Santa Ana College Presentation Hypothesis

Testingby Richard Corp

Where we have been:

• Sampling techniques• Sample Statistics• Population parameters• Sample distributions• Calculating probabilities• Calculating and interpreting z-

scores.

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.

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.

Survey ActivityVisit www.CorpMath.com and take the survey there.

Survey Activity

Step1: Forming HypothesisKey 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 ≠}

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

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

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

p̂ = 21

32= 0.65625

Success (females) 32(0.5) = 16 Failures (males) 32(0.5) = 16

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 = p̂− pp(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.

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

Step 3: Analyzing Data

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)

Step 4: Drawing ConclusionHypothesis 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.

Step 4: Drawing ConclusionIf 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.

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.