Class 04 - Basic Concepts of Statistics and Probability (2 of 2)

7/30/2019 Class 04 - Basic Concepts of Statistics and Probability (2 of 2)

1/15

1

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


2/15

2


Samplingfrom

Distributions


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


3/15

3


Samplingfrom

Distributions


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



4/15

4


Samplingfrom

Distributions


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



5/15

5


Samplingfrom

Distributions


HypothesisTesting



6/15

6


Samplingfrom

Distributions


HypothesisTesting



7/15

7


Samplingfrom

Distributions


HypothesisTesting



8/15

8


Samplingfrom

Distributions


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



9/15

9


Samplingfrom

Distributions


Chi-square (2) Distribution for various degrees of freedom.

HypothesisTesting



10/15

10


Samplingfrom

Distributions


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



11/15

11


Samplingfrom

Distributions


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



12/15

12


Samplingfrom

Distributions

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.


13/15

13


Samplingfrom

Distributions


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


14/15

14


Samplingfrom

Distributions


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.


15/15

15


Samplingfrom

Distributions


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.

Documents

Class 04 - Basic Concepts of Statistics and Probability (2 of 2)