My Lecture for SAC interview

1. Santa Ana College
Presentation
Hypothesis
Testing
by Richard Corp

2. Where we have been:
• Sampling techniques
• Sample Statistics
• Population parameters
• Sample distributions
• Calculating probabilities
• Calculating and interpreting z-
scores.

3. General idea and logic behind
hypothesis testing.
In 2011 Atlanta state investigators accused teachers in Georgia of
cheating on student state test. A trial began in 2015.
There are two opposing claims in this case:
The teachers’ claim: “We did not cheat”
The state’s claim: “The Teacher did cheat”
Since the teachers are “innocent until proven guilty” the prosecutors
must present evidence supporting their claim.
•100% of students at certain schools who failed the test, passed when
retaking the test.
•Most erased answers were changed to the correct answers.
The jury decided it would be extremely unlikely to get evidence like
that if the teachers’ claim of not cheating had been true. In other
words, the jury agree that the prosecutor brought forward strong
enough evidence to reject the teachers’ claim, and conclude that the
teachers did cheat.

4. General idea and logic behind
hypothesis testing.
A hypothesis test is a procedure for testing a claim about a property
of the population.
4 Components of Hypothesis test:
1.State the claims
• State the null and alternative hypotheses.
1.Gather Evidence
• Collect relevant data from a random sample and summarize them
(using a sample and test statistic).
1.Analyze the Evidence
• Is it unusual under the assumption of the Null Hypothesis.
1.Draw Conclusions
• Based how likely our evidence is, decide whether we have enough
evidence to reject Ho (and accept Ha), and state our conclusions in
context.

5. Survey Activity
Visit www.CorpMath.com and take the survey there.

6. Survey Activity

7. Step1: Forming Hypothesis
Key concepts to forming a hypothesis
Null Hypothesis (H0): Sets the value of a equal to a claim.
Alternative Hypothesis (Ha):
Note that the null hypothesis always takes the symbolic form:
Note that the alternative hypothesis will take the form of one of the following:
population parameter
H0: parameter = some value
Ha: parameter < that value
Ha: parameter > that value
Ha: parameter ≠ that value
“The null states nothing has changed, no difference from status-quo, no relationship”
Opposes the Null Hypothesis suggesting a difference {<,>, or ≠}

8. Step1: Forming Hypothesis
Example: As an instructor at Santa Ana
College I have noticed that a majority of my
Math 219 students are female. Write a
hypothesis that I could test to see if this
claim is true.
H0:
Ha:
The proportion of females is not a majority.
The proportion of females is a majority.
p = 0.50
p > 0.50

9. Step 2: Choosing a sample
and collecting data
• Sample must be a random sample.
• Selections are independent (from a large enough population).
• We have enough expected success and failures:
n×p0 ≥10
n×(1− p0 ) ≥10

10. Step 2: Choosing a sample
and collecting data
• We found in a sample of 32 students 21 are female.
• Assume Math 219 at 2:45pm T/Th is a random sample of all
students taking Math 219.
• Population of Math 219 Students is > 320
• Do we have enough expected success and failures?
1- proportion Hypothesis Test
ˆp =
21
32
= 0.65625
Success (females) 32(0.5) =16 Failures (males) 32(0.5) =16

11. Step 2: Choosing a sample
and collecting data
• Test Statistic: Measure of how far a statistic is from the parameter.
z =
statistic- parameter
standard deviation of statisitic
z =
ˆp− p
p(1− p)
n
=
0.65625− 0.5
0.5(1− 0.5)
32
=1.7678
Our sample proportion is 1.7678 standard deviations above
the mean.

12. Step 3: Analyze the data
• We can use confidence intervals.
• We can use Critical value.
• P-value the probability of observing data like those observed
assuming that Ho is true.
P-value = normalcdf( 1.7678, 1000, 0, 1) = 0.039

13. Step 3: Analyzing Data

14. Step 3: Analyzing Data
What the P-value means:
Assuming that 50% of SAC Math 219
students are female, the probability
that we would get a sample that had
65.6% or more female is 3.9% .
P − value = P(z ≥1.7678| p = 0.5)

15. Step 4: Drawing Conclusion
Hypothesis testing step 4: Drawing conclusions.
If under the given assumptions, the probability of a particular
event is extremely small, we conclude the assumption is
probably .
The ___________________ is the cutoff for what we consider to
be extremely rare or extremely small.
If the P-value is < α
If the P-value is > α
Reject Ho
the significance level
false
Fail to reject Ho
P-value is statistically significant at α = ____ , reject H0. Strong evidence for Ha.
P-value is not statistically significant at α = ____ , fail to reject H0. Not enough
evidence for Ha.

16. Step 4: Drawing Conclusion
If I had asked you to use α = 0.05
•Our P-value of 0.039 is smaller than α = 0.05.
•A sample proportion of 65.6% or larger is
extremely unlikely to be collected under the
assumption the that 50% of SAC Math 219
Students are female.
•Our P-value is statistically significant. Reject
the Null Hypothesis. We have strong evidence
to support the claim that more than 50% of
Math 219 students are Female.

17. Step 4: Drawing Conclusion
If I had asked you to use α = 0.01
•Our P-value of 0.039 is larger than α = 0.01.
•A sample proportion of 65.6% or larger is not
large enough to be considered unusual.
•Our P-value is not statistically significant. Fail
to reject the Null Hypothesis. We do not have
enough evidence to support the claim that
more than 50% of Math 219 students are
Female.