Stats 120A

1
Stats 120A

• Review of CIs, hypothesis tests and more

2
Sample/Population

• Last time we collected height/armspan data. Is
  this a sample or a population?

3
Gallup Poll, 1/9/07

• "As you may know, the Bush administration is
  considering a temporary but significant increase
  in the number of U.S. troops in Iraq to help
  stabilize the situation there. Would you favor or
  oppose this?"

4
Results

• Results based on 1004 randomly selected adults (gt
  18 years) interviewed Jan 5-7, 2007.
• 61 are opposed.
• "For results based on this sample, one can say
  with 95 confidence that the maximum error
  attributable to sampling and other random effects
  is 3 percentage points. "

5
Pop Quiz

• Is the value 61 a statistic or a parameter?
• The margin of error is given as 3. What does
  the margin of error measure? a) the variability
  in the sample
• b) the variability in the population
• c) the variability in repeated sampling

6
Sampling paradigm

• In the U.S., the proportion of adults who are
  opposed to a surge is p, (or p100).
• We take a random sample of n 1004.
• The proportion of our sample ("p hat") is an
  estimate of the proportion in the population.

7
A simulation

• Choose a value to serve as p (say p .6)
• Our "data" consist of 1004 numbers 0's represent
  those in favor, 1's are those opposed.
• x 589 out of 1004 say "opposed", so p-hat
  589/1004 .5866
• mean(x) .5866
• sd(x) .4926

8
xbar.5866, s .493
9
How do we know sample proportion is a good
estimate of population proportion?

• Law of Large Numbers
• sample averages (and proportions) converge on
  population values
• implying that for finite values, the sample
  proportion might be close if the sample size is
  large

10
Coin flips sample proportion "settles down" to
0.5
11
So if we stop earlier, say n 10
p-hat .60
12
Which raises the question

• If we stop early, how far away will our sample
  proportion be from the true value?
• Or, in a survey setting, if we take a finite
  sample of n1004, how far off from the population
  proportion are we likely to be?

13
A simulation might help

• Assume p .60 (population proportion)
• Take sample of n 1004 and find p-hat.
• Save this value
• Repeat above 3 steps 10000 times.

14
The R code (for the record)

• phat lt- c()
• for (i in 110000)
• x lt- sample(c(0,1),1004,replaceT,probc(.4,
  .6))
• temp lt- sum(x)/1004
• phat lt- c(phat,temp)
• hist(phat)

15
each dot represents one survey of 1004 people
16
10,000 sample proportions, n 1004
17
Observe that...

• sample proportions are centered on the true
  population value p .60
• variability is not great smallest is .54,
  biggest is .66
• distribution is bell-shaped

18
We've just witnessed the Central Limit Theorem

• If samples are independent and random and
  sufficiently large
• means (and proportions) follow a nearly Normal
  distribution
• the mean of the Normal is the mean of the
  population
• the SD of the Normal (aka the standard error) is
  the population SD divided by sqrt(n)

19
CLT applied to sample proportions

• phat is distributed with an approx Normal
• mean is p
• SE is sqrt(p(1-p)/n)
• For our simulation, p .60 so our p-hats will be
  centered on .6 with a SD of sqrt(.6.4/1004)
  0.0155

20
We saw

• Normal
• mean(phat) 0.600(expected .6)
• sd(phat) 0.01554(expected 0.0155)

21
In practice, we don't know p

• but we can get a good approximation to the
  standard error using
• sqrt(phat (1-phat)/n)
• rather than
• sqrt(p(1-p)/n)

22
So if we take a random sample of n 1004

• and we see p-hat .61, we know that
• The true value of p can't be far away.
• SE sqrt(.61.39/1004) 0.0154
• So 68 of the time we do this, p will be within
  0.0154 of phat
• And 95 of the time it will be with 2.0154
  0.03

23
Which leads us to conclude

• that the true proportion of the population that
  opposes a surge is somewhere in the interval.61
  - .03 0.58
• to .61.03 0.64

24
Confidence intervals

• This is an example of a 95 confidence interval.
• Because 95 of all samples will produce a p-hat
  that is within 2 standard errors of the true
  value, we are 95 confident that ours is a "good"
  interval.

25
Formula

• A 95 CI for a proportion is
• estimate /- 2 (Standard Error)
• p-hat /- 2sqrt(phat(1-phat)/n)
• 0.61 /- 2sqrt(.61.39/1004)
• (.58, .64)
• note our replacing phat for p in SE means we get
  an approximate value

26
What does 95 mean?

• If we repeat this infinitely many times
• take a sample of n 1004 from population
• calculate sample proportion
• find an interval using /- 2 SE
• then 95 of these CIs will contain the truth and
  5 will not.
• We see only one (.58, .64). It is either good
  or bad, but we are confident it is good.

27
Where did the 95 come from?

• It came from the normal curve.
• The CLT told us that p-hat followed a (approx)
  normal distribution.
• For Normal's, 68 of probability is within 1
  standard deviation of mean, 95 within 2, 99.7
  within 3.
• A normal table gives other probabilities

28
Change confidence level by changing the width of
margin of error
.015
-0.015
1 SE
68
2 SEs
95
3 SEs
99.7
90
1.6 SE
phat 0.61
29
The CLT applies to

• any linear combination of the observations
• assuming observations are randomly sampled, and
  independent
• it does NOT matter what the distribution of the
  population looks like
• if n is small, the distribution will be only
  approximately normal, and this might be a very
  poor approximation

30
the CLT does NOT apply to

• non-linear combinations, such as the sample
  median or the standard deviation
• non-random samples
• samples that are dependent

31
simulation

• http//onlinestatbook.com/stat_sim/sampling_dist/i
  ndex.html

32
Summary

• Confidence Level is a statement about the
  sampling process, not the sample
• Margin of error is determined to achieve the
  desired confidence level
• We can calculate the confidence level only if we
  know the sampling distribution the probability
  distribution of the sample

33
Pop Quiz

• Is the value 61 a statistic or a parameter?
• The margin of error is given as 3. What does
  the margin of error measure? a) the variability
  in the sample
• b) the variability in the population
• c) the variability in repeated sampling

34
Pop Quiz

• Is the value 61 a statistic or a parameter?
• The margin of error is given as 3. What does
  the margin of error measure? a) the variability
  in the sample
• b) the variability in the population
• c) the variability in repeated sampling

35
For next time

• In WWII, German army produced tanks with
  sequential serial numbers. The allies captured a
  few tanks, and wanted to infer the total number
  of tanks produced.
• Suppose you had captured 10 tanks. Come up with
  three estimators for the total number of tanks.
• Data 911 5146 6083 944 11944 9365 6087
  6647 7076 12275