STATISTICAL INFERENCE :

STATISTICAL INFERENCE :
TEST OF HYPOTHESIS

COVERAGE Introduction : Statistical Inference Hypothesis Testing
Procedure of Hypothesis Testing Types of Error : Type I and Type II One-tailed and Two-tailed Tests Test of Significance for Large & Small Samples Student’s t-test Summary

STATISTICAL INFERENCE
‘Statistical Inference’ is the branch of Statistics, which is concerned with Degree of Uncertainty in Decision-making, using probability concept It refers to the process of selecting and using a sample statistics to draw meaningful inference about a parameter, based on the sample Treats two different classes of problems : Hypothesis Testing and Estimation

HYPOTHESIS TESTING Hypothesis Testing’ begins with an assumption, called ‘Hypothesis’ ‘Statistical Hypothesis’ is a statement about the probable distribution of the population It is based on some reasoning “A Hypothesis in statistics is simply a quantitative statement about a population” - Prof Morris Hamberg

HYPOTHESIS TESTING Generally, there are two types of Hypothesis used in the process of Hypothesis Testing:- Null Hypothesis : H0 Alternative Hypothesis : Ha

PROCEDURE OF HYPOTHESIS TESTING
Setting up a Hypothesis Setting up a suitable Significance Level Setting up a Test Criteria Computation Making Decisions

SETTING UP A HYPOTHESIS
Set up a Hypothesis about the Population Parameter Collect Sample data, produce sample Statistics and decide how likely our hypothesised population parameter is correct Conventional approach: Construct two hypothesis :- Null and Alternative If one is accepted other is rejected

Null Hypothesis is very useful tool in testing significance of difference ‘Null’ means no real difference in sample and population. For example, A man is innocent, until proved guilty The difference found is accidental and unimportant arising out of fluctuations in sampling Null Hypothesis constitutes a challenge and the function of experiment is to give facts a chance to refute it (or fail to refute)

For example, the statement “Extra coaching has not benefited students” should be the Null Hypothesis (H0) for the case of ascertaining whether extra coaching has helped students or not The rejection of Null Hypothesis (H0) indicates that differences have statistical significance, and acceptance indicates that differences are due to chance

Practical problem aim at establishment of statistical significance of differences, rejection of null hypothesis may thus indicate success in statistical project

Alternative Hypothesis (Ha), on the other hand, specifies those values, which researcher believes to be true, and hopes that sample data lead to acceptance of Ha as true Example. To test whether, or not, a certain population has Blood Pressure of more than 130:- H0 : µ < 130; and Ha : µ > 130 H0 is accepted even without any evidence in support, whereas, it can be rejected only when there is complete evidence against it

SETTING UP SIGNIFICANCE LEVEL
After setting up a hypothesis, test the validity of H0 against Ha at certain level of significance The confidence, with which, H0 is rejected or accepted, depends upon the significance level adopted Significance Level is customarily expressed in percentage (5%, 1%) Hypothesis accepted at 5%, risk that in long run wrong decision 5% of the time Hypothesis rejected at 5%, risk of rejecting a true hypothesis in 5 out of every 100 occasions Testing may be ‘One-tailed’ or ‘Two-tailed’

SETTING UP SIGNIFICANCE LEVEL

LEVEL OF SIGNIFICANCE There is no single standard Level of Significance for testing the Hypothesis Generally, 5% and 1% levels are used

SETTING UP A TEST CRITERIA
Select an appropriate Probability Distribution for particular test, which can be applied properly Common Probability Distributions are:- F, t, Z, Chi- Square Appropriate Probability Distribution is must to employ test criteria. For example, Normal Distribution is inappropriate for a small sample size

COMPUTATIONS After designing a statistical test, various computations are to be done from a random sample (size ‘n’) necessary for test These include testing Statistics and Standard Error of Estimate (SEE) of testing statistics

DECISION MAKING Final step is to draw statistical conclusions and take decisions Statistical Conclusions are whether to accept the hypothesis or reject it Decision will depend on whether computed value of testing criteria falls in the range of rejection or acceptance

DECISION MAKING If observed set of results has probability less than 5% for testing at 5% significance level, difference between sample statistics and hypothetical parameter is considered as ‘Significant’ Sample result is so rare that it can not be explained by chance variation alone. Thus, it is decided to reject H0 and concluded that “Null Hypothesis is False”, i.e., Sample observations are not consistent with H0

DECISION MAKING If observed set of results has probability more than 5% for testing at 5% significance level, difference between sample statistics and hypothetical parameter can be explained by chance variation, hence, is not considered as Significant statistically Thus, it is decided not to reject H0 and stated that “Null Hypothesis is True”, i.e., Sample observations are consistent with H0

DECISION MAKING Rejection statement is much stronger than acceptance. This means that in case of H0 not rejected, it can not be categorically concluded that hypothesis is true In logic it is always easier to prove something false than to prove it true

TYPES OF ERROR Decision to reject or accept H0 is made on basis of information supplied by sample data; hence, there is always a chance of making an error There are always four possibilities:- Hypothesis is TRUE, test rejects it (Type-I Error) Hypothesis is FALSE, test accepts it (Type-II Error) Hypothesis is TRUE, test accepts it (Correct Decision) Hypothesis is FALSE, test rejects it (Correct Decision)

TYPES OF ERROR In Statistical hypothesis testing, Type-I and II errors can be computed for probability as:- Prob. of Type-I error:- α = Prob. [Reject H0 / Ha is True] Prob. of Type-II error:- β = Prob. [Accept H0 / Ha is False] Aim to reduce both errors Fixed sample size

TYPES OF ERROR While deciding acceptance or rejection of H0, probability of Type I error to be minimised Type-I error is more dangerous than Type-II error, because ‘Physical Loss’ is involved in Type-I error Type-II error involves simple ‘Opportunity Loss’

TYPES OF ERROR For example, Occurrence and Non-occurrence of Thunderstorm can be tested with the Forecast Occurrence and Non-occurrence, as:- F/C Occ F/C Non-Occ Obs. Occ Correct Decision (PoD) Type-I Error (Miss Rate) Obs. Non-Occ Type-II Error (False Alarm) (Correct Non-occurrence)

TYPES OF ERROR It is quite clear that Type-I errors are much more serious than Type-II errors Type-I errors can be controlled by setting significance level Commonly used significance level is 5%, i.e., probability of acceptance of True Hypothesis is 95%, with chances of rejection of 5% Type-II error probability can not be set by experimenter

TYPES OF ERROR Higher the level of significance set, more likely having Type-II error If we set α high, Type-I errors are less likely, but Type-II errors are more likely On the other hand, by setting α low, Type-I errors are more likely, but Type-II errors are less likely Therefore, it is important to keep balance between two types of errors in Decision-making process

TYPES OF ERROR Though Type-II errors can not be controlled directly (except by lowering α-level), different statistics at same α-level are more resistant to cause Type-II error This characteristic feature of statistics is called as ‘Power of Statistics’ More powerful statistics is less likely to yield Type- II error It is the tendency of statistics not to make Type-II error

TYPES OF ERROR Both types of errors can be completely eliminated, if we test the entire population But it is practically not possible, especially in the field of weather prediction, as future state is not known in advance

How can Type I error be reduced?
Type of Errors in HT How can Type I error be reduced? By shifting CV outwards Decr α (as a result β will incr) How can Type II error be reduced? By bringing CV inwards Incr α Incr sample size

Type I and Type II Errors
Trade off between  and  becomes important here as  increases  decreases and vice versa.  fixed based on criticality of situation, And  is kept low (for a given ) by increasing sample size.

ONE-TAILED & TWO-TAILED TESTS

One tail test. Two tailed test. Left tail test. Right tail test.
Types of Tests One tail test. Left tail test. Right tail test. Two tailed test. CV HO CV HO CV CV HO

ONE-TAILED TESTS If Rejection region is located in only one tail of the Probability Distribution (either to left or to right, depending on Alternative Hypothesis formulation), it is called as ‘One-tailed Test’

ONE-TAILED TESTS For example, if we are interested in testing the hypothesis that ‘the average maximum temperature over a base is more than 40ºC’ against the alternative hypothesis that ‘the average maximum temperature is 40ºC’, we will place all the α-risk on the right side of one theoretical sampling distribution

ONE-TAILED TESTS On the other hand, if we are interested in testing the hypothesis that ‘the average maximum temperature over a base is 40ºC’ against the alternative hypothesis that ‘the average maximum temperature is less than 40ºC’, the α-risk is on the left side of theoretical sampling distribution In both cases, the test will be One-tailed

TWO-TAILED TESTS A Two-tailed test of hypothesis will reject the null hypothesis, if the sample statistics is significantly higher than, or lower than, the hypothesised population parameter Thus a Two-tailed test, the rejection region is located on both the tails If the testing is being carried out at 5% significant level, then the size of acceptance region on each side of mean would be and the size of rejection region on each side would be 0.025

TWO-TAILED TESTS A Two-tailed test is appropriate, when the null hypothesis is µ = µHO (µHO being some specific value) and the alternate hypothesis is µ ≠ µHO.

ONE-TAILED TESTS : EXAMPLE
A wholesaler buys bulbs from a factory. He buys large lots and does not want to accept lots unless their life is 1000 h At each shipment, he takes the sample to decide either to accept or reject the shipment He will reject if the sample mean is below 1000 h If he gets mean more than 1000 h, he will not reject: Null : µ = 1000 Alternate : µ ≠ 1000 He will reject the null hypothesis, if his sample mean is significantly below 1000 h

TWO-TAILED TESTS : EXAMPLE
A bulb making factory wants to produce bulb with a mean life of 1000 hours. So, µ = µHO = 1000 h If the life is shorter, the manager will loose customers, and if life is longer, his cost of production goes up In order to check his production process is working properly, he takes a sample of the output to test the hypothesis:- Null : µ = 1000 Alternate : µ ≠ 1000 He uses two tailed test. He rejects the null hypothesis, if his sample mean is too large or too smaller than 1000 h. There are situations when two-tailed test is not appropriate

ONE-TAILED & TWO-TAILED TESTS

SAMPLING DISTRIBUTION
STANDARD ERROR AND SAMPLING DISTRIBUTION Sampling distribution or probability distribution of statistic Standard deviation of sampling distribution is standard error Measures sampling variability due to chance or random forces

TEST OF SIGNIFICANCE FOR LARGE SAMPLES
Standard Error of Mean. When S.D. of population is known, and when S.D. of population is not known, but S.D. of sample is known, then S.E. will be:- and It should be noted that if S.D. of both, population and sample, are known, then S.D. of population should be used

TEST OF SIGNIFICANCE FOR LARGE SAMPLES
Standard Error of Median Standard Error of Mean Deviation Standard Error of S.D

CENTRAL LIMIT THEOREM If random samples of large size (n > 30) are taken from a universe, which may be normally distributed or not, and has arithmetic mean ‘µ’ and S.D. ‘σ’, the sampling distribution of arithmetic mean of samples of large size will very closely approximate a Normal Distribution, having arithmetic mean ‘µ’ and S.D. as ‘σ/(n) ½’.

TEST OF SIGNIFICANCE FOR SMALL SAMPLES
To deal with problems related to small samples (n < 30), assumptions made for large samples will not hold good However, methods and theory of small samples are applicable to large samples as well In this case, main interest is not to estimate the population values, rather lies in testing a given hypothesis (i.e., whether observed values could have arisen by sampling fluctuations from some value given in advance)

For example, if a sample with size 10 gives correlation coefficient of +0.55, our interest will be whether this value could have arisen from an uncontrolled population, i.e., whether it is significant of correlation in parent population. While dealing with small samples also, assumption is made that the parent population is normally distributed.

If given population is markedly skewed, these methods can not be applied with confidence Sir William Gosset introduced a test in 1905, known as ‘Student’s t-test’ RA Fasher introduced another such test, known as ‘z-test’ Both these tests are based on ‘t-distribution’ and ‘z-distribution’ respectively

STUDENT’S T-TEST It is used when sample size is 30 or less, and population S.D. is not known. ‘t-statistics’ is defined as:- where is the sample mean, ‘µ’ is the population mean (actual / hypothetical) and ‘n’ is sample size.

STUDENT’S T-TEST : PROPERTIES
The variable t-distribution ranges from minus infinity to plus infinity The constant ‘c’ is actually a function of ‘υ’ so that for a particular value of ‘υ’, the distribution of f(t) is completely specified. Thus, f(t) is a family of functions, one for each value of ‘υ’ Like the standard normal distribution, the t- distribution is symmetrical and has a mean zero

The variance of the t-distribution is greater than one, but approaches one as the number of degrees of freedom and, therefore, the sample size becomes large Thus the variance of the t-distribution approaches the variance of the standard normal distribution as the sample size increases

It can be demonstrated that from an infinite number of degrees of freedom (‘υ’=∞), the t- distribution and normal distribution are exactly equal Hence there is a widely practiced rule of thumb that samples of size ‘n > 30’ may be considered large and the standard normal distribution may appropriately be used as an approximation to t- distribution, where the latter is the theoretically correct functional form

STUDENT’S T-TEST Comparison of a Normal Distribution with two t-distributions of different sample sizes:-

STUDENT’S T-TEST The diagram shows two important characteristics of the t-distribution:- A t-distribution is lower at the Mean and higher at tails than a Normal Distribution The t-distribution has proportionately greater area in its tails than the Normal Distribution Interval widths from t-distributions are, therefore, wider than those based on the Normal Distribution

STUDENT’S T-TEST : T-Table
It is the probability integral of t-distribution and gives, over a range of values of ‘υ’ (Degree of freedom = n-1), the probabilities of exceeding by chance value of ‘t’ at different levels of significance For infinitely large values of Degree of Freedom, t- distribution is equivalent to the Normal Distribution

STUDENT’S T-TEST : APPLICATIONS
TEST SIGNIFICANCE OF MEAN OF A RANDOM SAMPLE In determining, whether the mean of a sample drawn from a normal population deviates significantly from a stated value (the hypothetical value of the populations mean), when variance of the population is unknown, we calculate the statistics as:-

TEST SIGNIFICANCE OF MEAN OF A RANDOM SAMPLE Here is the sample mean, ‘μ’ is the actual or hypothetical mean of the population, ‘n’ is the sample size and ‘S’ is the standard deviation of the sample, calculated as:-

TEST SIGNIFICANCE OF MEAN OF A RANDOM SAMPLE If the calculated value of |t| exceeds t0.05, we say that the difference between and μ is significant at 5% level. If it exceeds t0.01, the difference is said to be significant at 1% level. If |t| < t0.05, we conclude that the difference between and μ is not significant, and hence, the sample might have been drawn from the population having Mean = μ.

TESTING DIFFERENCE BETWEEN MEANS OF TWO (INDEPENDENT) SAMPLES Given two independent random samples of size ‘n1’ and ‘n2’ with means ‘μ1’ and ‘μ2’, and standard deviations ‘S1’ and ‘S2’, we may be interested in testing the hypothesis that the samples come from the same normal population.

TESTING DIFFERENCE BETWEEN MEANS OF TWO (INDEPENDENT) SAMPLES To carry out the test, we calculate the statistic as:- where ‘S’ is the combined Standard Deviation, given as:-

TESTING DIFFERENCE BETWEEN MEANS OF TWO (INDEPENDENT) SAMPLES Given two independent random samples of size ‘n1’ and ‘n2’ with means ‘μ1’ and ‘μ2’, and standard deviations ‘S1’ and ‘S2’, we may be interested in testing the hypothesis that the samples come from the same normal population.

SUMMARY Hypothesis testing begins with a Hypothesis (i.e., Assumption) about a population parameter Procedure of Hypothesis Testing includes setting of a Hypothesis, suitable Level of Significance, Test Criteria and Making Decisions based on Calculations Two types of errors are possible in Hypothesis Testing : Type-I and Type-II

SUMMARY Type-I error is more severe Tests are One-tailed or Two-tailed
Tests for large samples differ from small samples, due to different assumptions Student’s t-test is used for:- Testing significance of a Mean of a random sample (Small size) Comparison of Means of two independent samples (Small size)

ANY QUESTION ?

Steps in Hypothesis Testing
H not : H one : Step 2 Data n Population Mean Estimator (x bar) Population SD Population CL LOS z dist or t dist Draw diagram Step 3 LTT OR RTT OR TTT Step 4 Decision Rule Step 5 Standard Error Error of Estimate Step 6 Lower Critical Value Upper Critical Value Step 7 CV > x bar or CV < x bar Step 8 Statistical Inference : Step 9 Admin Decision :

STATISTICAL INFERENCE :

Similar presentations

Presentation on theme: "STATISTICAL INFERENCE :"— Presentation transcript:

Similar presentations

About project

Feedback

Log in

Auth with social network:

STATISTICAL INFERENCE :

Similar presentations

Presentation on theme: "STATISTICAL INFERENCE :"— Presentation transcript:

Similar presentations

About project

Feedback