1. Econ 3790: Business and Economics Statistics
Instructor: Yogesh Uppal

2. 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

3. 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).

4. 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 .

5. 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 (σ), then z is said to have a standard normal distribution.
s = 1 z

6. 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?

7. 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.

8. 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?

9. 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?

10. Chapter 7, Part A Sampling and Sampling Distributions
Simple Random Sampling
Point Estimation
Introduction to Sampling Distributions
Sampling Distribution of

11. 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.

12. 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.

13. 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.

14. Simple Random Sampling: Finite 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.

15. 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.
We refer to as the point estimator of the population
mean .
s is the point estimator of the population standard
deviation .
is the point estimator of the population proportion p.

16. 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.

17. Sampling Error The sampling errors are: for sample mean
for sample standard deviation
for sample proportion

18. 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.

19. 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?

20. Summary of Point Estimates Obtained from a Simple Random Sample
Population
Parameter
Parameter
Value
Point
Estimator
Point
Estimate
m = Population mean
SAT score
40.9
= Sample mean
SAT score ….
s = Population std.
deviation for
SAT score
20.5
s = Sample std.
deviation for
SAT score
…..
p = Population pro-
portion
.62
= Sample pro-
portion wanting
campus housing ….

21. of n elements is selected
Process of Statistical Inference
Population
with mean
m = ?
A simple random sample
of n elements is selected
from the population.
The value of is used to
make inferences about
the value of m.
The sample data
provide a value for
the sample mean .

22. Sampling Distribution of
The sampling distribution of is the probability
distribution of all possible values of the sample
mean .
Expected Value of
E( ) = 
where:
 = the population mean

23. Sampling Distribution of
Standard Deviation of
Finite Population
Infinite Population
A finite population is treated as being
infinite if n/N < .05.
is the finite correction factor.
is also referred to as the standard error of the mean.

24. 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.

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

26. Sampling Distribution of for the air quality example when the population is finite

27. Relationship Between the Sample Size
and the Sampling Distribution of
E( ) = m regardless of the sample size. In our
example, E( ) remains at 40.9.
Whenever the sample size is increased, the standard
error of the mean is decreased. With the increase
in the sample size to n = 100, the standard error of the
mean decreases.

28. Relationship Between the Sample Size
and the Sampling Distribution of

29. Sampling Distribution of
If we use a large random sample (n>30), 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.

30. 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?

31. 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?

32. Relationship Between the Sample Size
and the Sampling Distribution of
Because the sampling distribution with n = 100 has a
smaller standard error, the values of have less
variability and tend to be closer to the population
mean than the values of with n = 5.
Basically, a given interval with smaller standard error
(larger n) will cover more area under the normal curve
than the same interval with larger standard error (smaller n).

33. Chapter 7, Part B Sampling and Sampling Distributions
Sampling Distribution of

34. of n elements is selected
Sampling Distribution of
Making Inferences about a Population Proportion
Population
with proportion
p = ?
A simple random sample
of n elements is selected
from the population.
The value of is used
to make inferences
about the value of p.
The sample data
provide a value for the
sample proportion .

35. Sampling Distribution of
The sampling distribution of is the probability
distribution of all possible values of the sample
proportion .
Expected Value of
where:
p = the population proportion

36. Sampling Distribution of
Standard Deviation of
Finite Population
Infinite Population
is referred to as the standard error of the
proportion.

37. Form of Sampling Distribution of
The sampling distribution of can be approximated
by a normal distribution whenever the sample size
is large: Central Limit Theorem (CLT).
The sample size is considered large whenever these
conditions are satisfied:
np > 5
and
n(1 – p) > 5

38. Chapter 8: Interval Estimation
Population Mean: s Known
Population Mean: s Unknown

39. 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.

40. 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.

41. Interval Estimate of a Population Mean: s Known
Interval Estimate of m
where: is the sample mean
1 - is the confidence coefficient
z/2 is the z value providing an area of
/2 in the upper tail of the standard
normal probability distribution
s is the population standard deviation
n is the sample size

42. Interval Estimation of a Population Mean: s Known
There is a 1 -  probability that the value of a
sample mean will provide a margin of error of
or less.
Sampling
distribution of
1 -  of all
values
/2
/2 

43. Summary of Point Estimates Obtained from a Simple Random Sample
Population
Parameter
Parameter
Value
Point
Estimator
Point
Estimate
m = Population mean
40.9
= Sample mean
s = Population std.
deviation
20.5
s = Sample std.
deviation
…….
p = Population pro-
portion
.62
= Sample pro-
portion

44. Example: Air Quality
Consider our air quality example. Suppose the population is approximately normal with μ = 40.9 and σ = This is σ 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?

45. 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 …...

46. Interval Estimation of a Population Mean:s 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.
(We’ll assume for now that the population is normally distributed.)

47. 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.

48. 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.

49. t Distribution t distribution (20 degrees Standard of freedom) normal
z, t

50. t Distribution For more than 100 degrees of freedom, the standard
normal z value provides a good approximation to
the t value.
The standard normal z values can be found in the
infinite degrees ( ) row of the t distribution table.

51. t Distribution
Standard normal
z values

52. Interval Estimation of a Population Mean: s 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

53. Example: Air quality when σ is unknown
Now suppose that you did not know what σ 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 s = ……

54. 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 σ
s Known
Case
Use
Use
s Unknown
Case

55. Interval Estimation of a Population Proportion
The general form of an interval estimate of a
population proportion is

56. Interval Estimation of a Population Proportion
Interval Estimate
where:  is the confidence coefficient
z/2 is the z value providing an area of
/2 in the upper tail of the standard
normal probability distribution
is the sample proportion