Ch 10 – Intro To Inference
10.1: Estimating with Confidence
10.2 Tests of Significance
10.3 Making Sense of Statistical Significance
10.4 Inference as a Decision
Definitions

Statistical Inference
– Provides methods for drawing conclusions
about a population from sample data
– What happens if I do this many times?

Formal Inference
– Using probability to express the strength of
our conclusions (probability takes chance
variation into account)

Margin of Error
– How accurate we
believe our guess
is, based on the
variability of the
estimate
– What you always
see in the fine print
Confidence Intervals

Use facts about sampling distributions (what would
happen in the long run) to express our confidence
in the results of any one sample

“We got these numbers by a method that gives
correct results 95% of the time.”

In the form of estimate ± margin of error

Given at confidence level C which gives probability
the interval will capture true parameter value in
repeated samples

Fathom Demo #29:
Capturing with
Confidence Intervals
– Black is a hit, red is a
miss
– What happens to CIs
when….?
p542 #10.1 to 10.3
 What CIs Say WS

Confidence Intervals Toolbox
I Create Fabulous Confidence Intervals

Identify
– Population
– Parameter
– Procedure
Conditions
 Formula
 Calculations
 Interpret in Context

Many CIs
Collect More Measures
Confidence Intervals for μ

Conditions
– SRS
– sampling distribution of
 Given
 CLT
 NPP


is approximately normal
± z*
“We are __% confident that the true mean ___
is between __ and __. By __% confident, we
mean that we arrived at this conclusion by a
method that gives correct results __% of the
time.”
z* (Critical Values)
Same as z-scores in Table A (standard
normal curve)
 Most common are 1.645, 1.96 and 2.576
(90, 95 and 99%)
 90% is really 95% to left when you use
the table (95 is 97.5, 99 is 99.5)
 Sketches
 p548 #10.5-10.6

Ways to Decrease Margin of Error

Make z* smaller (decrease confidence level C)

Increase sample size n

Decrease standard deviation σ
– High Confidence – our method almost always gives
correct answers
– Small Margin of Error – we have pinned down the
parameter quite precisely
Choosing Sample Size

A wise user of statistics never plans data
collection without planning the inference
at the same time.

Chapter 9 - the size of the sample
determines the margin of error, not the
size of the population (soup).

p551 #10.10-10.11
Cautions
Data can’t be from anything more
complicated than an SRS
 Beware of outliers and skewness
 Using σ is unrealistic, but we’re using it
now to understand the process – this
entire chapter is about the process!
 Read last paragraph on p554 about what
statistical confidence does not say.

YMS - 10.2
Hypothesis Testing
Tests of Significance – A Few Ways

Used to assess evidence about a claim while CIs
were estimating a parameter

An outcome that would rarely happen if a claim
were true is good evidence the claim is not true

“Does our sample result reflect a true change or did
our result occur just by chance; How unlikely is our
outcome if the null hypothesis were really true?”

Uses knowledge of how the sample mean
vary in repeated samples
would
Hypotheses

Null Hypothesis (Ho)
– Statement saying there is no effect or change in the
population
– If true, the sample result is just chance at work

Alternative Hypothesis (Ha)
– Alternative we believe to be true
– It is cheating to first look at data and then frame Ha

One-sided vs. Two-sided tests
– <, > or ≠
– Should know before you collect sample
– Choose two-sided to be safe
P-Value
Probability of a result
at least as far out as
the result we actually
got
 Evidence against Ho
 Lower p-value =
stronger evidence
 Probability from Ch 2!

Calculating Two-Sided P-Values

Calculate the same way as one-sided and then
double

Alternative hypothesis stated some difference,
not in any particular direction

Must consider both differences – greater than
and less than (even though your sample only
produces one or the other)

Diagram on p573
Statistically Significant
Chance alone would rarely produce so
extreme a result
 Significance level alpha α
 Reject null hypothesis when p < α

Stats in Dating
ha ha ha…
p564 #10.27 to 10.35 odds
Significance Tests Toolbox
1. Identify population and parameter AND state the null
and alternative hypotheses in words and symbols.
2. Choose and verify the procedure (conditions are still SRS
and normal).
3. Carry out the inference procedure.
- Calculate the test statistic (one-sample z-statistic).
- Find the p-value.
4. Interpret your results in the context of the problem.
- Reject or do not reject the null hypothesis.
- Include p-value and statement assessing strength.
p576 #10.38-10.39
Rejecting is not Accepting
Just because you can’t prove that something is
false, doesn’t mean that you believe it to be true
 Not rejecting the null hypothesis and accepting
the null hypothesis are not the same conclusion
 Examples

– Shakespeare Video
– OJ
 Ho: Person did not commit crime.
 Ha: Person did commit crime.
 If there is enough evidence, we find the person guilty. If
there is not, we proclaim they are not guilty. We aren’t
saying the person is innocent, just that we didn’t have
enough evidence to find them guilty.
Fixed Significance Level
Z Tests for μ

Use the z-score associated with chosen
significance level to make the decision
– You don’t need to find the p-value to make
your decision.
– More standard deviations from the mean
yields a smaller and smaller p-value/tail area

Example for One-Sided Tests with Ha >
– If z > 1.645 you can reject at 0.05
– If z > 1.96 you can reject at 0.025
– If z > 2.576 you can reject at 0.005
CIs and Two-Sided Tests

Reject if the value of μo falls outside a
level 1 – α confidence interval for μ
– You’re 99% confident the true mean is
captured in a particular interval, but the
interval doesn’t contain μo

Why use a CI over a test?
– CIs give an estimate of the parameter while
tests just reject/accept values
p580 #10.42-10.43
Test Review p583 #10.46 to 10.54 evens
YMS – 10.3
Making Sense of Statistical Significance
Choosing a Level of Significance
What are the ramifications for rejecting Ho?
Practical vs. Statistical Significance
Who cares if your scab falls off half of a
day sooner?
 Pay attention to actual data as well as
p-value.

Inference is Not Valid for All Sets of Data
Inference cannot correct a poorly designed
experiment or survey.
Beware of Multiple Analyses
Every once in a while the result
will show by chance.
p589 #10.58-10.59, 10.62 and 10.64
YMS – 10.4
Inference as Decision
Acceptance Sampling

When circumstances
call for a decision or
action at the end
result of inference

When we must accept
instead of just not
rejecting
Type I Errors
If we reject Ho (accept Ha) when in fact Ho
is true
 Probability of Type I error is equal to α

Type II Errors
If we accept Ho (reject Ha) when in fact Ha
is true
 Calculated based on the alternative for μ

Truth about the
Population
Ho True
Decision
based on
Sample
Type I Error
Correct
Decision
Correct
Decision
Type II Error
Reject Ho
Accept Ho
Ha True
p 595
p597 Examples 10.21 - 10.22
p598 #10.67
Type I & II Errors for HW
Power (1-β)
Choose your interpretation:
- Probability the test will reject Ho when an
alternative is true
- Probability of rejecting Ho when it is in fact false
- Probability of making a correct decision (to reject
Ho) when Ho is false
- Probability the test will pick up on an effect that is
present
- Probability the test will detect a deviation from
the null hypothesis should such a deviation exist
Ways to Increase Power

Increase α (less evidence is required to reject)

Consider an alternative that is farther away from μo
Increase the sample size n (less overlap
because spread decreases)
 Decrease σ (less overlap because spread
decreases)

Example 10.23
p603 #10.72-10.75