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```AP Statistics
Chapter 11 Notes
Significance Test & Hypothesis

Significance test: a formal procedure for
comparing observed data with a hypothesis
whose truth we want to assess.

Hypothesis: a statement about a population
parameter.
Null (Ho) and Alternative (Ha)
Hypotheses

The null hypothesis is the statement being tested
in a significance test.
Usually a statement of “no effect”, “no difference”,
or no change from historical values.
 The significance test is designed to assess the
strength of evidence against the null hypothesis.


The alternative hypothesis is the claim about the
population that we are trying to find evidence
for.
Example: One-sided test

Administrators suspect that the weight of the
high school male students is increasing. They
take an SRS of male seniors and weigh them. A
large study conducted years ago found that the
average male senior weighed 163 lbs.
What are the null and alternative hypotheses?
 Ho: μ = 163 lbs.
 Ha: μ > 163 lbs.

Example: Two-sided test

How well do students like block scheduling? Students
block schedules and the block score was subtracted





What are the null and alternative hypotheses?
Ho: μ = 0
Ha: μ ≠ 0
*You must pick the type of test you want to do before
you look at the data.*
Be sure to define the parameter.
Conditions for Significance Tests


SRS
Normality (of the sampling distribution)

For means:
1. population is Normal or
 2. Central Limit Theorem (n > 30) or
 3. sample data is free from outliers or strong skew


For proportions:


np > 10, n(1 - p) > 10
Independence (N > 10n)
Test Statistic



Compares the parameter stated in Ho with the
estimate obtained from the sample.
Estimates that are far from the parameter give
evidence against Ho.
For now we’ll us the z-test.
P-Value

Assuming that H0 is true, the probablility that
the observed outcome (or a more extreme
outcome) would occur is called the p-value of the
test.


Small p-value = strong evidence against H0.
How small does the p-value need to be?

We compare it with a significance level (α – level)
chosen beforehand.

Most commonly α = .05
P-value continued



If the p-value is as small or smaller than α, then the data
are “statistically significant at level α”.
Ex: α = .05
If the p-value is < .05, then there is less than a 5% chance
of obtaining this particular sample estimate if H0 is true.


If the p-value is > .05, our result is not that unlikely to
occur.


Therefore we reject the null hypothesis.
Therefore we fail to reject the null hypothesis.
If done by hand, the p-value must be doubled when performing
a 2-sided test. The calculator will already display this doubled pvalue if you choose the 2-sided option.
Confidence vs. Significance

Performing a level α 2-sided significance test is
the same as performing a 1 – α confidence
interval and seeing if μ0 falls outside of the
interval.

e.g. If a 99% CI estimated a mean to be (4.27,
5.12), then a significance test testing the null
hypothesis H0: µ = 4 would be significant at α =
.01.

1. Don’t place too much importance on
“statistically significant”.




Smaller p-value = stronger evidence against H0
2.Statistical significance is not the same as
practical importance.
3. Don’t automatically use a test…examine the
data and check the conditions.
4. Statistical inference is not valid for badlyproduced data.
Mistakes in significance testing

Type I error:


Reject H0 when H0 is actually true.
Type II Error:

Fail to reject H0 when H0 is actually false.
Errors Continued
Errors continued



The significance level α is the probability of
making a Type I error.
Power: The probability that a fixed level α
significance test will reject H0 when a particular
alternative value of the parameter is true.
Ways to increase the power.
Increase α
 Decrease σ
 Increase n

```
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