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Basic Concepts of Statistics & Probability
Review ofStatisticalConcepts
SamplingfromDistributions
Hypothesis
Testing
Industrial Engineering
Define the following.
Probability Population
Sample Mean
Median Mode
Standard Deviation Variance
Range Box-plot
Histogram Descriptive Statistics
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Types of Distributions
Continuous Distributions
Normal Distribution
Chi-square (X2) Distribution
t-Distribution
F-Distribution
Exponential Distribution
Weibull Distribution
Discrete Distributions
Binomial Distribution
Poisson Distribution
Sampling from Distributions
Hypothesis
Testing
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Normal Distribution
The probability of the normal random variable
Probabilities for the normal random variable are given by areas under thecurve.
Where for Standard Normal Distribution
= 0
= 1
= 3.14159
e = 2.71828
222/)(
2
1)(
xexf
Hypothesis
Testing
Sampling from Distributions
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Normal Distribution
43210-1-2-3-4
x
For a population that is
normally distributed:
approx. 68% of the data will lie within +1standard deviation of the mean;
approx. 95% of the data will lie within +2
standard deviations of the mean, and
approx. 99.7% of the data will lie within +3standard deviations of the mean.
Hypothesis
Testing
Sampling from Distributions
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Review ofStatisticalConcepts
Samplingfrom
Distributions
Industrial Engineering
HypothesisTesting
Sampling from Distributions
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Review ofStatisticalConcepts
Samplingfrom
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Industrial Engineering
HypothesisTesting
Sampling from Distributions
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Review ofStatisticalConcepts
Samplingfrom
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Industrial Engineering
HypothesisTesting
Sampling from Distributions
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Review ofStatisticalConcepts
Samplingfrom
Distributions
Industrial Engineering
Chi-square (2) Distribution
If x1, x2, , xn are normally and independently distributed randomvariables with mean zero and variance one, then the random variable
is distributed as chi-square with n degrees of freedom.
Furthermore, the sampling distribution of
is chi-square with n 1 degrees of freedom when sampling from a normalpopulation
22
2
2
1... nxxxy
2
2
2
1
2
)1()(
Snxx
y
n
i
i
HypothesisTesting
Sampling from Distributions
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Review ofStatisticalConcepts
Samplingfrom
Distributions
Industrial Engineering
Chi-square (2) Distribution for various degrees of freedom.
HypothesisTesting
Sampling from Distributions
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Samplingfrom
Distributions
Industrial Engineering
t-distribution
Ifxis a standard normal random variable and ifyis a chi-square randomvariable with kdegrees of freedom, then
is distributed as t with k degrees of freedom.
k
y
xt
HypothesisTesting
Sampling from Distributions
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Review ofStatisticalConcepts
Samplingfrom
Distributions
Industrial Engineering
F-distribution
If w and y are two independent chi-square random variables with uand v
degrees of freedom, respectively, then
is distributed as F with unumerator degrees of freedom and vdenominator
degrees of freedom.
vy
uwF
/
/
HypothesisTesting
Sampling from Distributions
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Review ofStatisticalConcepts
Samplingfrom
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Industrial EngineeringHypothesis Testing
HypothesisTesting
Statistical Hypothesis
Statement (assumption) either about the parameters of probability distribution
or parameters of a model. This assumption may or may not be true.
Statistical Hypothesis Test
A method of making statistical decisions using experimental data. It istypically consists of examining a random sample from the population. Ifsample data are consistent with the statistical hypothesis, the hypothesis is
accepted; if not, it is rejectedThere are two types of statistical hypotheses:
Null Hypothesis. The null hypothesis, denoted by H0, is usually thehypothesis that sample observations result purely from chance.
Alternative Hypothesis. The alternative hypothesis, denoted by H1 , whichis the hypothesis that sample observations are influenced by some non-random cause.
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HypothesisTesting
The significance level
, is the maximum probability tolerated for rejecting a true nullhypothesis.
The p value is the probability of a more extreme departure from the nullhypothesis than the observed data
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HypothesisTesting
The hypotheses are stated in such a way that they are mutually exclusive.
That is, if one is true, the other must be false.
Errors in Hypothesis Testing
Type I error occurs when the null hypothesis is rejected when it is
true, an error, or a "false positive". Thus indicating a test of
poor specificity.
Type II error occurs when the null hypothesis is not rejected
when it is false, a error, or a "false negative". Thus indicating atest of poor sensitivity.
= P(type I error) = P(reject H0H0 is true)
= P(type II error) = P(fail to reject H0H0 is false)
Power = 1- = P(reject H0H0 is false)
Critical Region or Rejection RegionA set of values in which the null hypothesis is rejected or failed in the
test statistics.
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Review ofStatisticalConcepts
Samplingfrom
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Industrial EngineeringHypothesis Testing
HypothesisTesting
Steps for Hypothesis Testing
1. Formulate a null hypothesis, and the alternative hypothesis. Thehypotheses are statements about the population parameters
2. State the test statistic
3. Define the level of significance of the test (the probability of rejectingwhen it is true) and hence the critical region.
4. Collect the data and calculate the observed value of the test statistic
using the sample data and find the p-value.
5. Reject H0 if observed value of test statistic falls in critical region or p-value is less than H0 . Otherwise there is no evidence to reject .
6. State the conclusions clearly in non-technical terms.