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Hypothesis

Hypothesis is an assumption about a population parameter.

for example, the coin/ dice is unbiased.

There are two types of hypothesis:1. Simple2. Composite

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Description

Simple Hypothesis: A statistical hypothesis which specifies the population completely (i.e. the form of probability distribution and all parameters are known) is called a simple hypothesis.

Composite Hypothesis: A statistical hypothesis which does not specify the population completely (i.e. either the form of probability distribution or some parameters remain unknown) is called a Composite Hypothesis.

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Hypothesis Testing or Test of Hypothesis or Test of Significance

Hypothesis Testing is a process of making a decision on whether to accept or reject an assumption about the population parameter on the basis of sample information at a given level of significance.

Null Hypothesis: Null hypothesis is the assumption which we wish to test and whose validity is tested for possible rejection on the basis of sample information.

It asserts that there is no significant difference between the sample statistic (e.g. Mean, Standard Deviation (S), and Proportion of Sample (p)) and Population Parameter (e.g. Mean(µ), Standard Deviation (σ), Proportion of Population (P)).

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Symbol-It is denoted by Ho Acceptance- The acceptance of null

hypothesis implies that we have no evidence to believe otherwise and indicates that the difference is not significant.

Rejection- The rejection of null hypothesis implies that it is false and indicates that the difference is significant.

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Alternative Hypothesis

Alternative hypothesis is the hypothesis which differs from the null hypothesis. It is not tested.

Symbol-It is denoted by H1. Acceptance- its acceptance depends on the

rejection of the null hypothesis. Rejection- Its rejection depends on the

acceptance of the null hypothesis.

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Level of Significance

Level of significance is the maximum probability of rejection the null hypothesis when it is true.

Symbol- it is usually expressed as % and is denoted by symbol α (called Alpha)

Example- 5% level of significance implies that there are about 5 chances in 100 of rejecting the Ho when it is true or in other words , we are about 95% confident that we will make a correct decision.

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Test Statistic

Test statistic refers to a function of sample observations whose computed value determines the final decision regarding acceptance or rejection of null hypothesis (Ho)

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Test Statistic Used for

z-Test For test of Hypothesis involving large sample i.e.>30

t-Test For test of Hypothesis involving small sample i.e.≤30 and if σ is unknown

X2 –Test For testing the discrepancy between Observed frequencies and expected frequencies, without any reference to population parameter.

F- Test For testing the sample variances.

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Critical Region or Rejection Region

Critical region is the region which corresponds to a pre- determined level of significance. The set of values of the test statistic which leads to rejection of the null hypothesis is called region of rejection or Critical region of the test.

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Acceptance region- The set of values of the test statistic which leads to the acceptance of Ho is called region of acceptance.

Critical value- Critical value is that value of statistic which separates the critical region from the acceptance region. It lies at the boundary of the regions of acceptance and rejection.

Size of Critical Region- The probability of rejecting a true null hypothesis is called as size of critical region.

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Area under Normal Curve

The critical region may be represented by a portion of the area under the normal curve in the following two ways:

Two- tailed Test- The tests of hypothesis which are based on the critical region represented by both tails under the normal curve are called ‘Two-tailed Tests’.

One –tailed Test- The tests of hypothesis which are based on the critical region represented by one tail(on right side or on left side)only under the normal curve are called ‘One-tailed Tests’.

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Table showing the Type of Test &Critical Region

Alternative Hypothesis

H1

Null Hypothesis

Ho

Type of Alternative

Type of Test Critical region is represented by

H1: µ≠? Ho: µ=? Both sided Two-tailed Both tails

H1: µ>? Ho: µ≤? One sided One –tailed Right tail

H1: µ<? Ho: µ≥? One sided One –tailed Left tail

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Critical Value of Test –Statistic’Z’

Type of Test Test of significance

1% 5%

Two- tailed ±2.58 ±1.96

One –tailed

Right tail +2.33 +1.645

Left-tail -2.33 -1.645

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Type I and Type II Errors

Type I Error- This is error committed by the test in rejecting a true null hypothesis. The probability of committing type I error is denoted by α (the level of significance)

Type II Error- - This is error committed by the test in accepting a false null hypothesis. The probability of committing type II error is denoted by β.

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True situation Statistical Decision of the Test

Ho is True Ho is false

Ho is True Correct Decision Type I Error

Ho is False Type II Error Correct Decision

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Power of the Test

Power of the Test is the probability of rejecting a false null hypothesis. It can be calculated as

Power of the Test= 1-Probability of Type II error Degree of freedom- mean the number of variables

for which one has freedom to choose. In case of one sample-n-1 In case of two samples-n1+n2-2

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Practical steps involved in testing Hypothesis

Specify Ho and H1.(see slide12) Specify the appropriate test statistic to be used,( see slide 08) Compute the value of test statistic( z,t,x²,f) Compute the decision as follows:Since the computed value is less than the tabular value, we accept

the null hypothesis (Ho) and conclude that difference is not significant.

or Since the computed value is greater than the tabular value, we

reject the null hypothesis (Ho) and conclude that difference is significant.

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Test of specified Mean

ILLUSTRATION-01 Philips Company claims that the length of

the life of its electric bulb is 2000 hours with standard deviation of 30 hours . A random sample of 25 showed an average life of 1940 hours with a standard deviation of 25 hours. At 5% level of significance can we conclude that the sample has come from a population with mean of 2000 hours?

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ILLUSTRATION-02

An automatic machine was designed to pack exactly 2.0 kg of Vanaspati. A sample of 100 tins was examined to test the machine. The average weight was found to be 1.94kg with standard deviation 0.10 kg. Is the machine working properly?

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ILLUSTRATION-03

The record for last several years of applicants for admission into Management colleges for a test showed that their mean score was 115. An administrator is interested in knowing whether the caliber of the recent applicants has changed. For the purpose of testing this hypothesis the score of the last 100 students is obtained from the admission office. The mean for this turned out to be 118 & standard deviation30. Use 5% significance level.

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ILLUSTRATION-04

A pharma company hypothesizes that the effect of a certain sedative is 13 hrs with a known standard deviation of 2 hrs. From a sample of 16 patients, it is found that the sample mean to be 12 hrs. At 0.01 level of significance ,should the company conclude that the average effect of the sedative is less that or equal to 13 hrs.

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ILLUSTRATION-05

In order to test whether the average weekly maintenance cost of a fleet of buses is more than `500,a random sample of 49 buses was taken. The mean & standard deviation were found to be `506 &`42.

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ILLUSTRATION-06

The quality control department of a processing company specifies that the mean net weight per pack of its produce must be 20gms.Experience has shown that weights are approximately normally distributed with a S.D. of 1.5gms.A random sample of 15 packs yield a mean weight of 19.5gms. Is this sufficient evidence to indicate that the true mean weight of the packs has decreased? Use 5% significance level.

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ILLUSTRATION-07

A random sample of 900 members has a mean 3.4cm. Can it be reasonably regarded as sample from large population of mean 3.2cm &Standard Deviation2.3cm?

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Test of significance for Difference of Means of two Large Samples:

ILLUSTRATION-01Intelligence test were given to two groups of

students of the college: Mean S.D. Size

Group A 75 8 60Group B 73 10 100Is there a significant difference in mean scores

obtained by Group A and Group B?

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ILLUSTRATION-02

Intelligence test on two groups of boys and girls gave the following results:

Is there a significant difference in the mean scores obtained by boys and girls?

(a) At 5%level

(b) At 1% level

Mean S.D. N

Girls 61 2 64

Boys 60 4 100

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ILLUSTRATION-03

The mean yield of wheat from a district A was210 kgs with S.D.=10 kgs from a sample of 100 plots. In another district B,the mean yield was 220 kgs with S.D.=12 kgs from a sample of 150 plots. Assuming that the standard deviation of yield in the entire state was 11 kgs. Test whether there is any significant difference between the mean yield of crops in the two districts.

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Test of Significance for Single Prpportion

ILLUSTRATION-01

A Coin was tossed 400 times and the head turned up 216 times. Test the hypothesis that the coin is unbiased.

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ILLUSTRATION-02

A manufacuter claims that only 4% of his products supplied by him are defective. A random samples of 600 products contained 36 defectives . Test the claim of manufacture.

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ILLUSTRATION-03

A manufacturer claimed that atleast 90% of the components which he supplied, conformed to specifications.A random sample of 200 components showed that only 40 were faulty. Test his claim at 1% level of significance.

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ILLUSTRATION-04

A manufacturer claimed that at most 10% of the components which he supplied could be faulty. A random sample of 200 components showed that only 160 were upto the standard. Test his claim at 1% level of significance.

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Testing of significance for difference of proportions

Machine produced 14 defective articles in a batch of 500. After over hauling ,it produce 4 defectives in a batch of 100. Has the machine improved?