Sampling Distributions

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• Chapter 7
• Sampling Distributions

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Introduction

• In real life calculating parameters of
  populations is prohibitive (difficult to find)
  because populations are very large.
• Rather than investigating the whole population,
  we take a sample, calculate a statistic related
  to the parameter of interest, and make an
  inference.
• The sampling distribution of the statistic is the
  tool that tells us how close is the statistic to
  the parameter.

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population
sample
Sampling Techniques
Parameters
Statistics
Statistical Procedures
inference
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Sampling

• Example
• A pollster is sure that the responses to his
  agree/disagree question will follow a binomial
  distribution, but p, the proportion of those who
  agree in the population, is unknown.
• An agronomist believes that the yield per acre of
  a variety of wheat is approximately normally
  distributed, but the mean m and the standard
  deviation s of the yields are unknown.
• If you want the sample to provide reliable
  information about the population, you must select
  your sample in a certain way!

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Types of Sampling Methods/Techniques
The sampling plan or experimental design
determines the amount of information you can
extract, and often allows you to measure the
reliability of your inference.
Sampling
Probability Samples
Non-Probability Samples
Simple Random
Stratified
Judgement
Chunk
Cluster
Systematic
Quota
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Simple Random Sampling

• Sampling Plan
• Simple random sampling is a method of sampling
  that allows each possible sample of size n an
  equal probability of being selected.

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Example

• There are 89 students in a statistics class. The
  instructor wants to choose 5 students to form a
  project group. How should he proceed?

1. Give each student a number from 01 to 89.
2. Choose 5 pairs of random digits from the random
   number table.
3. If a number between 90 and 00 is chosen, choose
   another number.
4. The five students with those numbers form the
   group.

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Other Sampling Techniques

• There are several other sampling plans that still
  involve randomization

• Stratified random sample Divide the population
  into subpopulations or strata and select a simple
  random sample from each strata.
• Cluster sample Divide the population into
  subgroups called clusters select a simple random
  sample of clusters and take a census of every
  element in the cluster.
• 1-in-k systematic sample Randomly select one of
  the first k elements in an ordered population,
  and then select every k-th element thereafter.

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Examples

• Divide West Malaysia into states and
• take a simple random sample within each state.
• Divide West Malaysia into states and take a
  simple random sample of 5 states.
• Divide a city into city blocks, choose a simple
  random sample of 10 city blocks, and interview
  all who
• live there.
• Choose an entry at random from the phone book,
  and select every 50th number thereafter.

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Non-Random Sampling Plans

• There are several other sampling plans that do
  not involve randomization. They should NOT be
  used for statistical inference!

• Convenience sample A sample that can be taken
  easily without random selection.
• People walking by on the street
• Judgment sample The sampler decides who will and
  wont be included in the sample.
• Quota sample The makeup of the sample must
  reflect the makeup of the population on some
  selected characteristic.
• Race, ethnic origin, gender, etc.

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Types of Samples

• Sampling can occur in two types of practical
  situations

• Observational studies The data existed before
  you decided to study it. Watch out for
• Nonresponse Are the responses biased because
  only opinionated people responded?
• Undercoverage Are certain segments of the
  population systematically excluded?
• Wording bias The question may be too complicated
  or poorly worded.

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Types of Samples

• Sampling can occur in two types of practical
  situations

• 2. Experimentation The data are generated by
  imposing an experimental condition or treatment
  on the experimental units.
• Hypothetical populations can make random sampling
  difficult if not impossible.
• Samples must sometimes be chosen so that the
  experimenter believes they are representative of
  the whole population.
• Samples must behave like random samples!

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Sampling Distributions

• Numerical descriptive measures calculated from
  the sample are called statistics.
• Statistics vary from sample to sample and hence
  are random variables.
• The probability distributions for statistics are
  called sampling distributions.
• In repeated sampling, they tell us what values of
  the statistics can occur and how often each value
  occurs.

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Sampling Distributions
Definition The sampling distribution of a
statistic is the probability distribution for the
possible values of the statistic that results
when random samples of size n are repeatedly
drawn from the population.
Each value of x-bar is equally likely, with
probability 1/4
Population 3, 5, 2, 1 Draw samples of size n 3
without replacement
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1. Sampling Distribution of the Mean

• A fair die is thrown infinitely many times, with
  the random variable X of spots on any throw.
  The probability distribution of X is
• and the mean and variance are

x 1 2 3 4 5 6
P(x) 1/6 1/6 1/6 1/6 1/6 1/6
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Sampling Distribution of Two Dice

• A sampling distribution is created by looking at
• all samples of size n2 (i.e. two dice) and their
  means
• While there are 36 possible samples of size 2,
  there are only 11 values for , and some (e.g.
  3.5) occur more frequently than others
  (e.g. 1).

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Sampling Distribution of Two Dice

• The sampling distribution of is shown below

6/36
5/36
4/36
3/36
2/36
1/36
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Example
Thrown two fair dice. Based on all possible
samples, the calculation of mean and standard
deviation can also be done as.
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Sampling Distribution of the Mean
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Sampling Distribution of the Mean
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Central Limit Theorem
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How Large is Large?
If the population is normal, then the sampling
distribution of will also be normal, no
matter what the sample size. When the population
is approximately symmetric, the distribution
becomes approximately normal for relatively small
values of n. When the population is skewed, the
sample size must be at least 30 before the
sampling distribution of becomes
approximately normal.
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The Sampling Distribution of the Sample Mean

• A random sample of size n is selected from a
  population with mean m and standard deviation s.
• The sampling distribution of the sample mean
  will have mean m and standard deviation
  .
• If the original population is normal, the
  sampling distribution will be normal for any
  sample size.
• If the original population is nonnormal, the
  sampling distribution will be normal when n is
  large.

The standard deviation of x-bar is sometimes
called the STANDARD ERROR (SE).
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Sampling Distribution of the Mean
Central Limit Theorem
Given population with and the
sampling distribution will have
Mean
Variance
Standard Deviation Standard Error (mean)
As n increases, the shape of the distribution
becomes normal (whatever the shape of the
population)
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Example

•  

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Finding Probabilities for the Sample Mean

• If the sampling distribution of is normal
  or approximately normal, standardize or rescale
  the interval of interest in terms of
• Find the appropriate area using Z Table.

Example A random sample of size n 16 from a
normal distribution with m 10 and s 8.
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Example
A soda filling machine is supposed to fill cans
of soda with 12 fluid ounces. Suppose that the
fills are actually normally distributed with a
mean of 12.1 oz and a standard deviation of 0.2
oz. What is the probability that the average fill
for a 6-pack of soda is less than 12 oz?
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Exercise

• The time that the laptops battery pack can
  function before recharging is needed is normally
  distributed with a mean of 6 hours and standard
  deviation of 1.8 hours. A random sample of 25
  laptops with a type of battery pack is selected
  and tested. What is the probability that the mean
  until recharging is needed is at least 7 hours?

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• The characteristics of the sampling distribution
  of a statistic
• The distribution of values is obtained by means
  of repeated sampling
• The samples are all of size n
• The samples are drawn from the same population

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2. Sampling Distribution of a Proportion
The proportion in the sample is denoted "p-hat"
The proportion in the population (parameter) is
denoted p
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Types of response variables
Quantitative
Sums
Averages
Response type
Categorical
Counts
Proportions
Prior chapters have focused on quantitative
response variables. We now focus on categorical
response variables.
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The Sampling Distribution of the Sample Proportion
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Approximating Normal from the Binomial

• Under certain conditions, a binomial random
  variable has a distribution that is approximately
  normal.
• When n is large, and p is not too close to zero
  or one, areas under the normal curve with mean
  np and standard deviation can
  be used to approximate binomial probabilities.
• Make sure that np and n(1-p) are both greater
  than 5 to avoid inaccurate approximations!

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The Sampling Distribution of the Sample Proportion
The standard deviation of p-hat is sometimes
called the STANDARD ERROR (SE) of p-hat.
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Finding Probabilities for the Sample Proportion

• If the sampling distribution of is normal
  or approximately normal, standardize or rescale
  the interval of interest in terms of
• Find the appropriate area using Z Table.

If both np gt 5 and np(1-p) gt 5
Example A random sample of size n 100 from a
binomial population with p 0.4.
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Example
The soda bottler in the previous example claims
that only 5 of the soda cans are underfilled.
A quality control technician randomly samples
200 cans of soda. What is the probability that
more than 10 of the cans are underfilled?
n 200 U underfilled can p P(U) 0.05 q
0.95 np 10 nq 190
This would be very unusual, if indeed p .05!
OK to use the normal approximation
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Example

•  

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Sampling Distribution of the Difference Between
Two Averages

• Theorem If independent sample of size n1 and n2
  are drawn at random from two populations,with
  means ?1 and ?2 and variances ?12 and ?22,
  respectively, then the sampling distribution of
  the differences of means, is
  approximately normally distributed with mean and
  variance given by

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Example

• Starting salaries for MBA grads at two
  universities are normally distributed with the
  following means and standard deviations. Samples
  from each school are taken

University 1 University 2
Mean RM 62,000 /yr RM 60,000 /yr
Std. Dev. RM14,500 /yr RM18,300 /yr
sample size, n 50 60

• What is the sampling distribution of
• What is the probability that a sample mean of U1
  students will exceed the sample mean of U2
  students?

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Example

• mean
• 
  62,000 60,000 2000
• and standard deviation
• 3128.3

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Sampling Distribution of the Difference Between
Two Sample Proportions

• Theorem If independent sample of size n1 and n2
  are drawn at random from two populations, where
  the proportions of obs with the characteristic of
  interest in the two populations are p1 and p2
  respectively, then the sampling distribution of
  the differences between sample proportions,
  is approximately normally distributed
  with mean and variance given by

 
and
Then
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Sampling Distribution of the Difference Between
Two Sample Proportions

• Example It is known that 16 of the households
  in Community A and 11 of the households in
  Community B have internets in their houses. If
  200 households and 225 households are selected at
  random from Community A and Community B
  respectively, compute the probability of
  observing the difference between the two sample
  proportions at least 0.10?

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Sampling Distribution of S2

• Theorem If S2 is the variance of a random
  sample of size n taken from a normal population
  having the variance ?2, then the statistic
• has a chi-squared distribution with v n -1
  degrees of freedom.