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# 322-6-2 Statistical Inference

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## M Karimi

on 1 February 2018

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#### Transcript of 322-6-2 Statistical Inference

Data Analytics
Business Analytics
Descriptive Analytics
Predictive Analytics
Prescriptive Analytics
© 2016 Cengage
Descriptive Statistics
Data Visualization

Statistical Inference
(Statistical) Inference
Inference:
a conclusion reached on the basis of evidence and reasoning.
Statistical Inference:
the process of deducing properties of an underlying probability distribution by analysis of
data
.

Inferential statistical analysis infers properties about a population: this includes
testing hypotheses
and
deriving estimates
.
Deriving Estimates
Testing Hypothesis
More Sophisticated
But how do we select a sample?
Senario 1:
The director of personnel for Electronics Associates, Inc. has been assigned the task of developing a prole of the company’s 2,500 employees. The characteristics to be identied include the mean annual salary proportion of employees having completed the company’s management training program.
Senario 2:
Consider the population of customers arriving at a McDonald’s. An employee is asked to select and interview a sample of customers in order to develop a prole of customers who visit the restaurant.
Step 1: Assign a random number to each element of the population.
Step 2: Select the n elements corresponding to the n smallest random numbers.
McDonald's Solution:
The sampling procedure was based on the fact that some customers presented discount coupons. Whenever a customer presented a discount coupon, the next customer served was asked to complete a customer prole questionnaire.
Mean
Interval Estimation
Proportion
Electronics Associates, Inc. (EAI) Data
Example:
Sample
Not Very Accurate?!
What if we select another sample?
Let's select 500 more samples of 30
Is this normal? (pun intended)
Does this only apply to sample mean?
sample mean is a random variable!
so it has has an
expected value
(mean),
a
standard deviation
, and a
characteristic shape
so, what about the shape?
Population has a normal distribution
Population does not have a normal distribution
When the population has a normal distribution, the sampling distribution of x is normally distributed for any sample size.
When the population does not have a normal distribution, the sampling distribution of x is normally distributed for large sample sizes.
General statistical practice is to assume that, for most applications, the sampling distribution of x can be approximated by a normal distribution whenever the sample size is 30 or more. In cases in which the population is highly skewed or outliers are present, sample sizes of 50 may be needed.
How many is large?
So does the
sample size matters?
The answer is yes, but in varieance, not mean!
Suppose that
in the EAI sampling problem we select a simple random sample of 100 EAI employees instead of the 30 originally considered.
sample proportion is a random variable!
so it has has an
expected value
(mean),
a
standard deviation
, and a
characteristic shape
How about the characteristic shape?
For a simple random sample from a large population, p is a binomial random variable indicating the number of elements in the sample with the characteristic of interest.
Does the sample size matter?
Provide information about how close the point
estimate is to the value of the population parameter.
Point Estimate
Let's put everything in one place!
what is the problem with
this estimation?
We know
We do not know
This introduces an additional source of uncertainty
t-Distribution
How do we deal with this uncertainty?
which distribution does represent less certainty?
degrees of freedom= n-1
Example: in our EAI example, n=30.
Thus we have 29 degrees of freedom
Let's put everything in one place
again
!
Before
After
In Class Exercise
a study designed to estimate the mean credit card debt for the population of U.S. households. A sample of 70 households provided the credit card balances shown in Table 6.5.
Data: NewBalance
Compute an interval estimate of the population mean with 95%
confidence interval
Why 90%
a hypothesis is an
assumption
we make about a population parameter such as any quantity or measurement about this population that is fixed and that we can use it as a value to a distribution variable. Typical examples of parameters are the
mean
and the
variance
.
How many emils should I send out before someone signs up for our service?
Example:
10?
5?
What if we instead ask: What is the
mean
number of e-mails that we need to send before someone signs up to our product? We can define the population here as the recipients of our offer.
Null Hypothesis
Alternative Hypothesis
H
H
0
a
a tentative conjecture
about a population parameter
the exact opposite of the tenative conjecture
counter claim
claim
hard to state
easy to state
hard to reject
easy to reject
Research Hypothesis
so... which one to begin with?
Start with
H
0
Start with
Several new fuel injection units will be manufactured, installed in test automobiles, and subjected to research-controlled driving conditions. The new system provides more than 24 miles per gallon.
Example:
Hypothesis Test
The label on a soft drink bottle states that it contains 67.6 fluid ounces.
Example:
Hypothesis Test(s)
Challenging the Null hypothesis
H
a
Summary
one-tailed
one-tailed
two-tailed
Hypothesis Test of the Population Mean
Example
The label on a large can of Hilltop Coffee states that the can contains 3 pounds of coffee.
The FTC (Federal Trade Commission) interprets the label information on a large can of coffee as a claim by Hilltop that the population mean filling weight is at least 3 pounds per can.
Hypothesis
a sample of 36 cans of coffee is selecte.
what does an average of 2.92 mean?
how much less than 3 pounds is
significance
?
probability of making a Type I error by rejecting the null hypothesis by mistake
test-statistics
t = ?
How small must the test statistic t be before we choose to reject the null hypothesis?
Summary of one-tailed test
Example
Holiday Toys manufactures and distributes its products through more than 1,000 retail outlets.
Holiday’s marketing director is expecting demand to average 40 units per retail outlet
Holiday decided to survey a sample of 25 retailers to gather more information.
Hypothesis
The sample of 25 retailers provided a mean of 37.4
The sample has a standard deviaiton of 11.79
p-value is a probability used to determine whether the null hypothesis should be rejected
For a two-tailed test, values of the test statistic in either tail provide evidence against the null hypothesis.
Summary of two-tailed test
Summary
Hypothesis Test of the Population Proportion
one-tailed
one-tailed
two-tailed
Example
Pine Creek implemented a special promotion designed to attract women golfers.
Over the past year, 20% of the players at Pine Creek were women.
One month after the promotion was implemented, the course manager requested a statistical study to determine whether the proportion of women players at Pine Creek had increased.
random sample of 400 players was selected.
how much more than 0.25 is
significance
?
test-statistics:
Summary of one-tailed and two-tailed test (proportion)
what does sample proportion of 0.25 mean?
Lower tail
Upper tail
Excel
sample proportion is 0.25
Full transcript