Psyc 235: Introduction to Statistics

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Psyc 235Introduction to Statistics
http//www.psych.uiuc.edu/jrfinley/p235/

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Announcements (1of2)

• Early Informal Feedback
• https//webtools.uiuc.edu/formBuilder/Secure?id97
  48379
• Open until Sat March 15th
• Special Lecture Thurs March 13th Conditional
  Probability (incl. Law of Total Prob., Bayes
  Theorem)
• Mandatory for invited students
• Anyone can come
• No OH Go to lab for Qs/help.

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Announcements (2of2)

• Target Dates STAY ON TARGET!
• You should be finishing the Distributions slice
• VoD 5. Normal Calculations, 17. Binomial
  Distributions, and 18. The Sample Mean and
  Control Charts,
• Quiz 3 Thurs-Fri March 13th-14th

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Population
?
Standard Error
SamplingDistribution
(of the mean)
Sample
size n
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Shape of the Sampling Distribution?

• If population distribution is normal
• Sampling distribution is normal (for any n)
• If sample size (n) is large
• Sampling distribution approaches normal
• Central Limit Theorem
• As sample size (n) increases
• Sampling distribution becomes more normal
• Variance (and thus std. dev.) decreases

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Great, Normal Distributions!

• Can now calculate probabilities like
• Just convert values of interst to z scores
  (standard normal distribution)
• And then look up probabilities for that z score
  in ALEKS (calculator)
• Or vice versa

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So far

• Weve been doing things like
• Given a certain population, whats prob of
  getting a sample statistic above/below a certain
  value?
• Population---gtSample
• How can we shift to
• Using our Sample to reason about the POPULATION?
• Sample---gtPopulation

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INFERENTIAL STATISTICS!

• Estimating a population parameter (e.g., the mean
  of the pop. ? )
• How to do it
• Take a random sample from the pop.
• Calculate sample statistic (e.g., the mean of the
  sample )
• Thats your estimate.
• Class dismissed.

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No, wait!

• The sample statistic
• is a point estimate of
• the population parameter ?
• It could be off, by a little, or by a lot!

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Population
?
SamplingDistribution
We only have one sample statistic.
(of the mean)
And we dont know where in here it falls.
Sample
size n
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Interval Estimate

• Point estimate (sample statistic) gives us no
  idea of how close we might be to the true
  population parameter.
• We want to be able to specify some interval
  around our point estimate that will have a high
  prob. of containing the true pop parameter.

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Confidence Interval

• An interval around the sample statistic that
  would capture the true population parameter a
  certain percent of the time (e.g., 95) in the
  long run.
• (i.e., over all samples of the same size, from
  the same population)

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?
This is the meanfrom one sample.
Lets put a 90 Confidence Intervalaround it.
Lets consider other possible samples(of the
SAME SIZE)
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?
The meanfrom another possible sample.
This one capturesthe true mean too.
So does this one.
And this one.
This one too.
Yep.
This interval missesthe true mean!
But this ones alright.

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?
A 90 Confidence Interval means that for 90 of
all possible samples(of the same size),that
interval around the sample statistic will capture
the true population parameter(e.g., mean).
Only sample statistics in the outer 10 of the
sampling distribution have confidence intervals
that miss the true population parameter.

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But, remember
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But, remember
All that we have is our sample.
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Still, a Confidence Interval is more usefulin
estimating the population parameterthan is a
mere point estimate alone.
So, how do we make em?
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CONFIDENCE INTERVAL (1 - ?) confidence interval
for a population parameter
P( C. I. encloses true population parameter )
1 - ?
Note ? P(Confidence Interval misses true
population parameter )
Proportion of times such a CI misses the
population parameter
Point estimate


standard deviation ofsampling distribution
sample statistic
or
ex
(aka Standard Error)
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Decision Tree for Confidence Intervals
CriticalScore
n large? (CLT)
Population Standard Deviation known?
Pop. Distributionnormal?
z-score
Yes
Standard normaldistribution
z-score
Yes
Yes
No
No
Cant do it
Yes
t-score
No
t distribution
No
t-score
Yes
Note ALEKS
No
Cant do it
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C.I. using Standard Normal Distribution
?
When ? known.
For the Population Mean
First, choose an ? level.
For ex., a.05 gives us a 95 confidence
interval.
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C.I. using Standard Normal Distribution
?
When ? known.
For the Population Mean
First, choose an ? level.
For ex., a.05 gives us a 95 confidence
interval.


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C.I. using Standard Normal Distribution
?
When ? known.
For the Population Mean
First, choose an ? level.
For ex., a.05 gives us a 95 confidence
interval.


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C.I. using Standard Normal Distribution
?
When ? known.
For the Population Mean
First, choose an ? level.
For ex., a.05 gives us a 95 confidence
interval.


Lookup value (ALEKS calculator, Z tables)
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Handy Zs
(Thanks, Standard Normal Distribution!)
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C.I. using Standard Normal Distribution
?
When ? known.
For the Population Mean


Remember random variable
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C.I. using t Distribution
?
When ? unknown!
For the Population Mean


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C.I. using t Distribution
?
When ? unknown!
For the Population Mean


We use the standard deviation from our sample
(s)to estimate the population std. dev. (?).
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C.I. using t Distribution
?
When ? unknown!
For the Population Mean


Critical value taken from a t distribution, not
standard normal.
The goodness of our estimate of ? will depend on
our sample size (n). So the exact shape of any
given t distribution depends on degrees of
freedom (which is derived from sample size n-1,
here). Fortunately, we can still just LOOK UP the
critical values(just need to additionally plug
in degrees freedom)
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Behavior of C.I.

• As Confidence (1-?) goes UP
• Intervals get WIDER
• (ex 90 vs 99)
• As Population Std. Dev. (?) goes UP
• Intervals get WIDER
• As Sample Size (n) goes UP
• Intervals get NARROWER

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C. I. for Differences(e.g., of Population Means)

• Same approach.
• Key is
• Treat the DIFFERENCE between sample means as a
  single random variable, with its own sampling
  distribution everything.
• The difference between population means is a
  constant (unknown to us).

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Remember

• Early Informal Feedback
• Special Lecture Thursday
• No OH Go to lab for Qs/help.
• Stay on target
• Finish Distributions
• VoDs
• Quiz 3

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