APSTAT UNIT 4A

INFERENCE PART 1

APSTAT Chapter 18

Sampling Distributionsand Sample Means

Lets Just DO IT!!!!

1 2 3 4 5 6 7 8 9 10 11 12 13

.60 .65 .55 .70 .85 .80 .75 .70 .65 .75 .80 .85 .75

Proportion of correct answers on last AP Stat Exam

Regular Old Distribution:

.55-.59 .60-.64 .65-.69 .70-.74 .75-.79 .80-.84 .85-.89

Lets Just DO IT!!!!

1 2 3 4 5 6 7 8 9 10 11 12 13

.60 .65 .55 .70 .85 .80 .75 .70 .65 .75 .80 .85 .75

Proportion of correct answers on last AP Stat Exam

• Now, Everyone take a 5-person random sample

• Do randint(1,13,5) to choose your subjects

• Add their scores and divide by 5 to get x-bar (sample mean)

• Now we will do a distribution of our sample means – a SAMPLING DISTRIBUTION!!!!!

Lets Just DO IT!!!!

1 2 3 4 5 6 7 8 9 10 11 12 13

.60 .65 .55 .70 .85 .80 .75 .70 .65 .75 .80 .85 .75

Proportion of correct answers on last AP Stat Exam

Class Sample Means:

.55-.59 .60-.64 .65-.69 .70-.74 .75-.79 .80-.84 .85-.89

Sampling Distribution

We Just DID IT!!!!

Give me at least 2 things that are different between the regular distribution and the sampling distributions:

1.

2.

3.

4.

5.

Bias

Unbiased StatisticMean of Sampling distribution should

equal True population mean. How did ours look earlier? The true

mean of the population was about 72.3….

Sample Proportions

Mean of Sample Proportion In last section:

Proportion is just outcome divided by n, so….

ˆ

ˆ

p

p

np

np pnp

μ

μ

μ

=

= =

=

Standard Deviation of Sample Proportion In last section:

Proportion is just outcome divided by n, so throw down a little Algebra….

ˆ 22

ˆ

(1 )

(1 ) (1 ) (1 )

(1 )

X

p

p

np p

np p np p np p

n nn

p p

n

σ

σ

σ

= −

− − −= = =

−=

Now try it…Sample Proportions Find mean and standard deviation for:

60 samples of 10 coin flips, p=.5

60 samples of 50 coin flips, p=.5

What does this say about variability in regards to sample size?

2 Rules of Thumb - Assumptions/Conditions Population size large enough

Population should be at least 10 times the sample size

10% Condition Normalness

n should be large enough to produce an approximately normal sampling distribution.

np > 10 AND n(1-p) >10

Try them out

A San Jose firm decide to sample 25 residents to determine if they oppose off-shore oil drilling. They predict that P(oppose) = 0.4Large enough population?

Normalness?

Example….

If the true percentage of students who pass the APStat exam is .64, what is the probability that a random sample of 100 students will have at least 70 students pass?

=.64, n=100, 70 or more

Check Conditions - Briefly Explain10%np and n(1-p) > 10

Draw Picture- (Find SD too)

Find P-Value

Conclusion

Same Problem Data…

Within what range would we expect to find 95% of sample proportions of size 100.

Sample Means

Take a whole bunch of samples and find the means

Why sample means? Remember our sample of class

scores?Less variableMore normal

Mean and Standard Deviation of X-BAR If we take all the possible samples from a

population, the mean of the sampling distribution will equal the population mean (if the population mean was accurate in the first place, but more on that later)

Xx μ=

Mean and Standard Deviation of X-BAR Standard Deviation of a sampling

distribution is:

22

22

s n

s n

sn

σ

σ

σ

=

=

=

Let’s try it!

If adult males have height N(68,2) what would be the mean and standard deviation for the distribution if:n=10n=40

What happened to the Standard deviation when n was quadrupled?

What would happen to the standard deviation if n was multiplied by 9?

CLT – The Central Limit Theorem If the population we are sampling from is

already normal with N(μσ), the sampling distribution will be normal as well with mean μ and standard deviation

But what if the population we are sampling from is not normal?

nσ

Age of Pennies

Riebhoff has 50 pennies, he took the current year and subtracted it from the date on the penny to obtain the following data…

Penny Ages

1 0 11 11 21 2 31 2 41 3

2 1 12 1 22 8 32 1 42 28

3 2 13 12 23 9 33 7 43 7

4 4 14 2 24 5 34 3 44 4

5 14 15 5 25 15 35 17 45 7

6 3 16 5 26 6 36 1 46 4

7 3 17 6 27 0 37 19 47 0

8 7 18 0 28 4 38 4 48 0

9 8 19 1 29 1 39 23 49 10

10 0 20 0 30 1 40 2 50 5

Sample Size n=1

0

1

2

3

4

5

6

7

0 3 6 9 12 15 18

# 0f Pennies

Sample Size n=4(everyone do 3 SRS)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

Sample Size n=8(everyone do 3 SRS)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

What happened?

The distribution got “normaler” as the sample size increased. Cool?

Central Limit Theorem says that even if a distribution is not normal, the distribution of the sampling distribution will approach normalcy when n is large.

Allows us to use z-scores and such, even when the larger population is not normally distributed.

Assumptions/Conditions

Random Sample - Always describe Independence - Describe 10% Condition

Try it

If the APSTAT EXAM 2005 had a mean score of 3.2 with a standard deviation of 1.2…Old Skool - Find the probability that a

single student would have a score of 4 or higher?

New Skool – find the probability that an SRS of 20 students would have a score of 4 or higher?

Old Skool - Find the probability that a single student would have a score of 4 or higher?

New Skool – find the probability that an SRS of 20 students would have a score of 4 or higher?

Check Conditions - Briefly Explain10%Independence and Random Sample

Draw Picture- (Find SD too)

Find P-Value

Conclusion

Standard Error

Sometimes we do not have the population standard deviation. If we have to estimate it, we call it Standard Error and roll an SE.

APSTAT Chapter 19

Confidence Intervals for Proportions

Confidence Interval for a Proportion (aka One-Proportion Z-Interval)

At Woodside High, 80 students are surveyed and 32% of them had tried marijuana.

How confident am I that the true proportion of WH students that have tried marijuana is at or near 32%?

CONFIDENCE INTERVAL!!!

The Dealio…

If I do know the population mean• If I sample, I know the sample mean might be quite

different than the population mean

BUT…That difference is predictable. For instance, if N(0.70,0.1) and n=4

Sample Mean = 0.70, Sample SD=0.1/sqrt4=.05 We expect 95% of samples (Empirical Rule) to

fall between 2 SD of the mean Therefore 95% of samples will fall between 0.6

and 0.8.

Confidence Intervals

Work in reverse (From Woodside High Example) I

sampled 80 and got sample p = .32 I want to know the true population

proportion. The true population proportion will lie

within 2SD of the Sample Proportion in 95/100 samples of this size.

Let’s Do It!!!!

Do It!

List what you knowp-hat=.32, n=80

Conditions/Assumptions10% for Independence

• Woodside HS has over 800 studentsnp and nq > 10 to use Normal Model

• Both .32 x 80 and .68 x 80 > 10 Find Standard Error

SE(p-hat)=

Do It!

Draw the Picture

Conclusion:We are 95% confident that the TRUE

mean proportion of ____________ falls between ____ and ____

Do It!

We can also write confidence intervals in the form:

(estimate) ± (margin of error)

ˆ ˆ(1 )ˆ *

p pp z

n

−±

Standard error

What Does 95% Confidence Mean?

If we did a whole bunch of confidence intervals at this sample size, we would expect 95 out of 100 intervals to contain the true mean.

Picture of this:TRUE POPULATION

PROPORTION

AHOY!

We do not always want 95% confidence. Example, if a part on an airplane’s landing

gear needs to be a certain size to work, wouldn’t you want a little more confidence in the sample being within certain parameters?

Common Intervals are 90, 95 and 99% Denote as C=.90, C=.95, or C=.99

But 90 and 99% aren’t Empirically Cool

We need this z-score! It’s critical! So critical, it is called the critical value and denoted as z*

Area = 90%

p

Mas z*

Now check t distribution critical values chart (back of book or formula sheet)

Look at bottom. It gives you C and right above it is…..

Yeah!

Try it!

A poll asked who would you vote for if an election were held today between Sen. Barack Obama and Sen. John McCain. 115 of the 250 respondents chose Sen. McCain. Construct and interpret a 90% confidence interval for the proportion of voters choosing McCain.

Try It!

Conditions:

Mean, SE, z*

Calculate CI

Conclusion

Last thing

Finding sample size needed for a CI with a given level of confidence and a given margin of error

NBC News is doing a poll on who will be the next Governor of California. The want a 3% margin of error at a 95% confidence interval. What sample size should they use?

Sample size needed

ˆ ˆ(1 )ˆ *

p pp z

n

−±

Margin of Error

Sample size needed

ˆ ˆ(1 )* .03

0.5(1 0.5)1.960 .03

.251.960 .03

p pz

n

n

n

−=

−=

=( )2

.251.960 .03

1.960 0.5

.03

32.67

1067

n

n

n

n

=

=

=

=

g

Why 0.5? Gives us largest n value. Safety First!

OOPS! YOU SHOULD ALWAYS ROUND UP TO STAY WITHIN CONFIDENCE INTERVAL! SHOULD BE 1068.

APSTAT Chapter 20

One ProportionHypothesis Tests

Significance Tests

Example. AP Stat Exam 2005: National Proportion Who Passed = .58 Priory Students n = 32, p-hat=.78

Two Possibilities Higher WPS proportion just happened by

chance (natural variation of a sample) The likelihood of 78% of 32 students passing

is so remote we must conclude that Priory Students are likely better at APStat than national average.

Hypothesis Testing

Reflect our two possibilities from above: NOTHING IS STRANGE (difference could

have been by natural variation of sample) SOMETHIN’ IS GOIN’ ON (difference is so

improbable we must assume there is a difference)

Here is how we write them: H0: Null Hypothesis (Nothing Strange) Ha: Alternative Hypothesis (Somethin’ is

goin’ on)

In our WPS SAT Example

In practice, we describe the hypotheses in both symbols and words H0: p = .58, Priory students perform at the

same level as the National Average Ha: p > .58, Priory students perform better

than the National Average We will perform test(s) that give evidence

against the H0 (kinda like a trial)

What to do with the Hypothesis… After we conduct a test we will have

evidence based on our understanding of probability and sample variation. With this info we can: Reject H0 in favor of Ha

• if there is SIGNIFICANT evidence that the result did not likely happen by chance variation.

Fail to Reject H0

• if there is not enough evidence to reject it. The variation could likely have happened by chance

Be Carefull…

Notice we NEVER, NEVER, NEVERAccept either HypothesisSay one or the other is true or false

We only have evidence, we could still be wrong….

BUT….the stronger the evidence the more confident we can be!

Where do we get evidence?

One way, P-value from a z-score. What is the probability that this event happened given the population mean, standard deviation and # in our trial?

Our old friend, the z-score: We are using a sample here, so we throw in our sample standard deviation.

ˆ

(1 )

p pz

p pn

−=

−

Let’s Do It! WPS SAT Example Step 1 Define Parameter:

p the true passing proportion of WPS APstat test-takers

Step 2 Hypotheses H0: p= .58, Priory students perform at the

same level as the national proportion Ha: p> .78, Priory students perform better

than the national proportion

WPS SAT Example Continued Step 3 Assumptions:

SRS• No, but we will assume WPS Students

are a representative sample of the population of all AP Stat test-takers.

Independence• Priory sample of 32 is less than 10%

of population of AP Stat test-takers

• .58(32) and .42(32) both > 10

WPS SAT Example Continued Step 4 Name Test and DO IT

One Sample Z-Test for a Proportion

ˆ

(1 )

p pz

p pn

−=

−

WPS SAT Example Continued Step 5 P-value and sketch of normal curve:

P(z> 2.31)= .01053

Step 6 Interpret P-value and Conclusion A P-Value of .01053 indicates that there is about a

1 in 100 chance that a result this distant from the p happened merely by chance. Therefore, reject H0 in favor of Ha. It is very likely that WPS students performed far better on average than the National Average on the 2005 APStat exam

.58 .78

PHAT-PI (MUCH LOVE TO AL YOUNG)

P - Parameters (What are we studying) H - Hypothesis (In words and symbols) A - Assumptions (depends on type of test) T - Test (Name it. Do it.) P - P-Value (Calculate it-Draw it) I - Interpret (Reject/Fail to Reject, Why, ATQ)

Alternative Hypotheses

Can be: Greater Than (Ha ------>blah)

Less Than (Ha --------<blah)

Not (Ha --------≠blah)Double your one-sided P-value

μ0

μ0

μ0

On TI-83

Still have to do all of Phat Pi, but helps with calculations.

Stat>Test>1-PropZTestp0 - Population proportionx - successes in sample n(p-hat)n - sample size

Do it for AP Stat Example

Defective Products

A company claims that just 3% of its products are defective. A simple random sample of 400 of their products yielded 14 defective items. Do these sample data suggest that the company’s claim is too low?

PHAT-PI

P

H

A

PHAT-PI

T

P

I

How Much Evidence?

GTang (and many texts) give a rule of thumb of 5%. If there is a 5% probability or less that the outcome would happen by chance, you can throw down the “enough evidence to reject H0…”

If it is 1% or less, you can throw down the “very strong evidence against H0. Reject H0 in favor…”

Significance level…

Sometimes a problem will specify a certain amount of evidence that is needed.

= Significance Level Usually = 0.05 or 0.01 Basically, your P-value must be below that

level to reject the null hypothesis. Example your p-value is .03 and = 0.05

Be careful with one and two-sided alternatives and significance levels Your p-value doubles in a 2-sided.

APSTAT Chapter 21

More Stuff About Hypothesis Tests

Great Chapter

Make sure you read it Important concepts:

What a Null Hypothesis is and isn’tWhat P-Value MeansSignificance () LevelCritical Value - One v. Two sidedConfidence Intervals and Tests of

Significance - Relationship Between

Great Chapter…But….

Goes further than you need in explaining:Types of Error

• Type I• Type II

Power

Errors - Can We Make Mistakes?

Sure, Rejecting a “Good” Shipment For Example, I need batteries that work 99% of

the time. My significance test of a sample from a battery shipment tells me to “reject” the shipment, but it is actually ok.

We Can also Fail to Reject a “Bad” Statement If I had accepted a shipment that was actually bad

because my sample proportion ended up close to the mean I was looking for.

Which of these is worse in real life?

Errors

Type I – Reject H0 when it is actually true Usually not so bad Rejecting a “good” shipment Probability is equal to

Type II – Failing to Reject H0 when it is actually false Usually bad Accepting a “bad” shipment Probability () is a bear to calculate

• Check book to see how! Ooooo, fun!• Be happy you will NEVER be asked to do it

Errors - #2

Decrease both Type I and II errors by:Increasing n

Decrease Type II Errors by:Increasing

• You end up rejecting more/failing to reject less

• Causes an increase in Type I errors

POWER

Basically, how sure we are that we will not get a Type II error

Power = 1 – P(Type II) OR Power = 1 - P() Never will you be asked to compute (unless

the probability of a type II error is given) Increase Power by:

Increasing n (Sample size) Increase (say from .01 to .05)

Power and Error Wrap

What you have to know:Explain Power, Type I, and Type II

errors in context of the problem.Calculate P(Type I error) given How to Decrease:

• Type I Error• Type II Error

How to increase Power

APSTAT Chapter 22

Two Proportion Hypothesis Tests

Let’s Hop Right In…

A recent report found that men wash their hands 75% of the time after using the restroom and women 85% of the time. If SRS’s of 1200 men and 1100 women were surveyed, can we statistically say there is a significant difference between hand washing habits of men and women?

Handwashing

Parameter (group 1=female, 2=male)1-p2: Difference between female and male

hand washing proportions Hypotheses

H0: p1-p2=0 No difference in hand washing

Ha: p1-p2≠0 Is a “ “ “ “

Handwashing

AssumptionsSRS’s YepIndependent samples Safe to Assumen1p1>5 and n1(1-p1)>5

n2p2>5 and n2(1-p2)>5 YepPopulation 10X Sample

Yep

Handwashing

Test – Two Sample Proportion Z-Test

1 2

1 2

ˆ ˆ( ) .85 .755.965

1 11 1 .7978(1 .7978)ˆ ˆ(1 )1100 1200

p pz

p pn n

− −= = =

⎛ ⎞ ⎛ ⎞− +− + ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠

POOLED 1 2

1 2

935 900ˆ .7978

1100 1200

x xp

n n

+ += = =

+ +

Pool if variances are equal (since our null theorizes that the populations – and thus the variances - are equal)

Handwashing

P-Value2*P(Z>5.965)=Really Really Really Small

InterpretationP Value is so small, there is VERY

significant evidence against the assumption that males and females wash hands at the same proportion. Reject Null Hypothesis in favor of the Alternative. Males and females almost assuredly have different hand washing proportions.

Pooled vs. Non-Pooled

( )1 2 1 2

1 1 2 2

1 2

ˆ ˆ( )

ˆ ˆ ˆ ˆ(1 ) (1 )

p p p pz

p p p pn n

− − −=

− −+

•Use Pooled when you hypothesize populations have the same variance (in proportions, the same p = same variance)

• Use Non-pooled when populations are likely to have separate variances. (If your null shows a non-zero difference)

Confidence Interval

( ) 1 1 2 21 2

1 2

ˆ ˆ ˆ ˆ(1 ) (1 )ˆ ˆ *

p p p pp p z

n n

− −− ± +

Use Non-Pooled because there is no null to test for.

So, To Review…

PhatPi is on, but with these changes: P Parameter of interest is now the

difference between ___ and ___ H H0: p1=p2 (or p1-p2=0)

Ha: p1>p2 (or p1-p2>0)

or Ha: p1<p2 (or p1-p2<0)

or Ha: p1≠p2 (or p1-p2 ≠ 0)

Plus, you have to choose Pooled v. Non-Pooled (Pooled if Null is p1=p2)

Using TI 83

Stat>Test>2-PropZTest Can Also Do Interval:

Stat>Test>2-PropZInt• Put in C-Level (usually .9, .95, or .99)

Let’s do one!

Some scientist suggest that sickle-cell traits protect against malaria. A study in Africa tested 543 for sickle-cell trait and also for malaria. In all, 136 of the children had sickle-cell trait and 36 of these had malaria. The other 407 children lacked the sickle-cell trait and 157 of them had malaria. Is there evidence that malaria infection is lower among children with the sickle-cell trait.

Malaria v. Sickle Cell

P

H

A

Malaria v. Sickle Cell

T

P

I

Do Using a 95% C-Interval

Assumptions:

Interval Calculation

Interpretation

That is it!

Just one section left to go!