Download 10.2

Section 10.2
Hypothesis Testing for Population
Means (s Known)
With valuable content added
by D.R.S., University of Cordele
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Two ways to reach a conclusion
1. Use the rejection region to draw a conclusion.
2. Use the p-value to draw a conclusion.
There are two ways to get the answer and we learn BOTH.
• The “Critical Value” method, comparing your z Test Statistic
to the z Critical Value, which was determined by the chosen
α Level of Significance.
• The “p-Value method”, in which your z Test Statistic leads
to a “p-Value”, an area under the normal curve, which is
compared to the chosen α Level of Significance.
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Hypothesis Testing for Population Means
(s Known)
Test Statistic for a Hypothesis Test for a Population Mean (s
Known) When the population standard deviation is known, and
either the sample size is at least 30 or the population distribution
is approximately normal, THEN COMPUTE: the test statistic for a
hypothesis test for a population mean is given by
x 
z
 s 


n
𝑥 is _________________________________
𝜇 is ___________________ according to ____
𝜎 is _________________________________
𝑛 is _________________________________
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Hypothesis Testing for Population Means
(s Known)
THEN COMPUTE: the test statistic for a hypothesis test
for a population mean is given by
x 
z
 s 


n
My sample mean is how far away
from the H0 Null Hypothesis mean?
Gauged by the standard deviation for
a sample mean as defined in the
___________ _________ Theorem.
We will use this Test Statistic to decide between
___________ing H0 or ___________ing to _________ H0.
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Rejection Regions
Alternative Hypothesis, Ha
 Value
 Value
 Value
The “rejection region” is
the one-tail area of size α.
Type of Hypothesis Test
Left-tailed test
Right-tailed test
Two-tailed test
The “rejection region” consists
of two tails, each of them
of size α/2, left and right.
And the “Fail to Reject H0” region is the big area of size 1 – α.
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A Left-Tailed Hypothesis Test: Ha < μ
DECISION RULE: Reject the
null hypothesis, H0, if the
test statistic calculated
from the sample data falls
within the rejection region.
The critical z value is the negative z value which separates the left tail of area α.
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A Right-Tailed Hypothesis Test: Ha > μ
DECISION RULE: Reject the
null hypothesis, H0, if the
test statistic calculated
from the sample data falls
within the rejection region.
The critical z value is the positive z value which separates the right tail of area α.
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A Two-Tailed Hypothesis Test: Ha ≠ μ
DECISION RULE: Reject the null
hypothesis, H0, if the test statistic
calculated from the sample data
falls within the rejection region.
The critical z values are the +/- z values which separate the two tails, area α/2 each.
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Commonly-occurring critical values
• These values for the α
Level Of Significance (or
c Level Of Confidence)
are typical. You should
already know how to
find them using tables
and using invNorm(). It
is convenient to have
this special table for
reference.
Area in
tail(s),
α, alpha
(and c too)
OneTailed
Test
TwoTailed
Test
0.10 (0.90)
1.28
±1.645
0.05 (0.95)
1.645
±1.96
0.02 (0.98)
2.05
±2.33
0.01 (0.99)
2.33
±2.575
Example 10.10: Using a Rejection Region in a Hypothesis Test for
a Population Mean (Right-Tailed, s Known)
The state education department is considering
introducing new initiatives to boost the reading levels
of fourth graders. The mean reading level of fourth
graders in the state over the last 5 years was a Lexile
reader measure of 800 L. (A Lexile reader measure is a
measure of the complexity of the language that a
reader is able to comprehend.) The developers of a
new program claim that their techniques will raise the
mean reading level of fourth graders by more than 50 L.
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Example 10.10: Using a Rejection Region in a Hypothesis Test for
a Population Mean (Right-Tailed, s Known) (cont.)
To assess the impact of their initiative, the developers
were given permission to implement their ideas in the
classrooms. At the end of the pilot study, a simple
random sample of 1000 fourth graders had a mean
reading level of 856 L. It is assumed that the population
standard deviation is 98 L. Using a 0.05 level of
significance, should the findings of the study convince
the education department of the validity of the
developers’ claim?
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Example 10.10: Using a Rejection Region in a Hypothesis Test for
a Population Mean (Right-Tailed, s Known) (cont.)
Solution
Step 1: State the null and alternative hypotheses.
The developers want to show that their
classroom techniques will raise the fourth
graders’ mean reading level to more than 850
L.
H0: ____________
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Ha:______________
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Example 10.10: Using a Rejection Region in a Hypothesis Test for
a Population Mean (Right-Tailed, s Known) (cont.)
Step 2: Determine which distribution to use for the
test statistic, and state the level of significance.
Since the population parameter of interest is _______
and s is { known, unknown }, the sample is a simple
random sample, and the sample size is ≥ ______. Thus,
we will use a _______ distribution, and { z, t } . And
someone decided that in this experiment, the level of
significance is a = _________
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Example 10.10: Using a Rejection Region in a Hypothesis Test for
a Population Mean (Right-Tailed, s Known) (cont.)
.
DRAW A PICTURE!
Label the critical z
value.
Later, put your test
statistic at the
appropriate place
on the z axis.
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Example 10.10: Using a Rejection Region in a Hypothesis Test for
a Population Mean (Right-Tailed, s Known) (cont.)
Step 3: Gather data and calculate the necessary
sample statistics.
(They’ve done all the hard work for us.)
“At the end of the pilot study, a simple random
sample of 1000 fourth graders had a mean
reading level of 856 L. The population standard
deviation is 98 L.”
Observation: The 856 { is, is not } more than the
______ in the null hypothesis, but is it strongly enough
higher to be s____________ally s___________ ?
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Example 10.10: Using a Rejection Region in a Hypothesis Test for
a Population Mean (Right-Tailed, s Known) (cont.)
Calculate your test statistic. Write it in on the
appropriate place on the z-axis in your normal curve
picture. (a couple of slides ago)
𝑥−𝜇
𝑧=
𝜎
𝑛
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Example 10.10: Using a Rejection Region in a Hypothesis Test for
a Population Mean (Right-Tailed, s Known) (cont.)
Step 4: Draw a conclusion and interpret the decision.
Remember that we determine the type of test
based on the alternative hypothesis. In this
case, the alternative hypothesis contains “>,”
which indicates that this is a right-tailed test.
Recall that we picked α = _______
The critical value, the z value that has this area
to its right, is __________. (And update your picture.)
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Example 10.10: Using a Rejection Region in a Hypothesis Test for
a Population Mean (Right-Tailed, s Known) (cont.)
The z-value of 1.94 falls { inside or outside ? } the
rejection region. So the conclusion is to { Reject? or
Fail To Reject? } the Null Hypothesis.
Conclusion: Thus the evidence collected suggests that
the education department can be _____% sure of the
validity of the developers’ claim that the mean Lexile
reader measure of fourth graders will increase by more
than 50 points.
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p-Values
p-value
A p-value is the probability of obtaining a sample
statistic as extreme or more extreme than the one
observed in the data, when the null hypothesis, H0, is
assumed to be true.
We’ll give examples of how to find a p-value by hand.
But usually you’ll get the p-value for free from the TI-84’s ZTest
or TTest feature.
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Example: Calculating the p-Value for a
z-Test Statistic for a _______-Tailed Test
.
Calculate the p-value for a
hypothesis test with the
hypotheses
shown at right.. Assume that
data have been collected and
the test statistic was calculated
to be z = 1.34.
The p-value is the area to the { left
H0 :   0.15
Ha :   0.15
right } of z= _______
The p-value is ______________
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Example: Calculating the p-Value for a
z-Test Statistic for a _______-Tailed Test
Calculate the p-value for a
.
hypothesis test with the
hypotheses
shown at right.. Assume that
data have been collected and
the test statistic was calculated
to be z = 2.78.
The p-value is the area to the { left
H0 :   0.43
Ha :   0.43
right } of z= _______
The p-value is ______________
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Example Calculating the p-Value for a
z-Test Statistic for a ______-Tailed Test
Calculate the p-value for a
.
hypothesis test with the
hypotheses
shown at right.. Assume that
data have been collected and
the test statistic was calculated
to be z = – 2.15.
We find the area to the { left
H0 :   0.78
Ha :   0.78
right } of z= _______
The p-value is ______________ times _____ = __________.
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How to make a conclusion using the p-Value
Conclusions Using p-Values
• If p-value ≤ a, then reject the null hypothesis.
(because your small p-value, that small area that’s
more extreme than your extreme result, is inside the
larger rejection area α)
• If p-value > a, then fail to reject the null hypothesis.
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Z-Test for the Reading Lexile Scores problem
STAT, TESTS, 1:Z-Test. Since they gave us summary
statistics, we pick “Stats” (not “Data”, which is for a
problem where we have the individuals’ scores).
μ0 comes from H0
σ is the “known” stdev.
bar-x from our sample
n is our sample size
Give the Ha inequality
Highlight Calculate and press ENTER.
Z-Test for this problem.
STAT, TESTS, 1:Z-Test. Here are the results.
It reminds you of what you told it was the alternative
hypothesis. Have a look and make sure it’s what you
wanted.
It tells you the Test Statistic.
It tells you the p-Value.
It reminds you of the sample
data, the sample mean and
sample size you told it.
Z-Test for this problem.
So for the reading example, the p-value was ________
and the level of significance was 𝜶 =________.
Therefore we { reject, fail to reject } the null
hypothesis.
Example 10.16: Performing a Hypothesis Test for
a Population Mean (Right-Tailed, s Known)
A researcher claims that the mean age of women in
California at the time of a first marriage is higher than
26.5 years. Surveying a simple random sample of 213
newlywed women in California, the researcher found a
mean age of 27.0 years. Assuming that the population
standard deviation is 2.3 years and using a 95% level of
confidence, determine if there is sufficient evidence to
support the researcher’s claim.
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Example 10.16: TI-84 Z-Test solution
• Null Hypothesis: ________ Alternative: ________
What is the α Level Of Significance? α = _______
What inputs do you give to the TI-84 Z-Test?
• 𝜇0 = _______, 𝜎 = ________
• 𝑥 = _______, 𝑛 = ________
• 𝜇: ≠ or < or >
The TI-84 Z-Test output gives us a p-value of ________.
Therefore we { reject or fail to reject } H0.
Draw a bell curve diagram of the story:
Example 10.16: Performing a Hypothesis Test for a
Population Mean (Right-Tailed, s Known) (cont.)
Draw a bell curve diagram for this problem.
Could you calculate the z test statistic and
find the critical z value if the problem asked
for those, too?
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Examples 10.16 and 10.17 complete solutions by
hand with all the details
Complete by-hand detailed solution for Example 10.16:
and the
Complete by-hand detailed solution for Example 10.17,
the next problem
1) 2205.drscompany.com
2) click on Slides
3) The links are there, immediately following the link
for these 10.2 slides.
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Example 10.17: Performing a Hypothesis Test for
a Population Mean (Two-Tailed, s Known)
A recent study showed that the mean number of
children for women in Europe is 1.5. A global watch
group claims that German women have a mean fertility
rate that is different from the mean for all of Europe. To
test its claim, the group surveyed a simple random
sample of 128 German women and found that they had
a mean fertility rate of 1.4 children. The population
standard deviation is assumed to be 0.8. Is there
sufficient evidence to support the claim made by the
global watch group at the 90% level of confidence?
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Example 10.17: TI-84 Z-Test solution
• Null Hypothesis: ________ Alternative: ________
What is the α Level Of Significance? α = _______
What inputs do you give to the TI-84 Z-Test?
• 𝜇0 = _______, 𝜎 = ________
• 𝑥 = _______, 𝑛 = ________
• 𝜇: ≠ or < or >
The TI-84 Z-Test output gives us a p-value of ________.
Therefore we { reject or fail to reject } H0.
Draw a bell-curve diagram of the story:
Example 10.17: Performing a Hypothesis Test for
a Population Mean (Two-Tailed, s Known)
Draw a bell curve diagram for this problem.
Could you calculate the z test statistic and
find the critical z value if the problem asked
for those, too?
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Excel’s Z.TEST(data array, μ, σ) function
•
•
•
•
Give it the cells (“range”) of the sample data values.
And the μ mean from the null hypothesis.
And the σ population standard deviation.
It gives you back the one-tailed p-value.
• So it calculates a z = test statistic based on the
sample.
• And then it calculates the area in the tail beyond
that z
• Either left tail beyond negative z
• Or right tail beyond positive z, it knows which one.
Excel’s Z.TEST(data array, μ, σ) function
•
•
•
•
•
Give it the cells (“range”) of the sample data values.
And the μ mean from the null hypothesis.
And the σ population standard deviation.
It gives you back the one-tailed p-value.
You compare that p-value to the α level of
significance that was pre-selected in the design.
• If p-value < α, then reject H0.
• If p-value > α, then fail to reject H0.
Excel’s Z.TEST(data array, μ, σ) if you have
a Two-Tailed Situation
• Recall the two-tailed situation: Ha has a ≠.
• So the rejection regions are two tails of area α/2.
• Excel Help says to do this to get the p-value:
2 * MIN(Z.TEST(data , μ, σ), 1 – Z.TEST(data , μ, σ))
(whereas we tell TI-84 it’s two-tailed and TI-84 takes
care of doubling it transparently, thank you.)
• And then decide as usual:
• If p-value < α, then reject H0.
• If p-value > α, then fail to reject H0.