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APSTAT UNIT 4A INFERENCE PART 1. APSTAT Chapter 18 Sampling Distributions and Sample Means. Lets Just DO IT!!!!. 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!!!!.

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lets just do it
Lets Just DO IT!!!!

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 it1
Lets Just DO IT!!!!

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 it2
Lets Just DO IT!!!!

Proportion of correct answers on last AP Stat Exam

Class Sample Means:

Sampling Distribution

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

we just did it
We Just DID IT!!!!
  • Give me at least 2 things that are different between the regular distribution and the sampling distributions:
  • Unbiased Statistic
    • Mean 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
Sample Proportions
  • Mean of Sample Proportion
  • In last section:
  • Proportion is just outcome divided by n, so….
standard deviation of sample proportion
Standard Deviation of Sample Proportion
  • In last section:
  • Proportion is just outcome divided by n, so throw down a little Algebra….
now try it sample proportions
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
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
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) = 0.4
    • Large enough population?
    • Normalness?
  • 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
=.64, n=100, 70 or more
  • Check Conditions - Briefly Explain
    • 10%
    • np and n(1-p) > 10
  • Draw Picture- (Find SD too)
  • Find P-Value
  • Conclusion
same problem data
Same Problem Data…
  • Within what range would we expect to find 95% of sample proportions of size 100.
sample means
Sample Means
  • Take a whole bunch of samples and find the means
  • Why sample means?
  • Remember our sample of class scores?
    • Less variable
    • More normal
mean and standard deviation of x bar
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)
mean and standard deviation of x bar1
Mean and Standard Deviation of X-BAR
  • Standard Deviation of a sampling distribution is:
let s try it
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=10
    • n=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
CLT – The Central Limit Theorem
  • If the population we are sampling from is already normal with N(,), the sampling distribution will be normal as well with mean  and standard deviation
  • But what if the population we are sampling from is not normal?
age of pennies
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…
what happened
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
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?
new skool find the probability that an srs of 20 students 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 Explain
    • 10%
    • Independence and Random Sample
  • Draw Picture- (Find SD too)
  • Find P-Value
  • Conclusion
standard error
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.
confidence interval for a proportion aka one proportion z interval
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%?
the dealio
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
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
Do It!
  • List what you know
    • p-hat=.32, n=80
  • Conditions/Assumptions
    • 10% for Independence
      • Woodside HS has over 800 students
    • np and nq > 10 to use Normal Model
      • Both .32 x 80 and .68 x 80 > 10
  • Find Standard Error
    • SE(p-hat)=
do it1
Do It!
  • Draw the Picture
  • Conclusion:
    • We are 95% confident that the TRUE mean proportion of ____________ falls between ____ and ____
do it2
Do It!
  • We can also write confidence intervals in the form:

(estimate) ± (margin of error)

Standard error

what does 95 confidence mean
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:


  • 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

Area = 90%


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*
mas z
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 it1
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 it2
Try It!
  • Conditions:
  • Mean, SE, z*
  • Calculate CI
  • Conclusion
last thing
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
Sample size needed

Margin of Error

sample size needed1
Sample size needed

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


significance 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
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 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
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
Be Carefull…
  • Notice we NEVER, NEVER, NEVER
    • Accept either Hypothesis
    • Say 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
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.

let s do it wps sat example
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
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 continued1
WPS SAT Example Continued
  • Step 4 Name Test and DO IT
    • One Sample Z-Test for a Proportion
wps sat example continued2



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
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



Alternative Hypotheses
  • Can be:
  • Greater Than (Ha ------>blah)
  • Less Than (Ha --------<blah)
  • Not (Ha --------≠blah)

Double your one-sided P-value


on ti 83
On TI-83
  • Still have to do all of Phat Pi, but helps with calculations.
  • Stat>Test>1-PropZTest
    • p0 - Population proportion
    • x - successes in sample n(p-hat)
    • n - sample size
  • Do it for AP Stat Example
defective products
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?
how much evidence
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
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.
great chapter
Great Chapter
  • Make sure you read it
  • Important concepts:
    • What a Null Hypothesis is and isn’t
    • What P-Value Means
    • Significance () Level
    • Critical Value - One v. Two sided
    • Confidence Intervals and Tests of Significance - Relationship Between
great chapter but
Great Chapter…But….
  • Goes further than you need in explaining:
    • Types of Error
      • Type I
      • Type II
    • Power
errors can we make mistakes
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?
  • 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
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
  • 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
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
let s hop right in
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?
  • 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 “ “ “ “
  • Assumptions
    • SRS’s Yep
    • Independent samples Safe to Assume
    • n1p1>5 and n1(1-p1)>5
    • n2p2>5 and n2(1-p2)>5 Yep
    • Population 10X Sample Yep
  • Test – Two Sample Proportion Z-Test


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

  • P-Value
    • 2*P(Z>5.965)=Really Really Really Small
  • Interpretation
    • P 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
Pooled vs. Non-Pooled
  • 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
Confidence Interval

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

so to review
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
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
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.
do using a 95 c interval
Do Using a 95% C-Interval
  • Assumptions:
  • Interval Calculation
  • Interpretation
that is it
That is it!
  • Just one section left to go!