Chapter 5 Sampling Distributions. Chapter 5.1 Sampling Distributions of sample mean X-bar.

Chapter 5

Sampling Distributions

Chapter 5.1

Sampling Distributions of

sample mean X-bar

Example: The National Collegiate Athletic Association (NCAA) requires Division I

athletes to score at least 820 on the combined math and verbal SAT exam to

compete in their first college year. The SAT scores of 2003 were approximately

normal with mean 1026 and standard deviation 209. What proportion of all

students would be NCAA qualifiers (SAT ≥ 820)?

0.84 1) 0, E99, -0.99,normalcdf(

:find and Calculator Use

99.0209

206

209

)1026820()(

209;1026;820

x

z

x

Review: Chapter 1.3, Normal distribution

An important property of a density curve is that areas under the curve correspond to relative frequencies

Review Chap3: Population versus sample Sample: The part of the

population we actually examine

and for which we do have data.

A statistic is a number

describing a characteristic of a

sample.

• Population: The entire group

of individuals in which we are

interested but can’t usually

assess directly.

• A parameter is a number

describing a characteristic of

the population.

Population

Sample

Objectives (chapter 5.1)Sampling distribution of a sample mean

Sampling distribution of sample mean (x-bar)

For normally distributed populations

The central limit theorem

Question: In high school, when doing Physics and Chemistry

experiments, why do we need to repeat an experiment for multiple

times? Then take an average as our final experiment result. It sounds

to only waste our time, energy and materials on the repetition. Is it

correct?

Simple random sample (SRS)

Data are summarized by statistics (mean, standard deviation, median, quartiles, correlation, etc..)

Concerns:

1) Is sample mean related to population mean?

2) If yes, what will be the relationship? Or say, how far or how close is a sample mean away from the population mean?

Sampling Distribution of sample mean of 10 random digits

(1) Select 10 random digits from Table B, and then take the sample mean;

(2) Repeat this process 4 times for each student from Dr. Chen’s class.

More details with illustration:1. Based on Table B (random digit table), we randomly select a line, for example line 106 in

this case:

2. Take sample average of random digits of (6, 8, 4, 1, 7, 3, 5, 0, 1, 3). We will have sample mean as

sample mean #1=(6+8+4+1+7+3+5+0+1+3) /10=3.8;

Now we move forward to another set of 10 random digits of (1, 5, 5, 2, 9, 7, 2, 7, 6, 5). We will have the sample mean as

sample mean #2=(1+5+5+2+9+7+2+7+6+5) /10=4.9;

Repeat this procedure 4 times until you get sample mean #4.

(2)2.2 2.2 (7)2.9 2.9 3.0 3.0 3.0 3.0 3.0 (8)3.1 3.1 3.2 3.2 3.3 3.4 3.4 3.5 (12)3.6 3.6 3.7 3.8 3.8 3.8 3.8 3.8 3.8 3.8 3.9 4.0 (16)4.1 4.1 4.2 4.2 4.2 4.3 4.3 4.3 4.3 4.4 4.4 4.4 4.5 4.5 4.5 4.5 (10)4.6 4.7 4.7 4.7 4.8 4.9 4.9 4.9 4.9 4.9 (8)5.2 5.2 5.3 5.3 5.3 5.3 5.5 5.5 (4)5.9 6.0 6.0 6.0 (4)6.2 6.2 6.3 6.4 (1)6.8Q: Draw a histogram with classes as:


Class (2, 2.5] (2.5,3] (3, 3.5] (3.5, 4] (4, 4.5] (4.5, 5] (5, 5.5] (5.5, 6] (6, 6.5] (6.5, 7]

Counts

Sampling Distribution of sample mean of 10 random digitsClass (2, 2.5] (2.5,3] (3, 3.5] (3.5, 4] (4, 4.5] (4.5, 5] (5, 5.5] (5.5, 6] (6, 6.5] (6.5, 7]

Counts 2 7 8 12 16 10 8 4 4 1

Sampling distribution of “x bar”

Histogram of some sample

averages

Q: Write a journal about how to get the sampling distribution of Sample mean X-bar today, by answering the following questions:1)How to obtain X-bar’s from Table B for each student?2)How many X-bar’s did we have totally in the class?3)How to make a histogram for X-bar? What is the name of the histogram?4)What did the smooth curve represent?5)For the smooth curve, what did the horizontal axis and vertical axis present?

Pop-Up Quiz:

Q: How to get the sampling distribution of Sample mean X-bar, from our IN-class EX?1)How to obtain X-bar’s from Table B for each student?2)How many X-bar’s did we have totally in the class?3)How to make a histogram for X-bar? What is the name of the histogram?4)What did the smooth curve represent?5)For the smooth curve, what did the horizontal axis and vertical axis present?

= 6

0

8

3

74

29

5

16

Population

Sampling Distribution1st Sample

47

3

8

2

26

7 60

2nd Sample

25th Sample

51

9

8

0

69

3 41

Sample mean

= 4.5

= 4.6

Select 10 random digits from Table B

There is some variability in values of a statistic over different samples.

Population Distribution for 10 random digits

X 0 1 2 3 4 5 6 7 8 9

Prob 1/10 1/10 1/10 1/10 1/10 1/10 1/10 1/10 1/10 1/10

Population distribution of 0-9 random digits


(1) Select 10 random digits from Table B, and then take the sample mean;

(2) Repeat this process 25 times for each students Spring 2012.

(3) Make a histogram of sample mean’s from the class with 1098 X-bar’s. The probability distribution looks like a Normal distribution.

X

Sampling distribution of “x bar”


averages

The probability distribution of a statistic is called its sampling distribution.

For the histogram:Center of X-bar = 4.541SD of X-bar = 0.9

Min. 1st Qu. Median Mean 3rd Qu. Max. 1.400 3.600 4.400 4.451 5.400 7.800


08

374

29

5

16

Population

47

3

8

226

7 6 0

22

1

8

0

76

3 4 55

1

9

8

069

3 4 1

Center of X-bar = 4.5SD of X-bar = 0.9

For any population with mean and standard deviation :

The mean, or center of the sampling distribution of x bar, is equal to

the population mean

The standard deviation of the sampling distribution is /√n, where n

is the sample size :

Mean and standard deviation of a sample mean

Sample Mean’s are less

variable than individual

observations.

For normally distributed populationsWhen a variable in a population is normally distributed, the sampling

distribution of x bar for all possible samples of size n is also normally

distributed.

If the population is N()

then the sample means

distribution is N(/√n).Population

Sampling distribution

Exact N( , ) Exact N( , )

Not Exact Normal, but with Approximately N( , )

Mean and SD

Distribution of X, (n=1): Sampling distribution of , (n>1) :

,

X

n

n

(By Central Limit Theorem)

Standardize: Z-score of Reverse: ; *X Xn

XZ

n

17

Sampling distribution of a sample mean=distribution of XPopulation

Example: Soda Drink

Let X denote the actual volume of soda in a randomly selected can.

Suppose X~N(12oz, 0.4oz), 16 cans are to be selected.

a) The average volume is normally distributed with mean____ and standard deviation___.

b) Find the probability that the sample average is greater than 12.1 oz.

Mean of x-bar = 12;SD of x-bar = 0.1;P(Z>1) = 1-0.8413 = 0.1587.



distribution is N(/√n).

Exercise 5.21, page 310 Diabetes during pregnancy. A patient is classified as having

gestational diabetes if the glucose level is above 140 mg/dl one

hour after a sugary drink. Patient Sheila’s glucose level follows a

Normal distribution with mg/dlmg/dl.

(a) If a single glucose measurement is made, what is the probability

that Sheila is diagnosed as having gestational diabetes.

(b) If measurements are made instead on three separated days and

the mean result is compared with criterion 140 mg/dl, what is the

probability that Sheila is diagnosed as having gestational diabetes.

If the population is N() then

the sample means distribution

is N(/√n).

(b) n=3:If x is the mean of three measurements, then x-bar has aN(125, 10/√3 ) or N(125 mg/dl, 5.7735 mg/dl) distribution, and P(x > 140) = P(Z >2.60) = 0.0047.

(a) n=1:Let X be Sheila’s measured glucose level. (a) P(X > 140) = P(Z > 1.5) = 0.0668.



distribution is N(/√n).

For Normal distributed populations

Concern:What will happen when sample size gets bigger and bigger?

Review----Sampling Distribution of sample mean of 10 random digits

08

374

29

5

16

Population

47

3

8

226

7 6 0

22

1

8

0

76

3 4 55

1

9

8

069

3 4 1

Center of X-bar = 4.5SD of X-bar = 0.9

Central Limit Theorem (CLT)

Sampling distribution of

for n = 2 observations

Sampling distribution of

for n = 10 observations

Sampling distribution of for n = 25 observations

Population with strongly skewed

distribution

CLT says that:

Even if the population is

NOT Normal, but with

mean and SD when

sample size is large

enough, the sample means

distribution is N(/√n)

approximately.

For Non-Normal distributed populations

Concern:What will happen when sample size gets bigger and bigger?



Mean and SD


,

X

n

n



XZ

n

24


IQ scores: population vs. sampleIn a large population of adults, the mean IQ is 112 with standard deviation

20. Suppose 200 adults are randomly selected for a market research

campaign.

The distribution of the sample mean IQ is:

A) Exactly normal, mean 112, standard deviation 20

B) Approximately normal, mean 112, standard deviation 20

C) Approximately normal, mean 112 , standard deviation 1.414

D) Approximately normal, mean 112, standard deviation 0.1

C) Approximately normal, mean 112 , standard deviation 1.414

Population distribution : N(112; 20)

Sampling distribution for n = 200 is N(112; 1.414)

Examples #5.12, page 309

Songs on an iPod. An ipod has about 10,000 songs. The

distribution of the play time for these songs is highly skewed.

Assume that the standard deviation for the population is 280

seconds.

(a) What is the standard deviation of the average time when

you take an SRS of 10 songs from this population?

(b) How many songs would you need to sample if you wanted

the standard deviation of x-bar to be 15 seconds?

(b)In order to have σ/√n = 280/√n = 15 seconds, we need √n = 280/15 ~ 18.667, so n ~ (18.667)^2 = 348.5 — use n = 349.

(a)The standard deviation is σ/√10 = 280/√10 ~ 88.5438 seconds.

Example: children’s attitudes toward reading In the journal Knowledge Quest (Jan/Feb 2002), education

professors at the University of Southern California investigated

children’s attitudes toward reading. One study measured third

through sixth graders’ attitudes toward recreational reading on a

140-point scale. The mean score for this population of children

was 106 with a standard deviation of 16.4.

In a random sample of 36 children from this population,

a) what is the sampling distribution of x-bar?

b) find P(x<100).

Answer to Example 4

Z=-2.20

Probability=normalcdf(-E99, -2.20, 0, 1)=0.0139

follows Approximately N( , )=N(106, )= N(106, )

Standardize: Z-score of

16.42.7333

36

100 106 2.20

2.7333=

X

X

n

X

n

29

More Exercise on Chapter 5.1:

1. You were told that the weight of a new born baby follows normal distribution with mean 7 pounds and SD 0.5 pounds. The average weight of the next 16 new born in your local hospital is around ______, with SD _____.

what’s the prob that the average is between 7.2 and 7.5 pounds?2. The carbon monoxide in a certain brand of cigarette (in milligrams) follows

normal distribution with mean 12 and SD 1.8. For 40 randomly selected cigarettes, a) What is the sampling distribution of sample mean?b) Find the prob that the average carbon monoxide is between 10 and 13.

3. The amount of time that a drive-through bank teller spends on a customer follows normal distribution with mean 4 minutes and SD 1.5 minutes. For the next 50 customers, find the prob that the average time spent is more than 5 minutes

4. The rate of water usage per hour (in Thousands of gallons) by a community follows normal distribution with mean 5 and SD 2. For the next 30 hours,

a) What is the sampling distribution of sample mean?b) Find the probability that the average rate of usage per hour is less than 4?

EX: 5.7, 5.8, 5.18(a-c), 5.24, 5.21,5.12

Answer: 1. new SD=0.125, Z7.2=1.6, Z7.5=4, area=1-0.9452=0.05482. new SD=0.285, Z10=-7.02, Z13=3.5, area is almost 100%3. new SD=0.212, Z5=4.72, area is almost zero.4. new SD=0.365, Z4=-2.74, area=1-0.9452=0.0031.

Chapter 5.2

Sampling Distributions of

sample proportion p-hat

Review: Sampling proportion p-hatSample proportion: (p-hat, or relative frequency)

count in the samplep̂

Total

Population proportion: p

Reminder from Chapter 3: Sampling variabilityEach time we take a random sample from a population, we are likely to

get a different set of individuals and calculate a different statistic. This

is called sampling variability.

If we take a lot of random samples of the same size from a given

population, the variation from sample to sample—the sampling

distribution—will follow a predictable pattern.

Sampling Distribution of sample proportion of 10 random digits(1) Select 10 random digits from Table B, and then take the sample proportion of

EVEN numbers;

(2) Repeat this process 4 times for each student from Dr. Chen’s class.

More details with illustration:1. Based on Table B (random digit table), we randomly select a line, for example line 106 in

this case:

2. Take sample proportion of EVEN numbers of random digits of (6, 8, 4, 1, 7, 3, 5, 0, 1, 3). We will have sample proportion of EVEN #’s and gives

sample proportion #1 = 4/10=0.4;

Now we move forward to another set of 10 random digits of (1, 5, 5, 2, 9, 7, 2, 7, 6, 5), and we will have sample mean and gives

sample proportion #2 = 3 /10=0.3;

Repeat this procedure 4 times until you get sample proportion #4.

(1)0.1 (2)0.2 0.2(5)0.3 0.3 0.3 0.3 0.3 (21)0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 (17)0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5(17)0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 (7)0.7 0.7 0.7 0.7 0.7 0.7 0.7 (5)0.8 0.8 0.8 0.8 0.8 (1)0.9

Q: Draw a histogram with classes as: (for line 101-120 in Table B)


Class (0, 0.1] (0.1, 0.2] (0.2, 0.3] (0.3, 0.4] (0.4, 0.5] (0.5, 0.6] (0.6, 0.7] (0.7, 0.8] (0.8, 0.9]

Counts


Class (0, 0.1] (0.1, 0.2] (0.2, 0.3] (0.3, 0.4] (0.4, 0.5] (0.5, 0.6] (0.6, 0.7] (0.7, 0.8] (0.8, 0.9]

Counts 1 2 5 21 17 17 7 5 1

Sampling distribution of “p hat”


proportion

Q: Write a journal about how to get the sampling distribution of Sample proportion p-hat today, by answering the following questions:

1)How to obtain p-hat’s from Table B for each student?2)How many p-hat’s did we have totally in the class?3)How to make a histogram for p-hat? What is the name of the histogram?4)What did the smooth curve represent?5)For the smooth curve, what did the horizontal axis and vertical axis present?

0

8

3

74

29

5

16

Population

Sampling Distribution1st Sample

47

3

8

2

26

7 60

2nd Sample

25th Sample

51

9

8

0

69

3 41

Sample proportion

Select 10 random digits from Table B and find sample

proportion of even #

There is some variability in values of a statistic over different samples.

Sampling Distribution of sample proportion of even # of 10 random digits(1) Select 10 random digits from Table B, and then take the sample proportion of even #.

(2) Repeat this process a lot of times, say 10,000 times.

(3) Make a histogram of these 10,000 sample mean’s. The probability distribution looks like a Normal distribution.

Sampling distribution of “p-hat” Histogram

of some sample

proportion

The probability distribution of a statistic is called its sampling distribution.

Center of p-hat = 0.5018SD of p-hat = 0.1598Note: n=10.

(1 )p p

n

SD of p-hat =

Sampling distribution of the sample proportionThe sampling distribution of is never exactly normal. But as the sample size

increases, the sampling distribution of becomes approximately normal.

The normal approximation is most accurate for any fixed n when p is close to

0.5, and least accurate when p is near 0 or near 1.

p̂p̂

Sampling Distribution of If data are obtained from a SRS and np>10 and n(1-p)>10, then

the sampling distribution of has the following form:

For sample percentage: is approximately normal with mean p and standard deviation:

p̂

p̂

p̂

(1 )p p

n

p follows Approximately N( , )

Standardize: Z-score of p

(1 )p

(1 )

Reverse:

p

(1 );

*

p

p p

n

p p

n

p

p

p p

n

Z

40

Sampling distribution of a sample Proportion=distribution of p

Example 1Maureen Webster, who is running for mayor in a large city, claims that she is

favored by 53% of all eligible voters of that city. Assume that this claim is

true. In a random sample of 400 registered voters taken from this city.

Find Population proportion p= _________.

a.) What is the sampling distribution of p-hat?

b) What is the probability of getting a sample proportion less than 49% in which

will favor Maureen Webster?

c.) Find the probability of getting a sample proportion in between 50% and 55%.

(c) Z=(0.5-0.53)/0.02495 = -1.20;Z=(0.55-0.53)/0.02495 = 0.80;Pr(-1.20 <Z<0.80)=normalcdf(-1.20, 0.80, 0, 1)=0.673

(b) Z=(0.49-0.53)/0.02495 = -1.60Pr(Z<-1.60) =normalcdf(-E99, -1.6, 0, 1)= 0.0548

The Gallup Organization surveyed 1,252 debit cardholders in the

U.S. and found that 180 had used the debit card to purchase a

product or service on the Internet (Card Fax, November 12, 1999).

Suppose the true percent of debit cardholders in the U.S. that have

used their debit cards to purchase a product or service on the

Internet is 15%.

Calculate p hat (sample proportion ).

The sample proportion (p hat ) is approximately normal with mean

= ______ and standard deviation = ______.

Find the probability of getting a sample proportion smaller than

14.4%. ANS: Z=(0.144-0.15)/0.01=-0.6 Pr(Z<-0.6)= normalcdf(-E99, -0.6, 0, 1)

= 0.2743

Example 2

43

More Exercise on Chapter 5.2:1. 30% of all autos undergoing an emissions inspection at a city fail in

the inspection. Among 200 cars randomly selected in the city, the percentage of cars that fail in the inspection is around_____, with SD______. Find the prob that the percentage is between 31% and 35%.

2. 60% of all residents in a big city are Democrats. Among 400 residents randomly selected in the city, a) What is the sampling distribution of p-hat?b) Find Pr(sample percentage<58%)

3. In airport luggage screening it is known that 3% of people have questionable objects in their luggage. For the next 1600 people, use normal approximation to find the prob that at least 4% of the people have questionable objects.

4. It is known that 60% of mice inoculated with a serum are protected from a certain disease. If 80 mice are inoculated, a) What is the sampling distribution of p-hat?b) find the prob that at least 70% are protected from the disease.

HWQ: 5.22, 5.23(a,b) 5.73



Mean and SD


,

X

n

n



XZ

n

44


Sampling distribution of the sample proportionThe sampling distribution of is never exactly normal. But as the sample size

increases, the sampling distribution of becomes approximately normal.

The normal approximation is most accurate for any fixed n when p is close to

0.5, and least accurate when p is near 0 or near 1.

p̂p̂

Summary to Chapter 51. If X~N(µ, σ) exactly, thena) what is the mean of X-bar?b) what is SD of X-bar?c) what is the sampling distribution of X-bar? (You need to specify what the curve look like? What is the center/Mean? What is the Spread/SD? Is it EXACT, or Approximate by Central Limit Theorem.)

2. If X is NOT normal, but with population mean µ and population SD σ. When sample size is big enough, a) what is the mean of X-bar?b) what is SD of X-bar?c) what is the sampling distribution of X-bar? (You need to specify what the curve look like? What is the center/Mean? What is the Spread/SD? Is it EXACT, or Approximate by Central Limit Theorem.)

3. With population proportion p and sample size n,a) what is the mean of p-hat?b) what is SD of p-hat?c) what is the sampling distribution of p-hat? (You need to specify what the curve look like? What is the center/Mean? What is the Spread/SD? Is it EXACT, or Approximate by Central Limit Theorem.)

Summary to Chapter 5 (Popup Quiz)1. If X~N(µ, σ) exactly, thena) what is the mean of X-bar?b) what is SD of X-bar?c) what is the sampling distribution of X-bar? Is it EXACT, or Approximate by Central Limit Theorem?

2. If X is NOT normal, but with population mean µ and population SD σ. When sample size is big enough, a) what is the mean of X-bar?b) what is SD of X-bar?c) what is the sampling distribution of X-bar? Is it EXACT, or Approximate by Central Limit Theorem?

3. With population proportion p and sample size n,a) what is the mean of p-hat?b) what is SD of p-hat?c) what is the sampling distribution of p-hat? Is it EXACT, or Approximate by Central Limit Theorem?

Chapter 5 Sampling Distributions. Chapter 5.1 Sampling Distributions of sample mean X-bar.

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