Psyc 235:Introduction to Statistics

DON’T FORGET TO SIGN IN FOR CREDIT!

http://www.psych.uiuc.edu/~jrfinley/p235/

Announcements (1of2)

• Early Informal Feedback https://webtools.uiuc.edu/formBuilder/Secure?id=9748379

Open until Sat March 15th• Special Lecture Thurs March 13th:

Conditional Probability (incl. Law of Total Prob., Bayes’ Theorem)Mandatory for invited studentsAnyone can comeNo OH; Go to lab for Qs/help.

Announcements (2of2)

• Target Dates: STAY ON TARGET! You should be finishing the Distributions

sliceVoD “5. Normal Calculations, 17.

Binomial Distributions,” and “18. The Sample Mean and Control Charts,”

• Quiz 3: Thurs-Fri March 13th-14th

€

X

Population

Sample

SamplingDistribution

€

X

size = n

€

X

=σ

n

“Standard Error”

sample statistic (a random variable!)

(of the mean)

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 normalVariance (and thus std. dev.) decreases

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… €

z =x − μ

σ

So far…

• We’ve been doing things like:Given a certain population, what’s prob

of getting a sample statistic above/below a certain value?

Population--->Sample

• How can we shift to …Using our Sample to reason about the

POPULATION? Sample--->Population

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: ) That’s your estimate.

• Class dismissed.€

X

No, wait!

• The sample statisticis a point estimate of

the population parameter • It could be off, by a little, or by a lot!€

X

€

X

Population

Sample

SamplingDistribution

€

X

size = n

(of the mean) We only have one sample statistic.

And we don’t know where in here it falls.

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.

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)

€

X

This is the meanfrom one sample.

Let’s put a 90% Confidence Intervalaround it.

Note: True PopulationParameter is constant!Note: True PopulationParameter is constant!

Note that this particularinterval capturesthe true mean!

Note that this particularinterval capturesthe true mean!

Let’s consider other possible samples(of the SAME SIZE)

€

X

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 one’s alright.…

€

X

…

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.

€

X

…

But, remember…

But, remember…

€

X Sample

size = n

All that we have is our sample.

€

X Sample

size = n

Still, a Confidence Interval is more usefulin estimating the population parameter

than is a mere point estimate alone.

So, how do we make ‘em?

CONFIDENCE INTERVALCONFIDENCE INTERVAL(1 - )% confidence interval for a population parameter

Note: = P(Confidence Interval misses true population parameter )

Pointestimate ±

criticalvalue

Std. dev. ofpoint estimate ·

P( C. I. encloses true population parameter ) = 1 -

“Proportion of times such a CI misses the population parameter”

sample statistic

Margin of Error

ex:

€

X (aka “Standard Error”)

standard deviation ofsampling distribution

€

zα / 2

€

tα / 2

or

Decision Tree for Confidence Intervals

PopulationStandard Deviationknown?

Yes

No

Pop. Distributionnormal?

n large?(CLT)

Yes

No Yes

No

Yes

No YesNo

z-score

z-score

Can’t do it

Can’t do it

t-score

t-score

CriticalScore

Note: ALEKS…

Standard normaldistribution

t distribution

C.I. using Standard Normal Distribution

For the Population Mean First, choose an level.

For ex., α=.05 gives us a 95% confidence interval.

Pointestimate ±

criticalvalue

Std. dev. ofpoint estimate ·

Margin of Error

When known.

C.I. using Standard Normal Distribution

For the Population Mean First, choose an level.

For ex., α=.05 gives us a 95% confidence interval.

±criticalvalue

Std. dev. ofpoint estimate ·

Margin of Error

€

X

When known.

C.I. using Standard Normal Distribution

For the Population Mean First, choose an level.

For ex., α=.05 gives us a 95% confidence interval.

±criticalvalue ·

Margin of Error

€

X

€

n

When known.

C.I. using Standard Normal Distribution

For the Population Mean First, choose an level.

For ex., α=.05 gives us a 95% confidence interval.

± ·

Margin of Error

€

X

€

n

When known.

€

zα / 2

Lookup value(ALEKS calculator,Z tables)

.10 if =

.05 if =

.01 if =

Confidence %90

Confidence %95

Confidence %99

€

zα / 2 =1.645

€

zα / 2 =1.960

€

zα / 2 = 2.576

valuecritical

.05upper

valuecritical

.025upper

valuecritical

.005upper

Handy Zs

(Thanks, Standard Normal Distribution!)

C.I. using Standard Normal Distribution

For the Population Mean

± ·

Margin of Error

€

X

€

n

When known.

€

zα / 2

€

X ± zα / 2( )σ

n

⎛

⎝ ⎜

⎞

⎠ ⎟ is a 1−α confidence interval of μ

€

Furthermore, in that case,

P X − zα / 2( )σ

n

⎛

⎝ ⎜

⎞

⎠ ⎟≤ μ ≤ X + zα / 2( )

σ

n

⎛

⎝ ⎜

⎞

⎠ ⎟

⎡

⎣ ⎢

⎤

⎦ ⎥=1−α

Remember:random variable

C.I. using t Distribution

For the Population Mean

± ·

Margin of Error

€

X

When unknown!

C.I. using t Distribution

For the Population Mean

± ·

Margin of Error

€

X

€

s

n

When unknown!

We use the standard deviation from our sample (s)to estimate the population std. dev. ().

€

s =x i − x ( )

2∑n −1

The “n-1” is an adjustment tomake s an unbiased estimatorof the population std. dev.

C.I. using t Distribution

For the Population Mean

± ·

Margin of Error

€

X

€

s

n

When unknown!

€

tα / 2

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)

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

€

n

Std dev of samplingdistribution of the mean

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).

€

X 1 − X 2( )

Remember

• Early Informal Feedback• Special Lecture Thursday

No OH; Go to lab for Qs/help.

• Stay on target Finish DistributionsVoDs

• Quiz 3

€

X