understand that sampling error is really sample variability

Understand that Sampling error is• really sample variability • unavoidable and predictable

AP Statistics Objectives Ch18

Be able to check all conditions for a sampling distribution

Be able to describe the shape, center, and spread of a sampling distribution


Be able to check the conditions necessary to use the Central Limit Theorem

Describe the shape, center, and spread of sample means


Sampling Error Sampling distribution model Sampling distribution model for a

proportion Sampling distribution model for a

mean Central Limit Theorem Standard error

Vocabulary

Textbook Notes Classroom Notes

Chapter 18 Assignments

Chapter 18 Assignment Part I

pp. 428-429 #2&4,10,12&14,21&22

Chapter 18 Assignment Part II

pp. 430-431 #26,30,36,42

7

Modeling the Distribution of Sample Proportions

• Rather than showing real repeated samples, imagine what would happen if we were to actually draw many samples.

• Now imagine what would happen if we looked at the sample proportions for these samples. What would the histogram of all the sample proportions look like?

8

Modeling the Distribution of Sample Proportions (cont.)

• We would expect the histogram of the sample proportions to center at the true proportion, p, in the population.

• As far as the shape of the histogram goes, we can simulate a bunch of random samples that we didn’t really draw.

9


• It turns out that the histogram is unimodal, symmetric, and centered at p.

• More specifically, it’s an amazing and fortunate fact that a Normal model is just the right one for the histogram of sample proportions.

• To use a Normal model, we need to specify its mean and standard deviation. The mean of this particular Normal is at p.

10


• When working with proportions, knowing the mean automatically gives us the standard deviation as well—the standard deviation we will use is

• So, the distribution of the sample proportions is modeled with a probability model that is , pqN p

n

pqn

11


• A picture of what we just discussed is as follows:

12

How Good Is the Normal Model?

• The Normal model gets better as a good model for the distribution of sample proportions as the sample size gets bigger.

• Just how big of a sample do we need? This will soon be revealed…

13

Assumptions and Conditions

• Most models are useful only when specific assumptions are true.

• There are two assumptions in the case of the model for the distribution of sample proportions:

1. The sampled values must be independent of each other.

2. The sample size, n, must be large enough.

14

Assumptions and Conditions (cont.)

• Assumptions are hard—often impossible—to check. That’s why we assume them.

• Still, we need to check whether the assumptions are reasonable by checking conditions that provide information about the assumptions.

• The corresponding conditions to check before using the Normal to model the distribution of sample proportions are the 10% Condition and the Success/Failure Condition.

15

Assumptions and Conditions (cont.)1. 10% condition: If sampling has not been

made with replacement, then the sample size, n, must be no larger than 10% of the population.

2. Success/failure condition: The sample size has to be big enough so that both and are greater than 10.

So, we need a large enough sample that is not too large.

ˆnp ˆnq

16

A Sampling Distribution Model for a Proportion

• A proportion is no longer just a computation from a set of data.– It is now a random quantity that has a distribution.– This distribution is called the sampling distribution

model for proportions.• Even though we depend on sampling distribution

models, we never actually get to see them. – We never actually take repeated samples from the

same population and make a histogram. We only imagine or simulate them.

17

A Sampling Distribution Model for a Proportion (cont.)

• Still, sampling distribution models are important because– they act as a bridge from the real world of data to the

imaginary world of the statistic and – enable us to say something about the population

when all we have is data from the real world.

18

The Sampling Distribution Model for a Proportion (cont.)

• Provided that the sampled values are independent and the sample size is large enough, the sampling distribution of is modeled by a Normal model with – Mean:– Standard deviation:

p̂

( )p̂ p ( )ˆ pqSD p n

19

What About Quantitative Data?• Proportions summarize categorical

variables.• The Normal sampling distribution model

looks like it will be very useful.• Can we do something similar with

quantitative data?• We can indeed. Even more remarkable,

not only can we use all of the same concepts, but almost the same model.

20

Simulating the Sampling Distribution of a Mean

• Like any statistic computed from a random sample, a sample mean also has a sampling distribution.

• We can use simulation to get a sense as to what the sampling distribution of the sample mean might look like…

21

Means – The “Average” of One Die

• Let’s start with a simulation of 10,000 tosses of a die. A histogram of the results is:

22

Means – Averaging More Dice• Looking at the average of two dice after a simulation of

10,000 tosses:• The average of three dice

after a simulation of 10,000 tosses looks like:

23

Means – Averaging Still More Dice

• The average of 5 dice after a simulation of 10,000 tosses looks like:

• The average of 20 dice after a simulation of 10,000 tosses looks like:

24

Means – What the Simulations Show

• As the sample size (number of dice) gets larger, each sample average is more likely to be closer to the population mean.– So, we see the shape continuing to tighten

around 3.5• And, it probably does not shock you that

the sampling distribution of a mean becomes Normal.

25

The Fundamental Theorem of Statistics

• The sampling distribution of any mean becomes Normal as the sample size grows. – All we need is for the observations to be

independent and collected with randomization.

– We don’t even care about the shape of the population distribution!

• The Fundamental Theorem of Statistics is called the Central Limit Theorem (CLT).

26

The Fundamental Theorem of Statistics (cont.)

• The CLT is surprising and a bit weird:– Not only does the histogram of the sample

means get closer and closer to the Normal model as the sample size grows, but this is true regardless of the shape of the population distribution.

• The CLT works better (and faster) the closer the population model is to a Normal itself. It also works better for larger samples.

27

The Fundamental Theorem of Statistics (cont.)

The Central Limit Theorem (CLT)The mean of a random sample has a sampling distribution whose shape can be approximated by a Normal model. The larger the sample, the better the approximation will be.

28

But Which Normal?

• The CLT says that the sampling distribution of any mean or proportion is approximately Normal.

• But which Normal model?– For proportions, the sampling distribution is

centered at the population proportion.– For means, it’s centered at the population

mean.• But what about the standard deviations?

29

But Which Normal? (cont.)

• The Normal model for the sampling distribution of the mean has a standard deviation equal to

where σ is the population standard deviation.

SD yn

30

But Which Normal? (cont.)

• The Normal model for the sampling distribution of the proportion has a standard deviation equal to

ˆ pqSD pn

31

Assumptions and Conditions• The CLT requires remarkably few assumptions, so

there are few conditions to check:1. Random Sampling Condition: The data values must

be sampled randomly or the concept of a sampling distribution makes no sense.

2. Independence Assumption: The sample values must be mutually independent. (When the sample is drawn without replacement, check the 10% condition…)

3. Large Enough Sample Condition: There is no one-size-fits-all rule.

32

Diminishing Returns

• The standard deviation of the sampling distribution declines only with the square root of the sample size.

• While we’d always like a larger sample, the square root limits how much we can make a sample tell about the population. (This is an example of the Law of Diminishing Returns.)

33

Standard Error

• Both of the sampling distributions we’ve looked at are Normal.– For proportions

– For means

ˆ pqSD pn

SD yn

34

Standard Error (cont.)

• When we don’t know p or σ, we’re stuck, right?

• Nope. We will use sample statistics to estimate these population parameters.

• Whenever we estimate the standard deviation of a sampling distribution, we call it a standard error.

35

Standard Error (cont.)

• For a sample proportion, the standard error is

• For the sample mean, the standard error is

ˆ ˆˆ pqSE pn

sSE yn

36

Sampling Distribution Models

• Always remember that the statistic itself is a random quantity. – We can’t know what our statistic will be

because it comes from a random sample.• Fortunately, for the mean and proportion,

the CLT tells us that we can model their sampling distribution directly with a Normal model.

37

Sampling Distribution Models (cont.)

• There are two basic truths about sampling distributions:

1. Sampling distributions arise because samples vary. Each random sample will have different cases and, so, a different value of the statistic.

2. Although we can always simulate a sampling distribution, the Central Limit Theorem saves us the trouble for means and proportions.

38

The Process Going Into the Sampling Distribution Model

39

What Can Go Wrong?

• Don’t confuse the sampling distribution with the distribution of the sample.– When you take a sample, you look at the

distribution of the values, usually with a histogram, and you may calculate summary statistics.

– The sampling distribution is an imaginary collection of the values that a statistic might have taken for all random samples—the one you got and the ones you didn’t get.

40

What Can Go Wrong? (cont.)• Beware of observations that are not

independent.– The CLT depends crucially on the assumption

of independence.– You can’t check this with your data—you have

to think about how the data were gathered.• Watch out for small samples from skewed

populations.– The more skewed the distribution, the larger

the sample size we need for the CLT to work.

41

What have we learned?

• Sample proportions and means will vary from sample to sample—that’s sampling error (sampling variability).

• Sampling variability may be unavoidable, but it is also predictable!

42

What have we learned? (cont.)

• We’ve learned to describe the behavior of sample proportions when our sample is random and large enough to expect at least 10 successes and failures.

• We’ve also learned to describe the behavior of sample means (thanks to the CLT!) when our sample is random (and larger if our data come from a population that’s not roughly unimodal and symmetric).

Chapter 18

LatinSample Statistic

GreekPopulation or Model

ParameterMeanStandard DeviationProportion

𝒙 s

�̂�

𝝁

AlreadyTaken

Chapter 18

1. Sampling Error – natural variation from sample to sample (aka Sampling Variability)

Chapter 182. Sampling Distribution for a Proportion

Provided that the sampled values are independent and the sample size is large enough, the sampling distribution of is estimated by a Normal model with

mean =

And standard deviation

Where is the _______ proportion andsample is the ___________ proportion;population’s

is the sample _____ and = ______size 1 –

N( )

- Used with

Chapter 18Sampling Distribution for a Proportion

Provided that the sampled values are independent and the sample size is large enough, the sampling distribution of is modeled by a Normal model with …

3. So the two assumptions to use the Normal model are: independent values and the sample size is large enough.

4. Assumptions are difficult to check so you should always check the following conditions:

Chapter 184. Assumptions are difficult to check so you should always check the following conditions:

i) Randomization Condition: An SRS or at least a sampling method that is not biased and that is representative of the population

ii) 10% Condition: If sampling without replacement, then the sample size, n, must be no larger than 10% of the population.

i) Success/Failure Condition: The sample size, n, has to big enough that both np and nq are at least 10.

Independence Assumption

Large Enough Sample Assumption

5. M&M’s InvestigationConsider a “fun size” pack of plain M&M’s. Discuss the assumptions and conditions needed to estimate the sampling distribution using a ______ model.

We can assume the M&M’s are ___________ of each other because, each “fun size” pack can be considered ____________ of the entire _________ of M&M’s andthe number of M&M’s in a “fun size” pack of M&M’s is certainly ____ ____ ___ of all M&M’s.

Normal

independent

representativepopulation

less than 10%

5. M&M’s InvestigationWe can assume that the “fun size” pack is large enough, if np and nq are both __ _____ __ . at least 10

This requires that we know1) The number of M&M’s in our “fun size” pack

proportion color2) The _________ of a particular _____ of M&M.

We need to pick a color.

We’ll start with blue, but write each of the proportions down.

𝝆𝟎 .𝟐𝟒 𝟎 .𝟐𝟎 𝟎 .𝟏𝟔 𝟎 .𝟏𝟒 𝟎 .𝟏𝟑𝟎 .𝟏𝟑

5. M&M’s Investigation

𝝆𝟎 .𝟐𝟒𝟎 .𝟐𝟎𝟎 .𝟏𝟔𝟎 .𝟏𝟒𝟎 .𝟏𝟑𝟎 .𝟏𝟑

𝟏𝟖

𝟓𝟕

𝟒𝟐𝟗

n 𝟒 .𝟑𝟏𝟑 .𝟕

𝟑 .𝟔𝟏𝟒 .𝟒

𝟐 .𝟖𝟏𝟓 .𝟐

𝟐 .𝟓𝟏𝟓 .𝟓

𝟐 .𝟑𝟏𝟓 .𝟕

𝟐 .𝟑𝟏𝟓 .𝟕

Sample sizetoo small

nnn

𝟏𝟑 .𝟕𝟒𝟑 .𝟑

𝟏𝟏 .𝟒𝟒𝟓 .𝟔

𝟗 .𝟏𝟒𝟕 .𝟗

𝟖 .𝟎𝟒𝟗 .𝟎

𝟕 .𝟒𝟒𝟗 .𝟔

𝟕 .𝟒𝟒𝟗 .𝟔

too small for all but blue & orange

nn𝟏𝟎𝟑𝟑𝟐𝟔

𝟖𝟓 .𝟖𝟑𝟒𝟑 .𝟐

𝟔𝟖 .𝟔𝟑𝟔𝟎 .𝟒

𝟔𝟎 .𝟏𝟑𝟔𝟖 .𝟗

𝟓𝟓 .𝟖𝟑𝟕𝟑 .𝟐

𝟓𝟓 .𝟖𝟑𝟕𝟑 .𝟐

Sample sizelarge enough for all colors

6. Since the conditions are met for an “Individual Size” Bag of plain M&M’s, let’s use that bag as our sample.

We can now use a Normal model to estimate our sampling distribution of the proportion of blue , but which Normal model?

Mean: =

Standard Deviation: =

= 0.24

=

= 0.057𝑁 (𝟎 .𝟐𝟒 ,𝟎 .𝟎𝟓𝟕)

---NOTE: We can assume the M&M’s are independent of each other because, the conditions that worked for a “fun size” pack still apply for a “individual size” bag.

a. What is the probability that the number of blue M&M’s in an “individual size” bag of 57 M&M’s will be 10 or fewer?

z = =

P( z )

=

= 0.175

0.175 -1.14

-1.14 0.127<

0.240.057

𝑁 (𝟎 .𝟐𝟒 ,𝟎 .𝟎𝟓𝟕)

b. What is the probability that the number of blue M&M’s in an “individual size” bag of 57 M&M’s will be 20 or larger?

= 0.351

P( z ) 1.95 0.026>

z = = = 0.351 1.950.240.057

𝑁 (𝟎 .𝟐𝟒 ,𝟎 .𝟎𝟓𝟕)

7. Since the conditions are met for a “Medium Bag” of plain M&M’s, let’s use that bag as our sample. We can now use a Normal model to estimate our sampling distribution for the proportion of _________________ , but which Normal model?

On your own.I’ll assign a different color

to each group.

Chapter 18Sampling Distribution for a Proportion

Provided that the sampled values are independent and the sample size is large enough, the sampling distribution of is modeled by a Normal model with

mean =

And standard deviation

Where is the sample proportion and is the population’s proportion; is the sample size and = 1 –

- Used with

Sampling Distribution of the Sample Proportion 1. Suppose that 70% of all dialysis patients will survive for at least 5 years. If 100 new dialysis patients are selected at random, what is the probability that the proportion surviving for at least 5 years will exceed 80%?

Sampling Distribution of the Sample Proportion

1. Suppose that 70% of all dialysis patients will survive for at least 5 years. If 100 new dialysis patients are selected at random, what is the probability that the proportion surviving for at least 5 years will exceed 80%?

Independence is reasonable, because the problem states that 100 dialysis patients are selected at random and

100 patients would be less than 10% of all dialysis patients. (10% Condition)

Success/Failure Condition- np = 100(.70) = 70 > 10 nq = 100(.30) = 30 > 10so the sample size is reasonably large enough.



Since the conditions are satisfied, we can model the sampling distribution of with a Normal model with

𝝁(�̂� )=𝒑=.𝟕𝟎SD(

z = z = = 2.183



z = 2.183

P( > .80) = P(z > 2.183) = 0.015There is about a 1.5% chance that the proportion surviving for at least 5 years will exceed 80% of our sample.


2. It is estimated that 48% of all motorists use their seat belts. If a police officer observes 400 cars go by in an hour, what is the probability that the proportion of drivers wearing seat belts is between 45% and 55%?



Independence is reasonable, because the 400 cars seen on the highway could be considered representative of all cars on the highway and

400 cars would be less than 10% of all cars on the highways. (10% condition)

Success/Failure Condition- np = 400(.48) = 192 > 10 nq = 400(.52) = 208 > 10So the sample size is reasonably large enough.




𝝁(�̂� )=𝒑=.𝟒𝟖SD(

z = z = = -1.2

z = = 2.8



z = -1.2

P( < .55) = P(-1.2<z<2.8)=0.882

There is about a 88.2% chance that the proportion of belted motorist the officer see will be between 45% and 55% of the 400 he sees.

z = 2.8


3. In a large class of introductory Statistics students, the professor has each person toss a coin 16 times and calculate the proportion of his or her tosses that were heads. The students then report their results, and the professor plots a histogram of these several proportions.



a) What shape would you expect this histogram to be? Why?



b) Where do you expect the histogram to be centered?



c) How much variability would you expect among these proportions?



d) Explain why a Normal model should not be used here.



e) The students try again, but toss the coins 25 times each. Use the 68-95-99.7 Rule to describe the sampling distribution model.



f) Confirm that you can use a Normal model here.



g) They increase the number of tosses to 64 each. Draw and label the appropriate sampling distribution model. Check the appropriate conditions to justify your model.



h) Explain how the sampling distribution model changes as the number of tosses increases.

Chapter 18

1. Central Limit Theorem (CLT)-The mean of a random sample has a sampling distribution whose shape can be approximated by a Normal model. The larger the sample, the better the approximation.

2. Central Limit Theorem Assumptions

i) Randomization Condition: An SRS or at least a sampling method that is not biased and that is representative of the population

ii) 10% Condition: If sampling without replacement, then the sample size, n, must be no larger than 10% of the population.


i) Randomization Condition: An SRS or at least a sampling method that is not biased and that is representative of the population ii) 10% Condition: If sampling without replacement, then the sample size, n, must be no larger than 10% of the population.

i) Depends on distribution of population


Large Enough Sample Assumption

2. Central Limit Theorem Assumptions

a) If it is already unimodal & symmetric, then a small sample size is fine

b) If it has a Strong skew, thena larger sample size is needed… > 40

3. Sampling Distribution for a Mean

Provided that the sampled values are drawn at random from a population of mean and standard deviation , the sample distribution will have

- Used with

Mean

and standard deviation

The shape of the sample distribution will be approximately normal as long as the sample size is large enough. The larger the sample size is, the more closely the Normal model approximates the sample distribution.

Chapter 184. Standard Error –

Both sampling distributions request knowledge of the population as a whole. But sometimes we only have information on the sample itself.

For portions we may not know p, only

𝑺𝑬 ( �̂� )=√ �̂� �̂�𝒏For means we may not know , only s

Use what you have to estimate. Whenever we estimate the standard deviation of a sampling distribution, we call it a standard error.

1998 Question 15. Consider the sampling distribution of a sample mean obtained by random sampling from a infinite population. This population has a distribution that is highly skewed toward the larger values.(a) How is the mean of the sampling distribution related to the mean of the population?

1998 Question 15. Consider the sampling distribution of a sample mean obtained

by random sampling from a infinite population. This population has a distribution that is highly skewed toward the larger values.

(b)How is the standard deviation of the sampling distribution related to the standard deviation of the population?

1998 Question 15. Consider the sampling distribution of a sample mean obtained

by random sampling from a infinite population. This population has a distribution that is highly skewed toward the larger values.

(c)How is the shape of the sampling distribution affected by the sample size?

#256. Assume that the duration of human pregnancies can be

described by a Normal model with mean 266 days and standard deviation 16 days.

a) What percentage of pregnancies should last between 270 and 280 days?

#25

6. Assume that the duration of human pregnancies can be described by a Normal model with mean 266 days and standard deviation 16 days.

b) At least how many days should the longest 25% of all pregnancies last?



c) Suppose a certain obstetrician is currently providing prenatal care to 60 pregnant women. Let represent the mean length of their pregnancies. According to the Central Limit Theorem, what’s the distribution of this sample mean, ? Specify the model, mean, and standard deviation.

CONTINUED

Independence is reasonable, because the 60 women that are being treated can be considered representative of all pregnant women andthe 60 women would be less than 10% of all pregnant women. (10% condition)Since the duration of human pregnancies can be described by a Normal model, any sample size is reasonably large enough.



c) Suppose a certain obstetrician is currently providing prenatal care to 60 pregnant women. Let represent the mean length of their pregnancies. According to the Central Limit Theorem, what’s the distribution of this sample mean, ? Specify the model, mean, and standard deviation.

#317. The College Board reported the score distribution shown in the table

for all students who took the 2004 AP Statistics exam.

a) Find the mean and standard deviation of the scores.

SCORE% of STUDENTS

5 12.54 22.53 24.8

2 19.81 20.4



b) If we select a random sample of 40 AP Statistics students, would you expect their scores to follow a Normal model? Explain.

SCORE% of STUDENTS

5 12.54 22.53 24.8

2 19.81 20.4



c) Consider the mean scores of random samples of 40 AP Statistics students. Describe the sampling model for these means (shape, center, and spread).

SCORE% of STUDENTS

5 12.54 22.53 24.82 19.81 20.4

CONTINUED

Independence is reasonable, because the 40 student scores were selected at random and

the 40 scores is less than 10% of all scores. (10% condition)A sample size of 40 is reasonably large enough.



c) Consider the mean scores of random samples of 40 AP Statistics students. Describe the sampling model for these means (shape, center, and spread).

SCORE% of STUDENTS

5 12.54 22.53 24.8

2 19.81 20.4

8. Scores of students on the ACT college entrance exam in a recent year had the normal distribution with mean and standard deviation

a. What is the probability that a single student randomly chosen from all those taking the ACT scores 21 or higher?

z = z = .407

P(ACT score > 21) = P(z > 0.407) = 0.342

There is about a 34.2% chance that the ACT score from a random student will be 21 or higher.

8. Scores of students on the ACT college entrance exam in a recent year had the normal distribution with mean and standard deviation b. What is the probability that the mean score for 36 students randomly selected from all those who took the ACT is 21 or higher? Hint: Standard deviation of the sampling distribution is

Independence is reasonable, because the problem states that the 36 students are selected at random and 36 students would be less than 10% of all students taking the ACT.Since the population of ACT scores had a “normal distribution”, any sample size would be large enough.



𝝁 (𝒙 )=𝝁=𝟏𝟖 .𝟔SD(

z = z = 2.44


z = 2.44P( > 21) = P(z > 2.44) = 0.007There is about a 0.7% chance that the mean ACT score from our sample of 36 students will be 21 or higher.

understand that sampling error is really sample variability

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