Econ 3790: Business and Economics Statistics

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Econ 3790 Business and Economics Statistics

• Instructor Yogesh Uppal
• Email yuppal_at_ysu.edu

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Normal Probability Distribution

• Characteristics

Probabilities for the normal random variable
are given by areas under the curve. The total
area under the curve is 1 (.5 to the left of the
mean and .5 to the right).
.5
.5
x
Mean m
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How to find probabilities of a random variable
(x) which has a normal distribution.

• Convert the x values into the z scores or more
  formally, standardize x.
• After the conversion, we can use the z-scores to
  find probabilities from a table (called table of
  standard normal probabilities).

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Standardizing the Normal Values or the z-scores

• Z-scores can be calculated as follows

• We can think of z as a measure of the number of
• standard deviations x is from ?.

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Standard Normal Probability Distribution
A standard normal distribution is a normal
distribution with mean of 0 and variance of 1.
If x has a normal distribution with mean (µ) and
Variance (s), then z is said to have a standard
normal distribution.
s 1
z
0
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Characteristics of Standard Normal Distribution

• It is a type of the normal distribution.
• Its mean is zero and variance is one.
• Z-values on the left side of the mean are
  negative and right side of the mean are positive.
• Important point is what symmetry means in this
  kind of distribution?
• How do you interpret the values in the Standard
  Normal Table?

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Example Air Quality

• I collected this data on the air quality of
  various cities as measured by particulate matter
  index (PMI). A PMI of less than 50 is said to
  represent good air quality.
• The data is available on the class website.
• Suppose the distribution of PMI is approximately
  normal.

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Example Air Quality

• Suppose I want to find out the probability of air
  quality being good?
• What is the probability that PMI is greater than
  80?
• What is the probability that PMI is with 2
  standard deviations from the mean?

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Computing x from a given z-score

• Suppose I tell you that in our air quality
  example, the probability is 40 that standardized
  value of the PMI is between -z and z.
• What are the corresponding x values?

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Chapter 7, Part ASampling and Sampling
Distributions

• Simple Random Sampling

• Point Estimation

• Introduction to Sampling Distributions

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Statistical Inference
The sample results provide only estimates of
the values of the population characteristics.
With proper sampling methods, the sample
results can provide good estimates of the
population characteristics.
A parameter is a numerical characteristic of a
population.
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Statistical Inference
The purpose of statistical inference is to
obtain information about a population from
information contained in a sample.
A population is the set of all the elements of
interest.
A sample is a subset of the population.
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Simple Random Sampling Finite Population

• A simple random sample of size n from a finite
• population of size N is a sample selected
  such
• that each possible sample of size n has the
  same
• probability of being selected.

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Simple Random SamplingFinite Population

• Replacing each sampled element before
  selecting
• subsequent elements is called sampling with
• replacement.

• Sampling without replacement is the procedure
• used most often.

• In large sampling projects, computer-generated
• random numbers are often used to automate
  the
• sample selection process.

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Point Estimation
In point estimation we use the data from the
sample to compute a value of a sample statistic
that serves as an estimate of a population
parameter.
s is the point estimator of the population
standard deviation ?.
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Sampling Error

• When the expected value of a point estimator
  is equal
• to the population parameter, the point
  estimator is said
• to be unbiased.

• The absolute value of the difference between
  an
• unbiased point estimate and the
  corresponding
• population parameter is called the sampling
  error.

• Sampling error is the result of using a subset
  of the
• population (the sample), and not the entire
• population.

• Statistical methods can be used to make
  probability
• statements about the size of the sampling
  error.

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Sampling Error

• The sampling errors are

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Air Quality Example

• Let us suppose that the population of air quality
  data consists of 191 observations.
• How would you determine the following population
  parameters mean, standard deviation, proportion
  of cities with good air quality.

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Air Quality Example

• How about picking a random sample from this
  population representing the air quality?
• We shall use SPSS to do this random sampling for
  us.
• How would you use this sample to provide point
  estimates of the population parameters?

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Summary of Point Estimates Obtained from a Simple
Random Sample
Population Parameter
Point Estimator
Point Estimate
Parameter Value
m Population mean SAT score
40.9
.
20.5
s Sample std. deviation for SAT
score
..
s Population std. deviation for
SAT score
.62
.
p Population pro- portion
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• Process of Statistical Inference

A simple random sample of n elements is
selected from the population.
Population with mean m ?
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Sampling Distribution of
Expected Value of
where ? the population mean
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Sampling Distribution of
Standard Deviation of
Finite Population
Infinite Population

• A finite population is treated as being
• infinite if n/N lt .05.

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The Shape of Sampling Distribution of

• If the shape of the distribution of x in the
  population is normal, the shape of the sampling
  distribution of is normal as well.
• If the shape of the distribution of x in the
  population is approximately normal, the shape of
  the sampling distribution of is approximately
  normal as well.
• If the shape of the population is not
  approximately normal then
• If n is small, the shape of the sampling
  distribution of is unpredictable.
• If n is large (n 30), the shape of the sampling
  distribution of can be assumed to be
  approximately normal.

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Sampling Distribution of for the air quality
example when the population is (almost) infinite
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Sampling Distribution of for the air quality
example when the population is finite
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Sampling Distribution of

• If we use a large random sample (ngt30), then the
  sampling distribution of can be approximated
  by the normal distribution.
• If the sample is small, then the sampling
  distribution of can be normal only if we
  assume that our population has a normal
  distribution.

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Sampling Distribution of for the air
quality Index when n 5.

• What is the probability that a simple random
  sample of 5 applicants will provide an estimate
  of the population mean air quality index that is
  within /-2 of the actual population mean, µ?
• In other words, what is the probability that
  will be between 38.9 and 42.9?

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Sampling Distribution of for the air
quality Index when n 100.

• What is the probability that a simple random
  sample of 100 applicants will provide an estimate
  of the population mean air quality index that is
  within /-2 of the actual population mean, µ?

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Chapter 7, Part BSampling and Sampling
Distributions
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• Making Inferences about a Population Proportion

A simple random sample of n elements is
selected from the population.
Population with proportion p ?
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Expected Value of
where p the population proportion
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Sampling Distribution of
Standard Deviation of
Finite Population
Infinite Population
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Form of Sampling Distribution of
np gt 5
n(1 p) gt 5
and
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Chapter 8 Interval Estimation

• Population Mean s Known

• Population Mean s Unknown

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Margin of Error and the Interval Estimate
A point estimator cannot be expected to provide
the exact value of the population parameter.
An interval estimate can be computed by adding
and subtracting a margin of error to the point
estimate.
Point Estimate /- Margin of Error
The purpose of an interval estimate is to
provide information about how close the point
estimate is to the value of the parameter.
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Margin of Error and the Interval Estimate
The general form of an interval estimate of a
population mean is

• In order to develop an interval estimate of a
  population mean, the margin of error must be
  computed using either
• the population standard deviation s , or
• the sample standard deviation s
• These are also Confidence Interval.

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Interval Estimate of a Population Mean s Known

• Interval Estimate of m

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Interval Estimation of a Population Means Known

• There is a 1 - ? probability that the value of a
• sample mean will provide a margin of error of
• or less.

?/2
?/2
?
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Summary of Point Estimates Obtained from a Simple
Random Sample
Population Parameter
Point Estimator
Point Estimate
Parameter Value
m Population mean
40.9
20.5
s Sample std. deviation
.
s Population std. deviation
.62
p Population pro- portion
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Example Air Quality

• Consider our air quality example. Suppose the
  population is approximately normal with µ 40.9
  and s 20.5. This is s known case.
• If you guys remember, we picked a sample of size
  5 (n 5).
• Given all this information, What is the margin of
  error at 95 confidence level?

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Example Air Quality

• What is the margin of error at 95 confidence
  level.
• We can say with 95 confidence that population
  mean (µ) is between 18 of the sample mean.
• With 95 confidence, µ is between . and ...

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Interval Estimation of a Population Means
Unknown

• If an estimate of the population standard
  deviation s cannot be developed prior to
  sampling, we use the sample standard deviation s
  to estimate s .

• This is the s unknown case.

• In this case, the interval estimate for m is
  based on the t distribution.

• (Well assume for now that the population is
  normally distributed.)

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t Distribution
The t distribution is a family of similar
probability distributions.
A specific t distribution depends on a
parameter known as the degrees of freedom.
Degrees of freedom refer to the number of
independent pieces of information that go into
the computation of s.
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t Distribution
A t distribution with more degrees of freedom
has less dispersion.
As the number of degrees of freedom increases,
the difference between the t distribution and
the standard normal probability distribution
becomes smaller and smaller.
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t Distribution
t distribution (20 degrees of freedom)
Standard normal distribution
t distribution (10 degrees of freedom)
z, t
0
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t Distribution
For more than 100 degrees of freedom, the
standard normal z value provides a good
approximation to the t value.
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t Distribution
Standard normal z values
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Interval Estimation of a Population Means
Unknown

• Interval Estimate

where 1 -? the confidence coefficient
t?/2 the t value providing an
area of ?/2 in the upper tail
of a t distribution with n - 1
degrees of freedom s the sample
standard deviation
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Example Air quality when s is unknown

• Now suppose that you did not know what s is. You
  can estimate using the sample and then use
  t-distribution to find the margin of error.
• What is 95 confidence interval in this case? The
  sample size n 5. So, the degrees of freedom for
  the t-distribution is 4. The level of
  significance ( ) is 0.05. s

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Summary of Interval Estimation Procedures for a
Population Mean
Can the population standard deviation s be
assumed known ?
Yes
No
Use the sample standard deviation s to estimate s
s Known Case
Use
Use
s Unknown Case
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Interval Estimationof a Population Proportion
The general form of an interval estimate of a
population proportion is
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Interval Estimation of a Population Proportion

• Interval Estimate