Download you will have to put the appropriate sign on the critical value, +, –, or

Chapter 10 – Hypothesis Tests Regarding a Parameter
Section 10.1 – The Language of Hypothesis Testing
A Hypothesis is a statement (claim) regarding a characteristic of one or more populations.
Hypothesis Testing is a procedure, based on sample evidence and probability, used to test
statements (claims) regarding a characteristic of one or more populations.
Because we use sample data to test hypotheses, we cannot say with 100% certainty that the
statement is true; we can only determine if the sample data support the claim or not. We
make an assumption about reality then look at sample evidence to determine whether it
contradicts our assumption.
STEPS IN HYPOTHESIS TESTING
Extract important information from the problem. Write down and label values such as x,
n, p̂, x , s .
1. State the claim in English. Use a complete sentence.
2. State H0 and H1 with symbols. (p,  ,  )
H0 should state the equality
H1 should include <, >, or 
3. Determine the  level.
4. Determine the critical value (Z, t,  2 ).
5. Draw the appropriate graph showing the critical (rejection) region or regions.
6. Calculate and state the appropriate test statistic (Z, t,  2 ). Label the test statistic on
the graph.
7. Use the calculator to find the P-value.
8. State the appropriate decision—either Reject H0 or Do Not Reject H0.
Classical:
P-Value:
If
If
If
If
the test statistic falls in the critical (rejection) region, then Reject H0
the test statistic does not fall in the critical region, Do Not Reject H0
P–value <  , then Reject H0
P–value >  , then Do Not Reject H0
9. Write the final conclusion, addressing the original claim
Reject H0: There is sufficient evidence to conclude/indicate that….(state claim)
Do not reject H0: There is not sufficient evidence to conclude/indicate that….
(state claim)
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1. State the claim in English. Use a complete sentence.
2. State H0 and H1 with symbols. (p,  ,  )
H0 should always state the equality (=)
H1 should include <, >, or 
The Null Hypothesis denoted Ho is a statement that the value of a population
parameter (μ, p, σ), is equal to some claimed value. During the test, you assume that
Ho is true until the evidence indicates otherwise.
Ho: μ = # (mean)
Ho: p = # (proportion)
Ho: σ = # (standard deviation)
The Alternative Hypothesis denoted H1 or HA is a statement that the parameter
somehow differs from Ho . It is the statement we are trying to find evidence to support.
It cannot contain equality, and it must be true if Ho is false.
H1: μ ≠ #
H1: p ≠ #
H1: σ ≠ #
H1: μ > #
H1: p > #
H1: σ > #
H1: μ < #
H1: p < #
H1: σ < #
(mean)
(proportion)
(standard deviation)
A Two-tailed test occurs when H1 contains ≠. Split  evenly into two tails.
A Left-tailed test occurs when H1 contains <. All of  goes in the left tail.
A Right-tailed test occurs when H1 contains >. All of  goes in the right tail.
Look for key phrases when forming the alternative hypothesis.
Examples:
Identify Ho and H1 then determine if the test is left tailed, right tailed, or two tailed.
a. The level of giving has changed from the mean charitable contribution per
household of $1,623.
b. A school nurse thinks that the percentage of 6- to 11-year olds who are overweight
is higher than 16%.
c. A teacher believes that the standard deviation of SAT math scores has decreased
from 113.
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3. Determine the  level.
The level of significance  is the probability of making a Type I error.
A Type I Error occurs when you reject Ho when it is actually true. (  )
A Type II Error occurs when you do not reject Ho when H1 is actually true. (β)
When a decision is made in a test of hypothesis, there is a chance of making a wrong
decision since it is based on incomplete (sample) information. These wrong decisions do
not refer to human errors in sampling, calculations, etc. They result from the fact that
with real data, occasionally a sample can occur that leads to an incorrect decision. The
types of errors are shown in table form:
H1 is true
(Ho is not true)
Ho is true
Do not reject Ho
Correct Conclusion
Type II Error
Reject Ho
Type I Error
Correct Conclusion
The probability of making a Type I error is  . So by setting  very small, .05 for
example, we are indicating that we are willing to accept this error only 5% of the time.
Setting the  -level involves the seriousness of the Type I error. In some cases, where
the Type I error is not so serious,  might be set at .10. A Type I error with more
serious consequences might mean and  of .01.
(Type I error: There is no change, but we decide that there is a change.)
The probability of making a Type I error is β. (This will not be in the hypothesis tests.)
Know that: P (Type I Error) = 
P (Type II Error) = β
As the probability of a Type I error increases, the probability of a Type II error decreases.
Practice: In any trial, the defendant is assumed to be innocent. The district attorney must
collect and present evidence proving the defendant is guilty beyond all reasonable doubt.
Determine the Type I and Type II error
Ho: The defendant is innocent
H1: The defendant is guilty
The two incorrect decisions are to convict an innocent person (Type I: Reject Ho when Ho is
really true) or let a guilty person go free (Type II: Do Not Reject Ho when H1 is really true).
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Claim: The proportion of cat owners who have spayed or neutered their cat is less than
40%.
Ho:
H1:
Type I: (Reject Ho when Ho is really true)
Type II: (Do Not Reject Ho when H1 is really true)
4. Determine the critical value (Z, t,  2 ).
The level of significance is used to determine the critical value. The Critical Value
represents the maximum number of standard deviations that the sample statistic can be
from the population parameter. It is the value that separates the probable from the
improbable.
The Critical or Rejection Region is the set of all values such that the null hypothesis is
rejected.
t: InvT(  , n - 1)
z: InvNorm(area to left)
 2 : Use Table VII
5. Draw the appropriate graph showing the critical (rejection) region or regions.
6. Calculate and state the appropriate test statistic (Z, t,  2 ). Label the test
statistic on the graph.
A Test Statistic is a value computed from the sample data that is used to make the
decision about whether or not Ho is rejected. It represents the number of standard
deviations that the sample statistic is from the assumed parameter in Ho.
x
t
s
n
mean
T-Test
z
p̂  p
(n  1)s2
 
2
2
pq
n
proportion
standard deviation
1-PropZ-Test
 2-Test
7. Use the calculator to find the P-value.
A P-value is the probability of observing a sample statistic as extreme or more extreme
than the one observed under the assumption that the null hypothesis is true. It is the
probability that a sample will result in a sample mean such as the one obtained if the null
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hypothesis is true.
The smaller the p-value is, the greater the evidence against the null hypothesis.
A small p-value (less than  ) implies that the results are unusual and there is a
significant difference from Ho. (so you should reject Ho)
A large p-value (more than  ) implies that the results are not unusual and there is no
significant difference from Ho. (so you should not reject Ho)
Calculating P-values:
(use the calculator)
Right-Tailed Test: The P-value is the area to the right of the test statistic
Left-Tailed Test: The P-value is the area to the left of the test statistic
Two-Tailed Test: The P-value is twice the area to the right of the positive test statistic
8. State the appropriate decision—either Reject H0 or Do Not Reject H0.
Classical:
P-Value:
If
If
If
If
the test statistic falls in the critical (rejection) region, then Reject H0
the test statistic does not fall in the critical region, Do Not Reject H0
p–value <  , then Reject H0
p–value >  , then Do Not Reject H0
If, under a given assumption, the probability of an observed event is exceptionally small
(unlikely), we say that the result is “statistically significant”. When results are found
to be statistically significant, we have evidence to reject the null hypothesis. Sample
evidence can never prove that the null hypothesis to be true. When we do not reject the
null hypothesis, we are saying that the evidence indicates that the null hypothesis could
be true. That is, there is not enough evidence to reject our assumption that the null
hypothesis is true.
9. Write the final conclusion, addressing the original claim
Reject H0: There is sufficient evidence to conclude/indicate that….(state claim)
Do not reject H0: There is not sufficient evidence to conclude/indicate that….
(state claim)
Practice: State the conclusion
a. Claim: The proportion of women who are overweight is less than 0.45.
Decision: Reject Ho
b. Claim: The mean number of peanuts in an 8 oz jar is not 200.
Decision: Fail to reject Ho
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Section 10.2 – Hypothesis Tests for a Population Proportion
Assumptions:
1. Simple random sample
2. Sample values are independent
3. npq ≥ 10
Symbols:
x, n, p
Critical Value:
z-score
invNorm (area to the left)
Test Statistic and p-value come from the calculator.
5:1-PropZ-Test
Page 488 Example 1
In order for a student to be prepared for college mathematics (75% probability of success),
he must make a minimum of a 22 on the ACT Exam. An official from Illinois wonders if less
than half of the students in her state are prepared for college. She obtains a simple random
sample of 500 records of students who have taken the ACT and find 219 are prepared for
college mathematics. Does this represent a significant evidence that less than half of the
Illinois student are prepared for college mathematics when they graduate high school?
Answer: Reject the null hypothesis
Example 2
When asked, “What do you think is more important – to protect the right of Americans to
own guns or to control gun ownership?’ 46 % of all Americans said protecting the right to
own guns is important. If 1267 people who had at least a bachelor’s degree were surveyed,
559 believed the protecting the right to own guns was more important. Does this result
suggest the proportion of Americans with at least a bachelor’s degree feel differently than the
general American population when it comes to gun control? Use a .01 level of significance.
Page 493
Test the hypothesis using (a) the classical approach and (b) the p-value approach.
8.
Ho : p  .6verses H1 : p  .6
n  250, x  124,   .01
Answers: P value = .0004
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Section 10.3 – Hypothesis Tests for a Population Mean – σ unknown
In a test for hypothesis, we are trying to distinguish between results that can easily occur by
chance and those that are highly unlikely to occur by chance.
Assumptions:
1. Simple random sample
2. σ is unknown
3. The population is normally distributed or n > 30
Symbols:
, x,s,n
Critical Value:
t-value
invT (  , n – 1)
Degrees of freedom = n – 1
For a right- or left-tail test, use  as area in the right tail
For a two-tailed test, use  /2 as area in the right tail
(you will have to put the appropriate sign on the critical value, +, –, or ±)
Test Statistic and p-value come from the calculator.
2:T-Test
Page 498 Example 1
The mean height of American males is 69.5 inches. The heights of 43 male U.S. presidents
have a mean height of 70.78 inches. Treating the 43 presidents as a simple random sample,
determine if there is evidence to suggest that the U.S. presidents are taller than the average
American male. Use the .05 level of significance.
Answer: t = 3.030. The test statistic lies in the critical region so we reject the null
hypothesis.
Page 500 Example 2
The “fun size” of a Snickers bar is suppose to weigh 20 grams. Because the penalty for
selling candy bars under their advertised weight is severe, the manufacturer calibrates the
machine so the mean weight is 20.1 grams. The quality-control engineer is concerned about
the calibration. He obtains a random sample of 11 candy bars and weighs them. The data is
as follows;
19.68, 20.66, 10.56, 19.98, 20.65, 19.61, 20.55, 20.36, 21.02, 21.5, 10.74
Should the machine be shut down and calibrated? Because shutting down the plant is very
expensive, he decides to conduct the test at the .01 significance level.
Answer: t= 1.042 which does not lie in the critical region so do not reject the null
hypothesis. P = .323
There is not sufficient evidence to conclude that the Snickers have a mean weight different
from 20.1 ounces at the .01 significance level. The machine does not have to be shut down.
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Section 10.4 – Hypothesis Tests for a Population Standard Deviation
Assumptions:
1. Simple random sample
2. Population is normally distributed
Symbols:
n, s or s2, σ or σ2
Critical Value:
Right Tail:
Left Tail:
Two Tails:
Use the χ2 table (Table VII). Degrees of Freedom: n – 1
Column 
Column 1 – 
Column  /2 and 1 –  /2
(n  1)s2
Test Statistic:  
will be calculated by hand.
0 2
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We will not calculate p-values for this section.
The Chi-Squared picture is always skewed right, centered at the degrees of
freedom, and contains NO negative values!!
Page 511
1. a. Determine the critical value for a right-tailed test of a population standard deviation
with 18 degrees of freedom and .05 significance level. Answer: 28.869
b.
Determine the critical value for a left-tailed test of a population standard deviation with
sample size of 23 and .01 significance level. Answer: 14.041
c.
Determine the critical values for a two-tailed test of a population standard deviation
with sample size of 30 and .05 significance level. Answer: 16.047 and 45.722
Page 512
11. The piston diameter of a certain hand pump is .5 inch. The quality-control manager
determines the diameters are normally distributed with a mean of .5 inch and a standard
deviation of .004 inch. The machine that controls the piston diameter is recalibrated in an
attempt to lower the standard deviation. After recalibration, the quality-control manger
randomly selects 25 pistons from the production line and determines the standard deviation
is .0025 inch. Was the recalibration effective? Use a significance level of .01.
Answer: Reject Ho
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