chapter 9wellsmat.startlogic.com/.../sitebuilderfiles/apstat_ch9_studynotes.pdf · parameter and...

Chapter 9

Sampling Distributions

Lesson 9-1, Part 1

Sampling Distribution


Suppose I randomly select 100 seniors in Sarasota County and record

each one’s GPA.

1.95 1.98 1.86 2.04 2.75 2.72 2.06 3.36 2.09 2.06

2.33 2.56 2.17 1.67 2.75 3.95 2.23 4.53 1.31 3.79

1.29 3.00 1.89 2.36 2.76 3.29 1.51 1.09 2.75 2.68

2.28 3.13 2.62 2.85 2.41 3.16 3.39 3.18 4.05 3.26

1.95 3.23 2.53 3.70 2.90 2.79 3.08 2.79 3.26 2.29

2.59 1.36 2.38 2.03 3.31 2.05 1.58 3.12 3.33 2.04

2.81 3.94 0.82 3.14 2.63 1.51 2.24 2.22 1.85 1.96

2.05 2.62 3.27 1.94 2.01 1.68 2.01 3.15 3.44 4.00

2.33 3.01 3.15 2.25 3.34 2.22 2.29 3.90 2.96 2.61

3.01 2.86 1.70 1.55 1.63 2.37 2.84 1.67 2.92 3.29


These 100 seniors make up one possible sample.

The sample mean and the sample

standard deviation

All seniors in Sarasota County make up the

population.

The population mean (μ) and the population standard

deviation (σ) are unknown

We can use to estimate (μ) and we can use to

estimate (σ). Theses estimates may or may not be

reliable.

2.5470x 0.7150Xs

x xs

Parameter and Statistics

A number that describes the population is

called a parameter.

Therefore, (μ) and (σ) are both parameters.

A parameter is usually represented by (p).

A number that is computed from a sample

is called a statistics.

Therefore, and are both statistics.

A statistic is usually represent by

x Xs

p̂

Sampling Variability

If I had chosen a different 100 seniors, then I would have a different sample, but it would still represent the same population.

A different sample almost always produce different results.

If I compare many different samples and the statistic is very similar in each one.

The sampling variability is low.

If I compare many different samples and the statistic is very different in each one.

The sampling variability is high.


The sampling distribution of a statistic is

a distribution of the values of the statistic

from all possible samples of the same size

from the same population.

Rather than showing real repeated

samples, we can imagine what would

happen if we were to actually draw many

samples.

Example


One year, I had a small statistics class of 7 students, I asked

them the age of their cars and obtained the following

data: 2, 4, 6, 8, 4, 3, 7.

The population mean is:

2 4 6 8 4 3 7 344.857

7 7μ

years

Example


Construct a sampling distribution of the mean for sample

size n = 2.

There are total of 7 individuals in the population. We are

selecting them two at time without replacement. Therefore,

there are 7C2 = 21 samples of size n = 2.

Example

Sampling DistributionSample Sample Mean Sample Sample Mean Sample Sample Mean

2, 4 3 4, 8 6 6, 7 6.5

2, 6 4 4, 4 4 8, 4 6

2, 8 5 4, 3 3.5 8, 3 5.5

2, 4 3 4, 7 5.5 8, 7 7.5

2, 3 2.5 6, 8 7 4, 3 3.5

2, 7 4.5 6, 4 5 4, 7 5.5

4, 6 5 6, 3 4.5 3, 7 5

Mean of statistic values102

4.85721 21

x

x x x

Example


0.00

0.05

0.10

0.15

0.20

Pro

bab

ilit

yty

2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5

Sample Means

Probability Distribution of the Sample Mean

Example

Sampling DistributionSample Distribution of the Sample Mean

Sample Mean Frequency Probability

2.5 1 1/21

3 2 2/21

3.5 2 2/21

4 2 2/21

4.5 2 2/21

5 4 4/21

5.5 3 3/21

6 2 2/21

6.5 1 1/21

6.5 1 1/21

7 1 1/21

7.5 1 1/21

What is the probability of

obtaining a sample mean

between 4 and 6 years,

inclusive – (4 6)?P x

13

(4 6) 0.61921

P x

If we took 10 random samples

of size 2 from this population,

about 6 of them would result

in sample means between 4

and 6 years inclusive.

Example – Page 493, #9.6

Use your calculator to replicate Exercise 9.5 as follows. The command

randbin(20,0.50) simulates tossing a coin 20 times. The output is the

number of heads in 20 tosses. The command randbin(20,0.50,10)/20

simulates 10 repetitions of tossing the coin 20 times and finding the

proportions of heads. Go into your statistics/List editor and place

your cursor on top of L1/list1. Execute the command

randbin(20,0.50,10)/20.

20,0.50,10)/20.


A). Plot a histogram of the 10 values of p̂


B). Increase the number of repetitions to 100. The

command should read randbin(20,0.50,100)/20.

Describe the shape of the distribution

The center is close to 0.50, and the shape is approximately normal


C). Use PLOT 2 to be a boxplot. How close is the median

(in the boxplot) to the mean (balance point) of the

histogram?

The mean and median are extremely close.


D). Note that we didn’t increase the sample size, only

the number of repetitions. Did the spread of the

distribution change? What would you change to

decrease the spread of the distribution?

The spread change very little. To decrease the

spread, I would increase the number of trials, n.

For example, randbin(50,0.50)

Describing


Shape

Is the shape of the distribution symmetric or

approximately normal?

Center

Is the center of the distribution very close to the true

value?

Spread

Do the values of the sample have a large spread?

Outliers

Are there any deviations from the overall pattern?

Lesson 9-1, Part 2


Unbiased Statistic

The statistic used to estimate a parameter is

unbiased if the mean of its sampling

distribution is equal to the true value of the

parameter being estimated.

Sample proportion is an unbiased

estimator of the population proportion (p).

Sample mean is an unbiased estimator of

the population mean (μ).

p̂

x

Variability of a Statistic

The variability of a statistic is described by

the spread of its sampling distribution.

This spread is determined by the sampling

design and size of the sample.

A statistics can be unbiased and still have

high variability. To avoid this, increase the

size of the sample.

Larger samples give smaller spread.


The table below contains the results of simulating on a computer 100

repetitions of the drawing of an SRS of size 200 from a large lot of

ball bearings. Ten percent of the bearings in the lot do not conform

to the specifications. That is, p = 0.10 for this population. The numbers

in the table are the counts of nonconforming bearings in each sample

of 200.

17 23 18 27 15 17 18 13 16 18 20 15 18 16 21 17 18 19 16 23

20 18 18 17 19 13 27 22 23 26 17 13 16 14 24 22 16 21 24 21

30 24 17 14 16 16 17 24 21 16 17 23 18 23 22 24 23 23 20 19

20 18 20 25 16 24 24 24 15 22 22 16 28 15 22 9 19 16 19 19

25 24 20 15 21 25 24 19 19 20 28 18 17 17 25 17 17 18 19 18


A). Make a table that shows how often each count occurs. For each

count in your table, give the corresponding value of sample

proportion . Then draw a histogram for the values

of the statistic .

ˆ / 200p count

p̂


Count Count Count

9 9/200

= 0.045

1 18 18/200=

0.090

12 24 0.120 10

13 0.065 3 19 0.095 9 25 0.125 4

14 0.070 2 20 0.100 7 26 0.130 1

15 0.075 5 21 0.105 5 27 0.135 2

16 0.080 11 22 0.110 6 28 0.140 2

17 0.085 12 23 0.115 7 30 0.150 1

p p̂ p p̂ p p̂


The histogram actually does not appear to have a normal shape. The

sampling distribution is quite normal in appearance, but even a sample of

size 100 does not necessarily show it.


p̂The mean of is 0.0981

C). Find the mean of the 100 observations of . Mark the mean on your

histogram to show its center. Does the statistic appear to have a

large or small bias as an estimate of the population proportion p?

p̂p̂

The bias seems to be small


D). The sampling distribution of is the distribution of the values of

from all possible samples of size 200 from this population. What

is the mean of this distribution?

p̂

p̂

The mean of the sampling distribution should be p = 0.10

p̂

E). If we repeatedly selected SRSs of size 1000 instead of 200 from

this same population, what would be the mean of the sampling

distribution of the sample proportion ? Would the spread be larger,

smaller, or about the same when compared with the spread of your

histogram in (a).

The mean would still be 0.10, but the spread would be smaller

Bias and Variability

High bias means that our aim is off and we consistently miss the bull’s eye in the same direction.

Low bias means the shots are center on the bull’s eye.

High variability means that repeated shots are widely scattered on the target.

Low variability means that the shots are close together.


Figure 9.10 shows histograms of four sampling distributions of

statistics intended to estimate the same parameter. Label each

distribution relative to the others as having large or small bias and

as having large or small variability.


Large bias and large variability


Small bias and small variability


Small bias and large variability


Large bias and small variability

Lesson 9-2

Sample Proportions

Sample Proportions

The parameter p is the population proportion. In practice, this value is always unknown. (If we know the population proportion, then there is no need for a sample.)

The statistics is the sample proportion

We use to estimate the value of (p).

The value of the statistic changes as the sample changes.

p̂

p̂

p̂

Sampling Distribution of

Sample Proportion

p̂

p̂

ˆx

pn

If our sample is an SRS of size n, then the following statements describe

the sampling distribution for

1. The shape is approximately normal.

ASSUMPTION: Sample size is sufficiently large

CONDITION: np 10 and n(1 – p) 10

2. Let be the proportion of the sample having that characteristic.

The mean of the sampling distribution is exactly p

Wherex is the count “success in a sample

n is the size of the sample

Sampling Distribution of

Sample Proportion

3. The standard deviation is

ASSUMPTION: Sample size is sufficiently large

CONDITION: The population is at least 10 times as large as the

sample.

(1 )x

p p pqs

n n

Summary

Select a large SRS from a population of which the proportion p are success.

The sampling distribution of the proportion of success in the sample

is approximately normal.

p̂

Sample Proportions

If we have categorical data, then we must use sample proportions to construct a sampling model.

Example – Suppose we want to know how many seniors in Florida plan to attend college. We want to now how many seniors answer, “Yes” to the question, “Do you plan to attend college?” These responses are categorical.

So p (our parameter) is the proportion of all seniors in Florida who plan to attend college

Let (our statistic) be the proportion of Florida students in an SRS of size 100 who plan to attend college.

To calculate the value of , we divide the number of “Yes” responses in our sample by the total number of students in the sample.

p̂

p̂

Sampling Model

If I graph the value of for all possible samples of

size 100, then I have constructed a sampling model.

What will the sample model look like?

It will be approximately normal. In fact, the larger my

sample size, the closer it will be a normal model.

So how large is large enough to ensure that the

sampling model is close normal?

Both np 10 and nq 10 in order for normal

approximations to be useful.

p̂

Sampling Model

The mean of sample model will equal the true

population

The standard deviation (if the population is

least 10 times as large as the sample) will be

pqσ

n


The Gallup Poll asked a probability sample of 1785 adults whether they attended

church or synagogue during the past week. Suppose that 40% of the adult

population did attend. We would like to know the probability that SRS of size

1785 would come within plus or minus 3 percentage points of this true value.

A). If is the proportion of the sample who did attend church or

synagogue, what is the mean of the sampling distribution of ?

What is its standard deviation?

p̂

p̂

0.40μ p

0.40 1 0.40(1 )0.0116

1785

p pσ

n


B). Explain why you can use the formula for the standard deviation of

in this setting (rule of thumb 1)p̂

0.40

1785

0.0116

μ p

n

σ

The population (U.S adults) is considerably larger than 10 times

the sample size


C). Check that you can use the normal approximation for the

distribution of (rule of thumb 2).p̂0.40

1785

0.0116

μ p

n

σ

10 (1 ) 10np and n p

714 10 1071 10


p̂

0.40

1785

0.0116

μ p

n

σ

D). Find the probability that takes a value between 0.37 and 0.43.

Will an SRS of size 1785 usually give a result within plus or

minus 3 percentage points of the true population proportion?

Explain.

p̂

ˆ(0.37 0.43) (0.37,0.43,.40,0.0116) 0.99P p normalcdf

Over 99% of all samples should give within ±3% of the

true population proportion

p̂


Harley-Davidson motorcycles make up 14% of all the motorcycles in the

United States. You plan to interview an SRS of 500 motorcycles owners.

A) What is the approximate distribution of your sample who

own Harleys?

The distribution is approximately normal with mean

μ = p = 0.14

Standard deviation is

(1 ) 0.14(0.86)0.0155

500

p pσ

n


B) How likely is your sample to contain 20% or more who own

Harley’s. Do a normal probability calculation to answer

this question.

μ = p = 0.14

σ = 0.0155ˆ( 0.20) 0.00005P p

5(0.20, 99,0.14,0.0155) 5.42 10 0.00005normalcdf E

20% or more Harley owners is unlikely


C) How likely is your sample to contain 15% or more who own

Harley’s. Do a normal probability calculation to answer

this question.

μ = p = 0.14

σ = 0.0155ˆ( 0.15) 0.2594P p

(0.15, 99,0.14,0.0155) 0.2594normalcdf E

There is a fairly good chance of finding at least 15%

Harley owners.

Lesson 9-3, Part 1

Sample Means

Sample Means

If we have quantitative data, then we must use sample means to construct a sampling model.

Example – Suppose I randomly select 100 seniors in Florida and record each one’s GPA. I am interested in knowing the average GPA of a senior in Florida.

These 100 seniors make up one possible sample.• Sample mean is

• Sample standard deviation is

So p (our parameter) is the true mean GPA of a senior in Florida.

Let (our statistic) is the mean GPA of a senior in Florida in an SRS of size 100.

To calculate the value of , we find the mean our sample .

p̂

p̂

x xs

x

Sample Means

If pick different samples, then value our statistic will chance.

If I graph the values of for all possible samples of size 100, then I have construct a sampling modelof sample means.

What will the sampling model look like?

Remember that each value is a mean.

Means are less variable then individual observations because we are looking only at means, then we don’t see any extreme values, only averages.

p̂ x

p̂ x

p̂

Sample means

The lager the sample size, the less variation we will

see in the values of .

So the standard deviation decreases as the sample

size increases.

So what will the sampling model look like?

If the sample size if large, it will be approximately

normal.

It can never be perfectly normal, because our data is

discrete, and normal distribution are continuous.

p̂

Sample Means

The mean of the sampling model of will

equal the true population mean

The standard deviation will be (if the

population is at least 10 times as large as

the sample)

x

σσ

n

x

xμ μ

Example – Page 519, #32

The scores of individual students on the American College Testing (ACT)

composite college entrance examination have a normal distribution

with mean 18.6 and standard deviation 5.9.

A). What is the probability that a single student randomly chosen

from all those taking the test scores 21 or higher?

( 21) 0.3421 P x

(21, 99,18.6,5.9) 0.3421normalcdf E

18.6

5.9

μ

σ

The probability of choosing a single student at random whose

test score exceeds 21 is about 0.34.


B). Know take an SRS of 50 students who took the test. What are the mean

and standard deviation of the average (sample mean) score for the

50 students? Do you results depend on the fact that individual scores

have a normal distribution?

18.6X 5.9 0.8344

50X

n

This result is independent of distribution shape.


C). What is the probability that the mean score of these students is 21

or higher?

( 21) 0.0020 P x

(21, 99,18.6,0.8344) 0.0020normalcdf E

18.6

0.8344

x

x

μ μ

σ

It is very unlikely (less than 1% chance) that we would

draw an SRS of 50 students whose average score exceeds

21.


A study of the health of teenagers plans to measure the blood

cholesterol level of an SRS of youth of ages 13 to 16 years. The

researchers will report the sample mean from their sample as the

estimate of the mean cholesterol level μ in this population.

A). Explain to someone who knows no statistics what it means to say

that x is “unbiased” estimator of μ.

If we choose many samples, the average of the x values

from these samples will be close μ.


B). The sample result x is an unbiased estimator of the population

parameter μ no matter what the size SRS the study chooses.

Explain to someone you knows no statistics why a large sample

gives more trustworthy results than a small sample.

The larger sample will give more information, and therefore

more precise results; that is x is more likely to be close

to the true population.

Lesson 9-3, Part 2

Central Limit Theorem


For any population, regardless of

its shape, as the sample size

increases, the shape of the

distribution becomes more “normal”


Draw SRS of size n from any population with mean μand finite standard deviation σ.

When n is large (n ≥ 30) the sampling distribution of sample mean is close to the normal distribution.

Mean

• The mean of the sample is equal to the population mean

Standard Deviation

X

Xn

,σ

N μn

Sampling Distribution of a

Sample Mean


A company that owns and services a fleet of cars for its

sale force has found that the service lifetime of disc brake

pads varies from car to car according to a normal distribution

with mean μ = 55,000 miles and standard deviation σ = 4500

miles. The company installs a new brand of brake pads on

8 cars.

A). If the new brand has the same lifetime distribution as

the previous type, what is the distribution of the sample

mean lifetime for the 8 cars.

4500

55000, 55000,15918

N N


B). The average life of the pads on these 8 cars turn out be

51,800 miles. What is the probability that the sample

mean lifetime is 51,800 miles or less if the lifetime

distribution is unchanged? (The company takes this

probability as evidence that the average lifetime of the

new brand of pads is less than 55,000 miles).

55000

4500

1591X

( 51800) 0.02215P x

( 99,51800,55000,1591) 0.02215normalcdf E


Children in kindergarten are sometimes given the Ravin Progressive Matrices

Test (RPMT) to assess their readiness for learning. Experience at Southwark

Elementary School suggests that the RPMT scores for its kindergarten

pupils have mean 13.6 and standard deviation 3.1. The distribution is close

to normal. Mr. Lavin has 22 children in his kindergarten class this year. He

suspects that their RPMT scores will be unusually low because the test was

interrupted by a fire drill. To check this suspicion, he wants to find the level L

such that there is probability only 0.05 that the mean score of 22 children

falls below L when the usual Southwark distribution remains true. What is

the value of L?

L

0.05

13.6

3.10.05,13.6, 12.51322

invorm

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