Sampling Distributions

1
Chapter 5

• Sampling Distributions

2
Chapter 5.1

• Sampling Distributions of sample mean X-bar

3
Review Chapter 1.3, Normal distribution
An important property of a density curve is that
areas under the curve correspond to relative
frequencies
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)?
4
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 cant usually assess
  directly.
• A parameter is a number describing a
  characteristic of the population.

Population
Sample
5
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?
6
Simple random sample (SRS)
Data are summarized by statistics (mean, standard
deviation, median, quartiles, correlation, etc..)
7
Sampling Distribution of sample mean of 10 random
digits

• Select 10 random digits from Table B, and then
  take the sample mean
• Repeat this process 4 times for each student from
  Dr. Chens 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(6841735013) /103.8
• 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(1552972765) /104.9
• Repeat this procedure 4 times until you get
  sample mean 4.

8
Sampling Distribution of sample mean of 10 random
digits
(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)6.8 Q 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
9
Sampling Distribution of sample mean of 10 random
digits
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 2 7 8 12 16 10 8 4 4 1

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

10
Pop-Up Quiz

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

11
Sampling Distribution
1st Sample
Sample mean
 
Select 10 random digits from Table B
 
6
2nd Sample
4.5
 
25th Sample
4.6
 
There is some variability in values of a
statistic over different samples.
12
Population Distribution for 10 random digits
Population distribution of 0-9 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
 
13
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 25 times for each students
   Spring 2012.
3. Make a histogram of sample means from the class
   with 1098 X-bars. The probability distribution
   looks like a Normal distribution.

The probability distribution of a statistic is
called its sampling distribution.
For the histogram Center of X-bar 4.541 SD 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
14
Sampling Distribution of sample mean of 10 random
digits
 
 
2
8
7
6
2
5
4
1
3
0
 
 
 
Center of X-bar 4.5 SD of X-bar 0.9
 
15
Mean and standard deviation of a sample mean
Sample Means are less variable than individual
observations.
16
For normally distributed populations

• When 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.

Sampling distribution
If the population is N(m, s) then the sample
means distribution is N(m, s/vn).
Population
17
Sampling distribution of a sample
meandistribution of
Population
18
Example Soda Drink

• Let X denote the actual volume of soda in a
  randomly selected can.
• Suppose XN(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.

If the population is N(m, s) then the sample
means distribution is N(m, s/vn).
Mean of x-bar 12 SD of x-bar 0.1 P(Zgt1)
1-0.8413 0.1587.
19
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 Sheilas glucose level
  follows a Normal distribution with m125 mg/dl,
  s10 mg/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(m, s) then the sample
means distribution is N(m, s/vn).
(b) n3 If x is the mean of three measurements,
then x-bar has a N(125, 10/v3 ) or N(125 mg/dl,
5.7735 mg/dl) distribution, and P(x gt 140) P(Z
gt2.60) 0.0047.
(a) n1 Let X be Sheilas measured glucose
level. (a) P(X gt 140) P(Z gt 1.5) 0.0668.
20
For Normal distributed populations
If the population is N(m, s) then the sample
means distribution is N(m, s/vn).
21
Review----Sampling Distribution of sample mean of
10 random digits
 
 
2
8
7
6
2
5
4
1
3
0
 
 
 
22
Central Limit Theorem (CLT)
23
For Non-Normal distributed populations
CLT says that Even if the population is NOT
Normal, but with mean m and SD s, when sample
size is large enough, the sample means
distribution is N(m, s/vn) approximately.
24
Sampling distribution of a sample
meandistribution of
Population
25
IQ scores population vs. sample

• In 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)
26
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 s/vn 280/vn 15 seconds,
we need vn 280/15 18.667, so n (18.667)2
348.5 use n 349.
(a) The standard deviation is s/v10 280/v10
88.5438 seconds.
27
Example childrens attitudes toward reading

• In the journal Knowledge Quest (Jan/Feb 2002),
  education professors at the University of
  Southern California investigated childrens
  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(?xlt100).

28
Answer to Example 4

• Z-2.20
• Probabilitynormalcdf(-E99, -2.20, 0, 1)0.0139

 
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 _____.
• whats 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,
• What is the sampling distribution of sample mean?
• Find the probability that the average rate of
  usage per hour is less than 4?

Answer 1. new SD0.125, Z7.21.6, Z7.54,
area1-0.94520.0548 2. new SD0.285, Z10-7.02,
Z133.5, area is almost 100 3. new SD0.212,
Z54.72, area is almost zero. 4. new SD0.365,
Z4-2.74, area1-0.94520.0031.
EX 5.7, 5.8, 5.18(a-c), 5.24, 5.21,5.12
30
Chapter 5.2

• Sampling Distributions of sample proportion p-hat

31
Review Sampling proportion p-hat

• Sample proportion (p-hat, or relative frequency)

Population proportion
32
Reminder from Chapter 3 Sampling variability

• Each 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 samplethe sampling distributionwill
  follow a predictable pattern.

33
Sampling Distribution of sample proportion of 10
random digits

• Select 10 random digits from Table B, and then
  take the sample proportion of EVEN numbers
• Repeat this process 4 times for each student from
  Dr. Chens 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 4/100.4
• 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 /100.3
• Repeat this procedure 4 times until you get
  sample proportion 4.

34
Sampling Distribution of sample mean of 10 random
digits
(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
35
Sampling Distribution of sample mean of 10 random
digits
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

• Q Write a journal about how to get the sampling
  distribution of Sample proportion p-hat today, by
  answering the following questions
• How to obtain p-hats from Table B for each
  student?
• How many p-hats did we have totally in the
  class?
• How to make a histogram for p-hat? What is the
  name of the histogram?
• What did the smooth curve represent?
• For the smooth curve, what did the horizontal
  axis and vertical axis present?

36
Sampling Distribution
1st Sample
Sample proportion
 
Select 10 random digits from Table B and find
sample proportion of even
 
 
2nd Sample
 
25th Sample
 
There is some variability in values of a
statistic over different samples.
37
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 means.
   The probability distribution looks like a Normal
   distribution.

The probability distribution of a statistic is
called its sampling distribution.
 
38
Sampling distribution of the sample proportion

• The 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.

39
Sampling Distribution of

• If data are obtained from a SRS and npgt10 and
  n(1-p)gt10, then the sampling distribution of
  has the following form
• For sample percentage
• is approximately normal with mean p and
• standard deviation

40
Sampling distribution of a sample
Proportiondistribution of
41
Example 1

• Maureen 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 ltZlt0.80) normalcdf(-1.20,
0.80, 0, 1) 0.673
(b) Z(0.49-0.53)/0.02495 -1.60 Pr(Zlt-1.60)
normalcdf(-E99, -1.6, 0, 1) 0.0548
42
Example 2

• 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(Zlt-0.6) normalcdf(-E99, -0.6, 0, 1) 0.2743
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 percentagelt58)
• 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
44
Sampling distribution of a sample
meandistribution of
Population
45
Sampling distribution of the sample proportion

• The 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.

46
Summary to Chapter 5

• 1. If XN(µ, s) exactly, then
• a) what is the mean of X-bar?
• b) what is SD of X-bar?
• c) what is the sampling distribution of X-bar?
• 2. If X is NOT normal, but with population mean µ
  and population SD s. 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?
• 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?