Quantitative Methods:
Module 8: Statistical Inference
Introduction
Statistical inference is the use of sample data to predict, or infer, further pieces of information about the population form which the sample or samples came,
Forms of Inference:
|
Estimation: |
A sample is the basis for predicting the values of
population |
|
|
|
|
Significance
Testing |
Procedure for deciding whether sample evidence supports or
rejects a hypothesis ( a statement about a population) |
Application of Statistical Inference
Statistical Inference has two main parts:
|
Market
research: |
A sample of consumers and then generalizing the results to
the whole of the potential market.. |
|
|
|
|
Medicine |
Usually given to a sample of sufferer’s and their progress
compared to that of patients not given the treatment. .A significant test helps to decide
whether the results are significant enough to support a new treatment as
beneficial |
Confidence Levels
Ø From the
variability of data within the sample, it is possible to calculate what is in
effect the probability that conclusions are correct.
Ø A
confidence level attached to a statement is in effect the probability that the statement
is true.
Ø By
convention, 95% is the usual confidence level. One in twenty (95%) is generally
regarded as an acceptable risk.
Sampling Distribution of the Mean
The sampling distribution of the mean has particular
predictable characteristics:
1) After several random samples are taken, and means
recorded, a distribution of sample means is built up.
2) The new distribution will be normal, have the same mean
as the original, and will be narrower. (Narrowness is intuitive--it’s due to
the tendency for outliers to be averaged out by those of more moderate value.)
3) Extent of narrowing is ‘Standard deviation / 
Relationship Between Sampling Distribution of the Mean, and
Original distribution
|
|
Individual
Distribution |
Sampling
Distribution |
|
Shape |
Normal |
Normal |
|
|
|
|
|
Mean |
|
|
|
Standard Deviation |
S |
__________ s/Ö Sample
size |
Relationship Between Sampling Distribution of the Mean, and
Original distribution (Case in which Original Distribution is Non-Normal)
|
|
Individual
Distribution |
Sampling
Distribution |
|
Shape |
Non-Normal |
Normal |
|
|
|
|
|
Mean |
|
|
|
Standard Deviation |
S |
|
Rule of Thumb: Sample Size is Greater Than 30.
The normalization property is a consequence of the Central
Limit Theorem. States that as the sample size is increased the sampling
distribution of the mean becomes progressively more normal.
Key Benefit: Even though the distribution of a variable is unknown,
the taking of samples enables the distribution of the sample mean is known.
Estimation
The sampling distribution of the mean allows you to estimate
the mean of a population by taking a sample. Procedure is as follows:
1) Take a random sample of at least size 30. Let the sample
size be labeled n. the minimum of 30 is so that the central limit theorem holds
and the standard deviation approximation is valid. A smaller sample can be used
if the distribution is normal, and the population standard deviation is already
known. Even if these two conditions do not hold, a sample smaller than 30 can
sometimes still be used, but more advanced theory is needed.
2) Calculate the sample mean (
) and the sample standard deviation (s).
3) The standard deviation of the sampling distribution of
the mean (the standard error) is calculated as 
4) The point
estimate of the population mean is 
.
5) The 95% confidence limits for the population mean
are: 
2 
This formula can also be used to answer questions in the
reverse, such as “What sample size is needed to estimate the average within +/-
x?
Answer:
Sample mean 
= 
2 
Plug in sample mean, standard deviation, and desired x, then solve for n to determine sample size.
Note again that accuracy is related to square root of sample
size.
Basic Significance Test
Significance tests are a methodology by which to judge
whether a particular piece of sample evidence is consistent with a
hypothesis. 5 steps:
|
1) Formulate the
hypothesis: |
Any idea or hunch, the truth of which is to be
investigated. A null hypothesis,
more or less that the new whatever is the same result as the old. Alternative hypothesis is what we
conclude if the null hypothesis is disproved. |
|
|
|
|
2) Collect a sample
of evidence |
Concerned with the validity of the hypothesis and
calculate the statistics needed to test the hypothesis. |
|
|
|
|
3) Decide on a
significance level: |
It is supposed
that if an event (the sample statistic) occurs which has a probability
greater than the significance level, then this is not especially unusual and
it is entirely believable that it happened purely by chance. If an event
occurs with a probability of less than the significance level, then this is
deemed to be an unusual event and it is not believable that it happened
purely by chance. (Conventionally it is 5%) |
|
|
|
|
4) Calculate the
Probability of the Sample Evidence: |
Using the mean and standard deviation of the sample, then,
assuming sample size is greater than 30, and that the sample is normal,
calculate the SE using the SD divided by the square root of the sample size.
Then, determine how far away from hypothesized population mean is from the sample mean (z value). |
|
|
|
|
5) Compare the
Probability with the Significance Level : |
If it is higher than it is judged consistent with the
hypothesis, (the sample result is thought to have happened purely by chance),
and the hypothesis is accepted; if it is lower, it is judged inconsistent
with the hypothesis (the sample is thought to be too unusual to have happened
by chance), and the hypothesis is rejected.
Rejected hypothesis is said go be significant |
Example: Sample of 100 butter packs are taken to determine if
machine is packing slightly overweight
. Average weight is 500.4g,
SD is 1.5g. Is this data consistent with a population of average weight of
500g?
1) Hypothesis: True
population average weight produced by the machine is 500g.
2) Evidence spelled out in
the example.
3) Significance level %
4) Sample size exceeds 30,
so SD can be calculated from the sample.
SD=1.5/
=1.5. Sample had mean of 500.4. Then:
z=(500.4-500)/.15
z=2.67
Under normal curve,
z=2.67=.4962,
Probability therefore =.5-.4962=.38%
5) Much lower than 5% significance level, so hypothesis is rejected.
Critical Values
The critical value is at the point which leaves 5% (the significance
level) in the tail of the distribution. It therefore has a z value of 1.645. Using the example given above, critical value is
found by:
Critical value =500+(1.645
x .15)
=500.247g
Since mean was far more extreme, hypothesis is rejected.
One vs. Two Tailed Tests
The one tailed tests evaluates extremes in only one
direction (for instance, butter packs weigh too much). Two tailed tests
evaluate extremes in both positive and negative direction. The probability is
twice the single tailed since the area in the tail occurs on both ends of the
distribution.
Using the previous example, the probability would be 2 x
.38% or .76 --and this would be compared with the 5% probability.
Errors in Significance Tests
The probability of falsely
rejecting a hypothesis is equal to the significance level. This is
known as a type 1 error.
To accept a hypothesis even when it is false is a type 2 errors. A type 2 errors is the
probability of:
* Wrongly accepting the null hypothesis
* Wrongly rejecting the alternative hypothesis
The probability of wrongly rejecting the alternative
hypothesis (type 2 error) must be the
area in the tail of the distribution based on the
alternative, as marked by the critical value. For the alternative distribution,
the z value associated with the
critical value is:
z =
(Critical value -1 ) / SE (=
))
Then, P(type 2 error) = (.5-[area under curve associated
with z value above])
The power of the test is the probability of accepting the
alternative hypothesis when it is true. Its:
Power= 100% - P(type 2 error)
Balancing Type 1 and Type 2 Errors
Errors can be balanced by:
a) Changing significance level: Increases probability of
type 2, and reduces type 1.
b) Changing the sample size: This
* changes the SE (=
),
* which increases the critical value,
* which reduces the z value,
* which increases the P(type 2 error).
Significance Tests and Confidence Levels
Directly comparable to the estimation of confidence
levels.
Two-tailed test, the rejection of a hypothesis at the 5%
level form sample data is the same as the hypothesized mean Not falling with 95% confidence interval of
the population mean.
One tailed test, the rejection of a hypothesis at the 5% level
from sample data is the same as the hypothesis mean. Not falling with 90% confidence interval of the population mean
More Significance Tests
Difference in Means of Two Samples
Difference in Sample Means
Method:
|
1) Set a Hypothesis: |
Usually no difference in samples--which the observed
difference comes from a distribution with mean 0. |
|
|
|
|
2) Collect sample evidence: |
The two samples and their means |
|
|
|
|
3) Set the significance level: |
Usually 5%. Hypothesis accepted if probability exceeds the
significance level. Rejected if less than significance level. |
|
|
|
|
4) Calculate the probability of the sample evidence: |
Use normal curve, 0 mean, and SD of |
|
|
|
|
5) Could also use critical value approach. |
|
Example: Comparison of supermarket promotions. Promotion A tries
sample of 36 stores over 1 week. Promotion B tries 36 similar sample stores in
same period. For both samples, increase in sales is measured. Average increase
A: $12000, B is $53000. SD of all stores is $120000. Is there significant
difference?
1) Hypothesis: no difference
2) Evidence: See above
3) Significance level: 5%
4) Probability Test:
0 mean, SD = 
120000 / 
=$28,284. Distribution is normal due to central limit
theorem. The observed difference is (53000-12000)=41000. The z value is:
(41000-0)/28284 = 1.45
Area under curve corresponding to 1.45 = .4265. Probability is .5-.4265= 7.35%. Since each
promotion could be better than the other, the test is two tailed, and
probability is 2x (14.7%)
5) This exceeds the significance level, so no difference.
Pooled Estimate of the Standard Error
Use this formula to pool
SD figures from the individual samples assuming the SD of the overall
distribution is not known:
s= 
Difference between Paired Samples
In tests when two samples are matched one for one, then:
1) Create a single new sample consisting of differences between
the paired samples
2) Run a simple significance test
Tests on Proportions
Sample size needs to be reasonable large
Normal distribution can be applied to data in the form of
proportions and used to conduct significance tests.
Arithmetic mean = p
Standard deviation = 
Normal distribution of data in the form of proportions can
be used for all types of significance tests, including those for two
independent samples and two paired samples, and for estimation, such as point
estimates and confidence intervals.
Reservations about the Use of Significance Tests
Ideas underlying significance tests make them important and powerful aids in approaching management problems.
Used with full awareness of the reservations and limitations
to their use
Qualifications are to do with the tests themselves as well
as with wider issues that may be missed if the tests are used without thought.
a)
Test is only as good as the logics surrounding it.
b)
Many decisions it is simply not possible to collect sample
evidence of the sort that can be used in a significance test.
c)
Significance tests make no attempt to balance costs.
d)
Significance test is a black and white, all-or-nothing
process.
e)
Significance tests are based on assumptions, such as a
normal distribution, as ample size exceeding 30, a known population standard
deviation etc.
f)
Generally a good significance test balances type 1 and type
2 errors. Incorrect rejection and
incorrect acceptance of the hypothesis should have small and equal
probabilities.
Key Message from Module
Statistical Inference belongs with traditional statistical
theory, Its relevance with specialized management tasks, such as quality
control and market research. Only
occasionally be applied to general management problems. Major value is that it encompasses ideas and
concepts, which enable problems to be viewed in broader more structured ways.
Two areas discussed estimation and significance
testing.
New theory introduced:
·
Confidence levels
·
Sampling distribution of the mean
·
Central limit theorem
·
Variance sum theorem
Estimation |
Conceptual contribution – concentrate attention
on the range of a business forecast rather than merely the point
estimate. Confidence limit try and attain 95% |
|
|
|
|
Significance Testing |
Conceptual contribution – distinguishing real
form apparent differences. The discrepancy between a sample mean and what is
thought to be the mean of the whole population is judged in the context of
inherent variation. Apparent Difference – arisen purely by chance i.e.
inherent variation Real Difference – one that is unlikely to have
arisen purely by chance and some other explanation is supposed. Significance level draw a dividing line between the
two |
Significance Testing 3 types
Single Sample |
Basic significance test – Evidence from one sample is used
to test a hypothesis relating to the population. |
|
|
|
|
Two independent samples |
Two independently drawn samples are compared, usually with
the hypothesis that there is no difference between them. |
|
|
|
|
Paired Samples |
Two samples are compared but they are not drawn
independently. Each is observed in
one sample had a partner in the other..
|
Three factors to consider
- the test
can be conducted with probabilities or critical values – mater of
preference for tester both would produce the same results
- the
test can be one-tailed or two tailed.
This decision is not a matter of preference and depends on the
purpose of the test and what outcome is wanted
- the
test can use data in the form of proportions. Depends on the nature of the data, whether proportional or
not
Both estimation and significance testing can improve the way
a manger thinks about particular type of numerical problems. Help o show the manger what to look for in a
management report
- Does
tan estimate or forecast also include a measure of accuracy?
- In
making comparisons, are the differences real or apparent?
Form the point of view of day-to-day management this is where
their importance lies.