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Symbols & graphs quiz
• Look through book and be
ready to ask questions.
Page 367 1-36
Types of Errors
• A type I error reject 𝐻0 , and 𝐻0 is true.
• A type II error fail to reject 𝐻0 , and 𝐻0 is false.
Larson/Farber 4th ed.
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EXAMPLE
• Claim : It will snow 3 inches tonight.
• 𝐻0 = 3 inches
• 𝐻𝑎 ≠ 3inches
• Type I error: we reject 𝐻0 and then it snows 3in
• Type II error: we fail to reject 𝐻0 and it does not snow 3in
Example
• Claim: a company claims their parachutes
failure rate is not more than 1%.
• 𝐻0 : p ≤ .01
• 𝐻𝑎 : p > .01
• Type I error: reject p ≤ .01 and it is true
• Type II error: fail to reject p ≤ .01 and it is false
• Unusual events: p ≤ .05 or 5%
• Reject null hypothesis when sample statistic is unusual.
• Because samples vary , there is always a chance you
reject 𝐻0 when it is true.
• Can decrease the probability of this happening by
lowering the “level of significance”
• Level of significance: maximum probability that we
make a type I error (reject 𝐻0 )
• Symbol : α ( alpha)
Level of Significance
Level of significance
Denoted by , the lowercase Greek letter alpha.
• By setting the level of significance at a small
value, you are saying that you want the
probability of rejecting a true null hypothesis
to be small.
• Commonly used levels of significance:
 = 0.10  = 0.05
 = 0.01
• Smaller the value of α says you want a small
probability of rejecting a true 𝐻0 .
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• P(type II error) = β (beta)
• Power of test: 1 – β ( probability of rejecting
𝐻0 when it is false)
Statistical Tests
• 1.After stating the 𝐻0 and 𝐻𝑎 and specifying the level
of significance,
• 2. then a random sample is taken from the population
and find sample statistics ( x-bar, 𝑠 2 , s, p, etc).
• Then pick a statistic that is compared with the
parameter in the null hypothesis is called the test
statistic.
Population
parameter
Test statistic
μ
x
p
σ2
p̂
s2
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Standardized test
statistic
z (Section 7.2 n  30)
t (Section 7.3 n < 30)
z (Section 7.4)
χ2 (Section 7.5)
P-values
P-value (or probability value)
• The probability, if the null hypothesis is true,
of getting a sample statistic with a value as or
more extreme than the one determined from
the sample data.
• So if you did another sample : P-value is probability
the sample statistic would be the same or farther out
than the 1st statistic you got.
Larson/Farber 4th ed.
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Nature of the Test
• Three types of hypothesis tests
– left-tailed test
– right-tailed test
– two-tailed test
• The type of test depends on the region of the
sampling distribution that favors a rejection of
H0.
• This region is indicated by the Ha .
Larson/Farber 4th ed.
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Left-tailed Test
• The alternative hypothesis Ha contains the
less-than inequality symbol (<).
H0: μ  k
Ha: μ < k
P is the area to
the left of the
test statistic.
z
-3
-2
-1
0
Test
statistic
Larson/Farber 4th ed.
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1
2
3
Right-tailed Test
• The alternative hypothesis Ha contains the
greater-than inequality symbol (>).
H0: μ ≤ k
Ha: μ > k
P is the area
to the right
of the test
statistic.
z
-3
-2
-1
0
1
2
Test
statistic
Larson/Farber 4th ed.
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3
Two-tailed Test
• The alternative hypothesis Ha contains the not equal
inequality symbol (≠). Each tail has an area of ½P.
H0: μ = k
Ha: μ  k
P is twice the
area to the right
of the positive
test statistic.
P is twice the
area to the left of
the negative test
statistic.
z
-3
-2
-1
Test
0
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1
Test
2
3
• The smaller the p-value, then the more
evidence to reject H0
• A small P-value shows an unusual event.
• Still can’t say 𝐻0 is false for sure.
Example: Identifying The Nature of a
Test
For each claim, state H0 and Ha. Then determine
whether the hypothesis test is a left-tailed, righttailed, or two-tailed test. Sketch a normal sampling
distribution and shade the area for the P-value.
1. A university publicizes that the proportion of its
students who graduate in 4 years is 82%.
Solution:
H0: p = 0.82
Ha: p ≠ 0.82
½ P-value
area
Two-tailed test
-z
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½ P-value
area
0
z
z
Example: Identifying The Nature of a
Test
For each claim, state H0 and Ha. Then determine
whether the hypothesis test is a left-tailed, righttailed, or two-tailed test. Sketch a normal sampling
distribution and shade the area for the P-value.
2. A water faucet manufacturer announces that
the mean flow rate of a certain type of faucet is
less than 2.5 gallons per minute.
Solution:
H0: μ ≥ 2.5 gpm
Ha: μ < 2.5 gpm
P-value
area
-z
Left-tailed test
Larson/Farber 4th ed.
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0
z
Example: Identifying The Nature of a
Test
For each claim, state H0 and Ha. Then determine
whether the hypothesis test is a left-tailed, righttailed, or two-tailed test. Sketch a normal sampling
distribution and shade the area for the P-value.
3. A cereal company advertises that the mean
weight of the contents of its 20-ounce size
cereal boxes is more than 20 ounces.
Solution:
H0: μ ≤ 20 oz
Ha: μ > 20 oz
P-value
area
0
Right-tailed test
Larson/Farber 4th ed.
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z
z
Making a Decision
Decision Rule Based on P-value
• Compare the P-value with .
If P  , then reject H0.
(probability less than or = to level of significance)
If P > , then fail to reject H0.
(probability greater than level of significance)
*failing to reject 𝐻0 does not mean it is true, just tells you
there is not enough evidence to say it is false.
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Making a Decision
Decision Rule Based on P-value
Claim
Decision
Claim is H0
Claim is Ha
Reject H0
There is enough evidence to
reject the claim
There is enough evidence to
support the claim
Fail to Reject H0
There is not enough evidence There is not enough evidence
to reject the claim
to support the claim
Example: Interpreting a Decision
You perform a hypothesis test for the following
claim. How should you interpret your decision if
you reject H0? If you fail to reject H0?
1. H0 (Claim): A university publicizes that the
proportion of its students who graduate in 4
years is 82%.
Larson/Farber 4th ed.
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Solution: Interpreting a Decision
• The claim is represented by H0.
• If you reject H0 you should conclude “there is
sufficient evidence to indicate that the
university’s claim is false.”
• If you fail to reject H0, you should conclude “there
is insufficient evidence to indicate that the
university’s claim (of a four-year graduation rate
of 82%) is false.”
Larson/Farber 4th ed.
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Example: Interpreting a Decision
You perform a hypothesis test for the following
claim. How should you interpret your decision if
you reject H0? If you fail to reject H0?
2. Ha (Claim): Consumer Reports states that the
mean stopping distance (on a dry surface) for
a Honda Civic is less than 136 feet.
Solution:
• The claim is represented by Ha.
• H0 is “the mean stopping distance…is greater than or
equal to 136 feet.”
Larson/Farber 4th ed.
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Solution: Interpreting a Decision
• If you reject H0 you should conclude “there is
enough evidence to support Consumer Reports’
claim that the stopping distance for a Honda Civic
is less than 136 feet.”
• If you fail to reject H0, you should conclude “there
is not enough evidence to support Consumer
Reports’ claim that the stopping distance for a
Honda Civic is less than 136 feet.”
Larson/Farber 4th ed.
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Steps for Hypothesis Testing using P-value
1. State the H0: ?
Ha: ?
2. Specify the level of significance.
α= ?
3. Determine the standardized
sampling distribution and
draw its graph.
4. Calculate the test statistic
and its standardized value.
Add it to your sketch.
Larson/Farber 4th ed.
This sampling distribution
is based on the assumption
that H0 is true.
z
0
0
z
Test statistic
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Steps for Hypothesis Testing
5. Find the P-value.
6. Use the following decision rule.
Is the P-value less
than or equal to the
level of significance?
Fail to reject H0.
No
Yes
Reject H0.
7. Write a statement to interpret the decision in
the context of the original claim.
Larson/Farber 4th ed.
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ASSIGNMENT
• Page 367 37-51, 57,58.