symbols: <, >, or
 Ex: H1 : p > 0.5
H1 :  < 98.6

H1 :
 
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 Ex: Use the given claims to express the corresponding null and alternative
hypothesis in symbolic form.
a. The proportion of workers who get jobs
through networking is greater than 0.5.
Claim:
H 1:
H 0:
b. The mean weight of airline passengers with
carry on baggage is at most 195 lb.
Claim:
H 1:
H 0:
c. The standard deviation of IQ scores of actors is equal to 15.
Claim:
H 1:
H 0:
3
 Forming Your Own Claims (hypotheses):
If you are conducting a study and want to use a hypothesis test to support
your claim, they claim must be worded so that it becomes the alternative
hypothesis (and can be expressed using only the symbols <, >, or ≠). You can
never support a claim that some parameter is EQUAL to a specified value.\
 Significance level (denoted by  ):
 If the test statistic falls in the critical region, we reject the null
hypothesis, so  is the probability of making the mistake of rejecting the
null hypothesis when it is true.
 Most common choices for  are _______________________
(They correspond with confidence levels of 90%, 95%, and 99%)
 Test Statistic:
Parameter
Proportion p
Mean µ
Mean µ
Standard
deviation σ or
variance σ2
Sampling
Distribution
Requirements
Test Statistic
np ≥ 5 and nq ≥ 5
z
σ is unknown and normally
distributed population
OR σ is unknown and n > 30
σ is known and normally
distributed population
OR σ is known and n > 30
normally distributed
population (strict
requirement)
pˆ  p
pq
n
x
t
s
z
n
x
2 

n
(n  1) s 2
2
 Critical region (or rejection region):
4
 Ex: A survey of n = 703 randomly selected workers showed that 61% ( p̂ =
0.61) of those respondents found their job through networking. Find the
value of the test statistic for the claim that most workers (more than
50%) get their jobs through networking.
 Ex #2: Using the claim about the XSORT method of gender selection,
we have n = 100 and x = 58 girls. Find the value of the test statistics for
the claim that more girls are born if couples use the XSORT method.
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2 Types of Hypothesis Tests:


The test statistic alone usually does not give us enough information to make a
decision about the claim being tested
There are 2 methods of interpreting the test statistic:
1. ______________________________ (or p-value or probability value)
2. ______________________________ (or traditional method)
 Both approaches require that we first determine whether our hypothesis test
is two-tailed, left-tailed, or right-tailed:
a. Two-tailed test: the critical region is in the two extreme regions (tails)
under the curve (the significance level is divided equally between the
two tails that constitute the critical region)
Ex: p  0.5 (so the critical region is in ___________ of the normal
distribution)
b. Left-tailed test: the critical region is in the extreme left region (tail)
under the curve
Ex: p < 0.5 (so the critical region is in the _________ of the normal
distribution)
c. Right-tailed test: the critical region is in the extreme right region (tail)
under the curve
Ex: p > 0.5 (so the critical region is in the _________of the normal
distribution)
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 Hint: The direction of the inequality symbol in the alternative hypothesis
points to the tail of the critical region:
>
____________________
<
____________________
 P-Value Method of Hypothesis Testing:
 P-value:
 The P-value that is used with the test statistic in order to make a decision
about the null hypothesis
 More common than the critical value method because technology can be
used to find P-values quickly
 Summary of procedure:
Critical region in the left tail:
P-Value = __________________________________
Critical region in the right tail:
P-Value = __________________________________
Critical region in 2 tails:
P-Value = __________________________________
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 Ex: You found a test statistic of z = 1.60 and the alternative hypothesis
states that the mean length is greater than 30.45mm.
 Ex: You found a test statistic of z = 1.78 and the alternative hypothesis
states that the population proportion is not equal to 75%.
 Don’t confuse the P-value (or probability that the test statistic is at least
as extreme as the one obtained) with the population proportion p
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 Critical Value Method of Hypothesis Testing:
 Critical value (or traditional) method:
 Depends on the nature of the null hypothesis, the sampling distribution
that applies, and the significance level of 
 Ex: Using significance level of  = 0.05, find the critical values for each
of the following alternative hypothesis (assuming that the normal
distribution can be used to approximate the binomial distribution).
o Left-tailed test: p < 0.5 (so the critical region is in the left tail of
the normal distribution)
o Right-tailed test: p > 0.5 (so the critical region is in the right tail of
the normal distribution)
o Two-tailed test: p  0.5 (so the critical region is in both tails of the
normal distribution)
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 Conclusions in Hypothesis Testing: We always test the null hypothesis. The
initial conclusion will always be one of the following:
1.
2.
 Decision to reject or fail to reject the null hypothesis is made using the
following four methods:
 P-value method:
--Reject H0 if the P-value   (where  is the significance level, such
as 0.05).
--Fail to reject H0 if the P-value > 
Ex: With a significance level of  = 0.05 and P-value = 0.0548, we
have a P-value > , so we should ____________
Ex: With a significance level of  = 0.10 and P-value = 0.075, we
have a P-value < , so we should ____________
 Critical Value method:
--Reject H0 if the test statistic falls within the critical region
--Fail to reject H0 if the test statistic does not fall within the
critical region
Ex: With a test statistic z = 1.60 and  = 0.05 where the critical
region is everything to the right of the critical value z = 1.645, the
test statistic _______fall within the critical region & we ________
___________
Ex: With a test statistic z = 2.45 and  = 0.05 so the critical
region is everything to the outside of the critical values of
 z / 2 = ______ and z / 2 = _______ so the test statistic ________
fall within the critical region & we ___________
 Confidence Intervals:
--Because a confidence interval estimate of a population parameter
contains the likely values of that parameter, reject a claim
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that the population parameter has a value that is not included
in the confidence interval.
-for a two-tailed test, construct a confidence interval with a
confidence level of 1 - 
-for a one-tailed hypothesis test with significance level ,
construct a confidence interval with a confidence level
of 1 - 2
--This method may give a conclusion that may be different from a
conclusion based on the other hypothesis tests (those testing
the population proportion p)
--We will not focus on using confidence intervals to test hypotheses
in this course
 Comprehensive Hypothesis Test:
P-value Method:
Critical Value Method:
Confidence Interval Method:
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 Restate the Conclusion Using Simple & Non-technical Terms:
--We always test the null hypothesis
Condition
Original claim does not include
equality and you reject H0
Original claim does not include
equality and you fail to reject H0
Original claim includes equality and
you reject H0
Original claim includes equality and
you fail to reject H0
Conclusion
“There is sufficient evidence to
support the claim that…(original
claim)”
“There is not sufficient evidence to
support the claim that…(original
claim)”
“There is sufficient evidence to
warrant rejection of the claim
that…(original claim)”
“There is not sufficient evidence to
warrant rejection of the claim that …
(original claim)”
--Recognize that we are not proving the null hypothesis. The term
“accept” is somewhat misleading because it seems to imply that the
null hypothesis has been proved. “Fail to reject” says more correctly
that the available evidence isn’t strong enough to warrant rejection
of the null hypothesis.
--When stating the final conclusion, it’s possible to get correct
statements with up to 3 negative terms. Such conclusions are
confusing & should be restated in a way that makes them more
understandable (but does not change the meaning).
--Ex: “There is not sufficient evidence to warrant rejection of the
claim of no difference between 0.5 and the population proportion.”
 Examples: First determine whether the given conditions result in a righttailed test, a left-tailed test, or a two-tailed test, then find the P-value
and the critical value(s). Use both methods to test the null hypothesis and
state conclusion.
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a. A significance level of  = 0.05 is used in testing the claim that p > 0.25,
and the sample data result in a test statistic of z = 1.18.
H1:
H 0:
P-Value method:
Critical value method:
Conclusion:
b. A significance level of  = 0.05 is used in testing the claim that p 
0.25, and the sample data result in a test statistic of z = 2.34
H1:
H 0:
P-Value method:
Critical value method:
Conclusion:
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 Errors in Hypothesis Tests:
 Conclusions are sometimes correct & sometimes wrong (even if we do
everything correctly)
--Type I error ():
--Type II error ():
--Mneumonic device: use the consonants from “RouTiNe FoR FuN”
Type I error: “Reject True Null” (RTN)
Type II error: “Failure to Reject a False Null” (FRFN)
 Descriptions of type I and II errors refer to the null hypothesis
being true or false, but when wording a statement representing a
type I or type II error, be sure that the conclusion addresses the
original claim (which may or may not be the null hypothesis)
 Ex: Assume that we are conducting a hypothesis test of the
claim that p < 0.5. Here are the null and alternative hypotheses:
H0: p = 0.5
a.
H1: p < 0.5
Type I error—the mistake of rejecting a true null hypothesis so
this is a type I error:
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b. Type II error—the mistake of failing to reject hypothesis when
it is false so this is a type II error:
 Controlling type I & II errors: Since , , and the sample size n
are all related, when you choose or determine any two of them, the
3rd is automatically determined. The usual practice is to select the
values of  and n so the value of  is determined.
1. For any fixed , an increase in the sample size n will cause a
decrease in . (A larger sample will lessen the chance
that you make the error of not rejecting the null
hypothesis when it’s actually false).
2. For any fixed sample size n, a decrease in  will cause an
increase in . (Conversely, an increase in  will cause a
decrease in .)
3. To decrease both  and , increase the sample size.
-Ex: In testing the claim that  = 0.8535 g. for M&M’s, we
might choose  = 0.05 and n = 100. In testing the claim that 
= 325 mg. for Bufferin brand aspirin tablets, we might choose
 = 0.01 and n = 500 because of the more serious
consequences associated with testing a commercial drug.
Part II: Beyond the Basics of Hypothesis Testing: The Power of a Test
Power of a Hypothesis Test:
 Computed by using a particular significance level  and a particular value of
the population parameter that is an alternative to the value assumed true
in the null hypothesis.
 The power of a hypothesis test is the probability supporting an alternative
hypothesis that is true.
 Used to gauge the test’s effectiveness in recognizing that a null hypothesis
is false
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 The values of power can be found using some software packages such as
Minitab or can be manually calculated (but calculations are complicated).
 Ex: Assume that we have the following null and alternative hypotheses,
significance level, and sample data:
H0: p = 0.5
H1: p  0.5
significance level:  = 0.05
sample size: n = 100
sample proportion: p̂ = 0.57
test statistic is z = 1.4
 We get the following examples of power values:
Specific
Alternative
Value of P

Power of
Test
(1 - )
0.3
0.013
0.987
0.4
0.484
0.516
0.6
0.484
0.516
0.7
0.013
0.987
We see that this hypothesis test has a power of
0.987 (or 98.7%) of rejecting H0: p = 0.5 when the
population is actually 0.3. Similarly, there is a
51.6% probability of rejecting p = 0.5 when the
true value of p is actually 0.4. It makes sense that
this test is more effective in rejecting the claim
of p = 0.5 when the population is actually 0.3 than
when the population proportion is actually 0.4.
 To increase power, you can…
---- A power of at least 0.80 is a common requirement for determining that a
hypothesis test is effective. (some argue that it should be higher—0.85 or
0.90)
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8.3
Testing a Claim about a Proportion
Claims about a population proportion are usually tested by using a normal distribution as
an approximation to the binomial distribution. Instead of using the same exact methods
of section 6.7, we use a different but equivalent form of the test statistic and don’t
include the correction for continuity (because its effect tends to be very small with
large samples).
Testing Claims about a Population Proportion p:
 Requirements:
1. The sample observations are a simple random sample
2. The conditions for a binomial distribution are satisfied (fixed # of
trials having consistent probabilities, and each trial has 2 outcome
categories—success and failure)
3. The conditions np ≥ 5 and nq ≥ 5 are both satisfied, so the binomial
distribution of sample proportions can be approximated by a normal
distribution with  = np and  = npq
--Note that p is the assumed proportion used in the claim, not the
sample proportion
 Notation:
n=
pˆ 
x
n
p=
q=
 Test Statistic for Testing a Claim about a Proportion: z 
pˆ  p
pq
n
 P-values: use the standard normal distribution (table A-2) and find the pvalues
Memory Trick: “If the P-value is low, the null must go”
“If the P-value is high, the null will fly”
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 Critical values: use the standard normal distribution (table A-2) and find
the z scores that correspond to the given significance level
 Ex: Among 703 randomly selected workers, 61% got their jobs through
networking. Use the following sample data with 0.05 significance level to
test the claim that most workers (more than 50%) get their jobs through
networking.
Claim: Most workers get their jobs through networking (p > 0.5)
Sample data: n = 703 and p̂ = 0.61
--Are the requirements satisfied?
1. simple random sample
2. fixed number of independent trials, 2 categories—got job
through networking or not
3. np ≥ 5 and nq ≥ 5 are both satisfied with n = 703, p = 0.5 and q =
0.5. (np = 351.5 and nq = 351.5)
--P-value method of hypothesis testing:
Step 1: The original claim in symbolic form:
Step 2: The opposite of the original claim:
Step 3: H0: __________
H1: __________
Step 4: Use a significance level of  = ________
Step 5: The normal distribution is used for this test (because we
are testing a claim about a population proportion p & the
sampling distribution of sample proportions p̂ is approximated
by a normal distribution)
Step 6: Calculate the test statistic
z
pˆ  p
pq
n
Because the hypothesis test is right-tailed, we find the
cumulative area to the right of z = _____ is ______
Step 7: Because the P-value of ______ is less than or equal to the
significance level of  = 0.05, we ______________________
Step 8: We conclude that
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--Critical value method of hypothesis testing:
Steps 1-5: same as above
Step 6: This is a right-tailed test, so the area of the critical region
is an area of  = ___ in the right tail. From table A-2, we find
z = ________
Step 7: Because the test statistic (z = 5.83) falls within the critical
region, we _____________________________
Step 8: We conclude that
--Confidence interval method of hypothesis testing:
Claim of p > 0.5 can be tested with a 0.05 significance level by
constructing a 90% confidence interval
Using the methods from section 7-2, we find the following
confidence interval: 0.580 < p < 0.640
Because we are 90% confident that the true value of p is contained
within the limits of 0.580 and 0.640, we have sufficient
evidence to support the claim that p > 0.5.
Note: The confidence interval uses an estimated standard deviation
based on the sample proportion & may often differ from results of
the traditional and P-value methods of hypothesis testing. A good
strategy is to use confidence intervals to estimate population
proportions but use the traditional and P-value methods for testing a
hypothesis.
 Ex: When Gregor Mendel conducted his famous hybridization experiments
with peas, one such experiment resulted in offspring consisting of 428
peas with green pods and 152 peas with yellow pods. According to Mendel’s
Theory, ¼ of the offspring peas should have yellow pods. Use a 0.05
significance level to test the claim that the proportion of peas with yellow
pods is equal to ¼.
--Are the requirements satisfied?
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--P-value method of hypothesis testing:
--Critical value method of hypothesis testing:
20
--Finding the number of successes if you’re given the sample proportion:
-Technology often requires the number of successes (x) in order to test a
claim about a population proportion
-Change the percent to a decimal and multiply it by the sample size (n)
-Ex: A Harris Interactive survey conducted for Gillette showed that
among 514 human resource professionals polled, 90% said that appearance
of a job applicant is most important for a good first impression. What is
the actual number of respondents who said that appearance of a job
applicant is most important for a good first impression?
--Using a TI-83 or TI-84 Graphing Calculator to find the test statistic when
testing a claim about a population proportion:
1. Press STAT and select TESTS
2. Select 1: PropZTest
3. Enter the claimed value of the population proportion for p0, then enter
the values for x and n, and then select the type of test.
4. Highlight Calculate and press ENTER
Ex: If you were testing the claim from the previous example that the
proportion of human resource professionals who said that appearance of a
job applicant is most important for a good impression is greater than 75%,
use the graphing calculator to find the test statistic.
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8.4
Testing a Claim about a Mean
Part I: Testing Claims about a Population Mean (with  not known):
More practical and realistic because we don’t usually know the value of 
 Requirements:
1.
2.
3.
 Test Statistic for Testing a Claim about a Mean (with  not known):
z
x  x
s
n
 P-values & Critical Values: use table A-3 and use df = n – 1 for the number
of degrees of freedom.
 Ex: Data set 20 in Appendix B includes weights of 13 red M&M candies
randomly selected from a bag containing 465 M&M’s. The sample weights
are listed below and they have a mean of x = 0.8635 and a standard
deviation of s = 0.0576. The bag states that the net weight of the
contents is 396.9 g, so the M&M’s must have a mean weight that is at least
396.9/465 = 0.8535 g in order to provide the amount claimed. Use the
sample data with a 0.05 significance level to test the claim of a production
manager that the M&M’s have a mean that is actually greater than 0.8535
g, so consumers are being given more than the amount indicated on the
label.
0.751
0.942
0.857
0.841
0.873
0.856
0.809
0.799
0.890
0.966
0.878
0.859
0.905
--Check the requirements: simple random sample, we are not using a known
value of , the sample size n = 13 is not greater than 30 but the normal
quantile plot suggests the weights are normally distributed.
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--Critical value method of hypothesis testing:
Step 1: The original claim:
Step 2: The opposite of the original claim:
Step 3: H0: _________
H1: _________
Step 4: Use a significance level of  = __________
Step 5: Because the claim is made about the population mean  and
because the requirements for using the t test statistic are
satisfied, we use the t distribution.
x  x
Step 6: Calculate the test statistic: t 
=
s
n
The correct df = ____________
The test is right-tailed with  = _______
Use table A-3 to find the critical value of t = ________
Step 7: Because the test statistic of t = 0.626 does not fall in the
critical region, we __________________________
Step 8: We conclude that
.
 Finding P-values with the t distribution:
 Use Table A-3 to identify a range of P-values as follows: use the
number of degrees of freedom to locate the relevant row of the
table, then determine where the test statistic lies relative to the t
values in that row. Based on a comparison of the t test statistic and
the t values in the row of table A-3, identify a range of values by
referring to the area values given at the top of table A-3.
--Ex: Use the previous example and the P-value method of
hypothesis testing:
-This is a right-tailed test, so the P-value is the area to the
right of the test statistic.
-Use table A-3 & locate the row where df =_____
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-The test statistic of t = _____ is smaller than every value in
that row of the table so the “area in one tail” is greater
than ______
-The area to the right of t = ______ is greater than _____
-- Ex: Use Table A-3 to find a range of values for the P-value
corresponding to a t test with these components: Test statistic is t
= 2.777, sample size is n = 20, significance level is  = 0.05,
alternative hypothesis is H1:   5
-Use table A-3 & locate the row where df = _____
-Because the test statistic of t = ____ lies between the table
values of _____ and ____, the P-value is between the
corresponding “areas in two tails” of _____ and ____
-Although we can’t find the exact P-value, we can conclude
that ______ < P-value < _______
 Using a Confidence Interval to test a claim about  when  is not known:
--For a two-tailed hypothesis test with a 0.05 significance level, we
construct a 95% confidence interval.
--For a one-tailed hypothesis test with a 0.05 significance level, we
construct a 90% confidence interval.
--Using the sample data from the previous example: n = 13, x =
0.8635 g, s = 0.0576 g. (with  not known), and a significance
level of 0.05
--By the methods learned in section 7-4, we construct this 90%
confidence interval: 0.8350 <  < 0.8920.
--Because the assumed value of  = 0.8535 is contained within the
confidence interval, we cannot reject that assumption.
--We don’t have sufficient evidence to support the claim that the
mean weight is greater than 0.8535 g. Based on the
confidence interval, the true value of  is likely to be any value
between 0.8350 and 0.8920 g, including 0.8535 g.
 Although the 3 methods of testing hypotheses can be different when you
know the value of  (section 8-3), when testing a claim about a population
mean, there is no such difference—all 3 methods are equivalent.
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--Ex #2: Listed below are the measure radiation emissions (in W/kg)
corresponding to a sample of cell phones. Use a 0.05 significance level to test
the claim that cell phones have a mean radiation level that is less than 1.00
W/kg. Assume the sample is a simple random sample and that the data is from
a population that is normally distributed.
0.38
0.55
1.54
1.55
0.50
0.60
0.92
0.96
1.00
0.86
1.46
-Critical value method:
Step 1: The original claim:
Step 2: The opposite of the original claim:
Step 3: H0: ___________
H1: ___________
Step 4: Use a significance level of  = _______
Step 5: Because the claim is made about the population mean  and
because the requirements for using the t test statistic are
satisfied, we use the t distribution.
x  x
Step 6: Calculate the test statistic: t 
s
n
The correct df = ___________
The test is right-tailed with an area of  = ______ in one tail
Use table A-3 to find the critical value of t = __________
Step 7: Because the test statistic of t = -0.486 does not fall in the
critical region, we ________________________________
Step 8: We conclude that
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-P-value method:
-This is a left-tailed test, so the P-value is the area to the left of
the test statistic.
-Use table A-3 & locate the row where df = _____
-The test statistic of t = +________ is smaller than every value in
that row of the table so the “area in one tail” (the area
to the right of 0.486) is greater than _______
-So, by symmetry, the area to the left of t = -0.486 is also greater
than 0.10
-The area to the left of t =______ is greater than ______
-Because the test statistic of t = -0.486 is greater than α = 0.05, we
________________________________
-We conclude that
--Using a TI-83 or TI-84 Graphing Calculator to test a claim about the
population mean when σ is unknown
1. Press STAT and select TESTS
2. Select 2: T-test
3. You can choose whether you want to enter data into the list L1 or you
can enter the statistics
4. Enter the claimed value of the population mean for µ0, then enter
the values for the sample mean, sample standard deviation, and
sample size, and then select the type of test.
4. Highlight Calculate and press ENTER
Ex: Find the test statistic t and the p-value using example #2:
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Part II: Testing Claims about a Population Mean (with  known):
Uses the normal distribution with the same components of hypothesis testing as section
8-3.
 Requirements:
1.
2.
3.
 Test Statistic for Testing a Claim about a Mean (with  known): z 
x  x

n
 P-values: use the standard normal distribution (table A-2) and find the pvalues
 Critical values: use the standard normal distribution (table A-2) and find
the z scores
 Ex: Data set 13 in Appendix B includes weights of 13 red M&M candies
randomly selected from a bag containing 465 M&M’s. The standard
deviation of the weights of all M&M’s in the bag is  = 0.0565 g. The
sample weights are listed below and they have a mean of x = 0.8635. The
bag states that the net weight of the contents is 396.9 g, so the M&M’s
must have a mean weight that is at least 396.9/465 = 0.8535 g in order to
provide the amount claimed. Use the sample data with a 0.05 significance
level to test the claim of a production manager that the M&M’s have a
mean that is actually greater than 0.8535 g, so consumers are being given
more than the amount indicated on the label.
0.751
0.942
0.857
0.841
0.873
0.856
0.809
0.799
0.890
0.966
0.878
0.859
0.905
--Check the requirements: simple random sample, value of  = 0.0565 g,
the sample size n = 13 is not greater than 30 but the normal quantile
plot suggests the weights are normally distributed.
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--P-value method of hypothesis testing:
Step 1: The original claim:
Step 2: The opposite of the original claim:
Step 3: H0: ______________
H1: ______________
Step 4: Use a significance level of  = _____
Step 5: Because  is known and the population appears to be
normally distributed, the central limit theorem indicates that
the distribution of sample means can be approximated by a
normal distribution.
x  x
Step 6: Calculate the test statistic: z 

n
Because the hypothesis test is right-tailed, we find the
cumulative area to the right of z = ____ is ______________
Step 7: Because the P-value of ________ is greater than the
significance level of  = 0.05, we _____________________
________________________
Step 8: We conclude that
--Critical value method of hypothesis testing:
Steps 1-5: same as above
Step 6: This is a right-tailed test, so the area of the critical region
is an area of  = ___ to the right so 0.95 is the area from the
left. From table A-2, we find z = _______
Step 7: Because the test statistic (z = 0.64) does not fall within the
critical region, __________________________________
Step 8:
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--Using a TI-83 or TI-84 Graphing Calculator to test a claim about the
population mean when σ is known
1. Press STAT and select TESTS
2. Select 1: Z-test
3. You can choose whether you want to enter data into the list L1 or you
can enter the statistics
4. Enter the claimed value of the population mean for µ0, then enter
the values for the sample mean, population standard deviation, and
sample size, and then select the type of test.
4. Highlight Calculate and press ENTER
Ex: Find the test statistic z and the p-value using the previous example:
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8.5
Testing a Claim about a Standard Deviation or Variance
Testing Claims about a Population Standard Deviation  or a Population Variance ²
Uses the chi-square distribution from section 7-4
 Requirements:
1.
2.
 Test Statistic for Testing a Claim about  or ²:  2 
where n =
s=
σ=
s2 =
σ2 =
(n  1) s 2
2
 P-values and Critical Values: Use table A-4 with df = n – 1 for the number
of degrees of freedom
*Remember that table A-4 is based on cumulative areas from the _______
 Properties of the Chi-Square Distribution:
1. All values of  2 are nonnegative and the distribution is not symmetric
2. There is a different  2 distribution for each number of degrees of
freedom
3. The critical values are found in table A-4 (based on cumulative areas
from the right)
--locate the row corresponding to the appropriate number of
degrees of freedom (df = n – 1)
--the significance level  is used to determine the correct column
--Right-tailed test: Because the area to the right of the critical
value is 0.05, locate 0.05 at the top of table A-4
--Left-tailed test: With a left-tailed area of 0.05, the area to the
right of the critical value is 0.95 so locate 0.95 at the top of
table A-4
--Two-tailed test: Divide the significance level of 0.05 between the
left and right tails, so the areas to the right of the two
critical values are 0.975 and 0.025. Locate 0.975 and 0.025 at
the top of table A-4
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 Ex #1: The industrial world shares this common goal: Improve quality by
reducing variation. Quality control engineers want to ensure that a
product has an acceptable mean, but they also want to produce items of
consistent quality so that there will be few defects. The Newport Bottling
Company had been manufacturing cans of cola with amounts having a
standard deviation of 0.051 oz. A new bottling machine is tested, and a
simple random sample of 24 cans results in the amounts (in ounces) listed
below. (Those 24 amounts have a standard deviation of s = 0.039 oz). Use
a 0.05 significance level to test the claim that cans of cola from the new
machine have amounts with a standard deviation that is less than 0.051 oz.
11.98 11.98 11.99 11.98 11.90 12.02 11.99 11.93
12.02 12.02 12.02 11.98 12.01 12.00 11.99 11.95
11.95 11.96 11.96 12.02 11.99 12.07 11.93 12.05
--Check requirements: simple random sample, histogram & normal quantile
plot show that the sample appears to come from a population with a
normal distribution
--Critical value method of testing hypotheses:
Step 1: The original claim: __________
Step 2: If the original claim is false, then __________
Step 3: H0: __________
H1: __________
Step 4: The significance level is ________
Step 5: Because the claim is made about , we use the chi-square
distribution and we have a _______________ test
(n  1) s 2
Step 6: The test statistic is  2 
2

Using table A-4, df = __ and the column corresponding to ___,
we find the critical value = _______
Step 7: Because the test statistic is not in the critical region, we
__________________________________
Step 8:
.
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--P-value method of testing hypotheses: When using table A-4, we usually
cannot find exact P-values because that chi-square distribution table
includes only selected values of . If using table A-4, we can identify limits
that contain the P-value. You will reach the same conclusion as with the
traditional method.
Step 1: Identify the df = n – 1 =
Step 2: Use the test statistic of _____ and see that in the ____ row of
table A-4, the test statistic is between _____ and ______ so the area
to the right is between ______ and _______
Step 3: Since we have a left-tailed test, the area to the left must be
between _____ and _____ Since our P-value is ______________ to 0.05,
we _____________________________________
Step 4: Conclusion:
--Confidence interval method of testing hypothesis: Using the methods from
section 7-5, we can use the sample data (n = 24, s = 0.039) to construct
this 90% confidence interval: 0.032 <  < 0.052. Again, we cannot conclude
that  is less than 0.051. (same conclusions as the critical value & p-value
methods)
--Note: The TI-83/84 graphing calculator does not test hypotheses about σ
or σ2
 Ex #2: The Skytek Avionics company uses a new production method to
manufacture aircraft altimeters. A simple random sample of new altimeters
resulted in the errors listed below. Use a 0.01 significance level to test the
claim that the new production method has errors with a standard deviation
greater than 32.2 feet, which was the standard deviation for the old
production method. If it appears that the standard deviation is greater, does
the new production method appear to be better or worse than the old method?
Should the company take any action?
-42
78
-22
-72
-45
15
17
51
-5
-53
-9
-109
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--Critical value method of testing hypotheses:
--P-value method of testing hypotheses:
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