### Math 10 Chapter 9 Notes: Hypothesis Testing

```Math 10 Chapter 9 Notes:
Hypothesis Testing
polling, education, etc.
•to do: set up 2 contradictory
statements
•first statement - often the accepted
belief
•conduct a test to see whether our data
supports or does not support the
first hypothesis
•
Math 10 Chapter 9 Notes:
Hypothesis Testing
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
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Null hypothesis:
Ho:
Alternate hypothesis: Ha:
Example:
Ho: John loves Marcia
Ha: John does not love
Marcia
Example:
Ho: m = 6
Ha: m < 6
Math 10 Chapter 9 Notes:
Hypothesis Testing

Decision:
Ho is:
True

False


Do not reject Ho Correct
Type II
Decision Error


Reject Ho
Type I
Error
Correct
Decision
Math 10 Chapter 9 Notes:
Hypothesis Testing
Type I error:
Reject the null
hypothesis when the null is TRUE.
P(Type I error) = a
Type II error:
Do not reject the null
hypothesis when the null is FALSE.
P(Type II error) = b
Goal: Minimize a and b.
Math 10 Chapter 9 Notes:
Hypothesis Testing
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
“ = ,  , or  ” are ALWAYS in the null
hypothesis Ho.
“  , > , or < ” are ALWAYS in the
alternate hypothesis Ha.
Math 10 Chapter 9 Notes:
Hypothesis Testing


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Examples: State the null hypothesis,
Ho, and the alternative hypothesis, Ha,
in terms of the appropriate parameter
(m or p).
At most 60% of Americans vote in
presidential elections. (Right-Tailed)
Ho:
p  0.60
Ha:
p > 0.60
Math 10 Chapter 9 Notes:
Hypothesis Testing
Fewer than 5% of adults ride the bus to
work in New York City. (Left-tailed)


Ho: p  0.05
Ha: p < 0.05
Math 10 Chapter 9 Notes:
Hypothesis Testing
The average number of cars a person
owns in his/her lifetime is not more
than 10. (Right-tailed)


Ho: m  10
Ha: m > 10
Math 10 Chapter 9 Notes:
Hypothesis Testing
Europeans have an average paid
vacation each year of six weeks. (Twotailed)


Ho: m = 10
Ha: m  10
Math 10 Chapter 9 Notes:
Hypothesis Testing
Private universities cost, on average,
more than \$20,000 per year for tuition,
room, and board. (Right-tailed)


Ho: m  20,000
Ha: m > 20,000
Math 10 Chapter 9 Notes:
Hypothesis Testing

What are Type I and Type II errors for
some of these problems?
Ho: m  20,000
Ha: m > 20,000
Type I: We believe that private
universities cost, on average, more than
\$20,000 per year for tuition, room, and
board when, in fact, the average cost is
no more than \$20,000.
Math 10 Chapter 9 Notes:
Hypothesis Testing
Ho: m  20,000
Ha: m > 20,000
Type II: We believe that private
universities cost, on average, no more
than \$20,000 per year for tuition, room,
and board when, in fact, the average
cost is more than \$20,000.
no more than = at most = less than or
equal to
Math 10 Chapter 9 Notes:
Hypothesis Testing
To perform a hypothesis test:
sample data is gathered
data typically favors one of the
hypotheses
Decisions
if data favors the null hypothesis
(Ho), we “do not reject the null”
if data favors the alternate hypothesis
(Ha), we “reject the null”
Math 10 Chapter 9 Notes:
Hypothesis Testing
NOTE: We are ALWAYS testing the null
hypothesis, never the alternate. Our
conclusion is ALWAYS in regards to the
null (Ho).
after a decision is made, an appropriate
null hypothesis
Math 10 Chapter 9 Notes:
Hypothesis Testing
sometimes, data favors neither
hypothesis (this implies an
“inconclusive” test result)
a test may be “left-tailed”, “right-tailed”,
or “two-tailed” depending upon the null
hypothesis
associated with the null hypothesis is a
“pre-conceived” a. P(Type I error ) = a
Math 10 Chapter 9 Notes:
Hypothesis Testing
If no pre-conceived a is given, it is
common practice to use a = 0.05. A
test may be “left-tailed”, “right-tailed”,
or “two-tailed” depending upon the null
hypothesis
data is collected to calculate what is
called the p-value, or level of
significance, or calculated a
Math 10 Chapter 9 Notes:
Hypothesis Testing
p-value = P(the information/data will
happen purely by chance GIVEN that
the null hypothes is true)
decision to reject or to not reject the
null is based upon whether a > p-value
or a < p-value
Reject Ho if a > p-value
Do not Reject Ho if a < p-value
Math 10 Chapter 9 Notes:
Hypothesis Testing
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Random Variable:
Xbar = average …
P’ = proportion …
Examples:
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Xbar = the average tuition for a
private college
P’ = the proportion of voters who
voted for the winning candidate
```