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Formulating HypothesesParametric Tests,[object Object],Business  Research Methodology,[object Object],MBA : 2nd Semester ,[object Object]
Formulating Hypotheses,[object Object],Begins with an assumption called Hypothesis.,[object Object],Hypothesis,[object Object],Claim (assumption) about a population parameter.,[object Object],An unproven proposition or supposition that tentatively explains certain facts or phenomena.,[object Object],A proposition that is empirically testable.,[object Object]
What is Hypothesis testing,[object Object],A set of logical and statistical guidelines used to make decisions from sample statistics to population characteristics.,[object Object]
Hypotheses Testing,[object Object],[object Object]
These two hypotheses are mutually exclusive and exhaustive.
Sample information is collected and analysed.,[object Object]
Null Hypothesis,[object Object],Specific statement about a population parameter made for the purposes of argument.,[object Object],States the assumption to be tested, is a status quo.,[object Object],Is always about a population parameter, not about a sample statistic.,[object Object]
Alternate Hypothesis,[object Object],Represents all other possible parameter values except that stated in the null hypothesis.,[object Object],Challenges the status quo.,[object Object],Hypothesis that is believed (or needs to be supported) by the researcher –a research hypothesis.,[object Object]
Level of Significance,[object Object],The critical probability in choosing between the null & alternative hypotheses.,[object Object],The probability of making a Type I error.,[object Object],The higher the significance level, the higher the probability of rejecting a null hypothesis when its true.,[object Object],Risk that a researcher is willing to take of rejecting the null hypotheses when it happens to be true.,[object Object],Confidence level:,[object Object],	A percentage or decimal value that tells how confident a researcher can be about being correct.,[object Object]
Critical Region,[object Object],The rejection region.,[object Object],If the value of mean falls within this region, the null hypothesis is rejected.,[object Object],Critical value,[object Object],	The value of a test statistic beyond which the null hypothesis can be rejected.,[object Object]
Decision Rule(Test of Hypothesis),[object Object],The rule according to which we accept or reject null hypothesis.,[object Object]
Type I Error & Type II Error ,[object Object],A Type I error is the mistake of rejecting the null hypothesis when it is true.,[object Object],α- Type I error.,[object Object],A Type II error is the mistake of failing to reject the null hypothesis when it is false.,[object Object],β- Type II error.,[object Object]
Type I Error & Type II Error ,[object Object],Probability  of Type I error is determined in advance.,[object Object],Level of significance of testing the hypothesis.,[object Object]
Type I Error & Type II Error ,[object Object],Accept H0		 Reject H0,[object Object],Correct Decision		Type I Error,[object Object],Type II Error	  Correct Decision	,[object Object],Ho (True),[object Object],Ho (False),[object Object]
Type I Error & Type II Error ,[object Object],Suppose there is a test for a particular disease. ,[object Object],[object Object]
If it is notdiagnosed and treated, the person will become severely disabled
If a person is erroneously diagnosed as having the disease and treated, no physical damage is done,[object Object]
Power of Test,[object Object],The ability of a test to reject a false null hypothesis.,[object Object],The probability of supporting an alternative hypothesis that is true.,[object Object],Power = 1- β,[object Object],High value of 1- β(near 1) means test is working fine, it is rejecting a null hypothesis when it is false.,[object Object]
One Tailed & Two Tailed Test,[object Object],Two-Tailed Tests,[object Object],If the null hypothesis is rejected for  values of the test statistic falling into either tail of its sampling distribution.,[object Object],A deviation in either direction would reject the null hypothesis,[object Object],Normally α is divided into α/2 on one side and α/2 on the other.,[object Object]
One Tailed & Two Tailed Test,[object Object]
One Tailed & Two Tailed Test,[object Object],One-Tailed Tests,[object Object],Only used when the other tail is nonsensical.,[object Object],If null hypothesis is rejected only for values of the test statistic falling into one specified tail of its sampling distribution.,[object Object]
One Tailed & Two Tailed Test,[object Object]
One Tailed & Two Tailed Test,[object Object],A manufacturer of a light bulb wants to produce bulbs with a mean life of 1000 hours. If the lifetime is shorter, he will lose customers to the competitors; if the lifetime is longer, he will have a very high production cost because the filaments will be very thick. Determine the type of test.,[object Object],The wholesaler buys bulbs in large lots & does not want to accept bulbs unless their mean life is at least 1000 hours. Determine the type of test.,[object Object]
One Tailed & Two Tailed Test,[object Object],A highway safety engineer, decides to test the load bearing capacity of a bridge that is 20 yrs old. Considerable data is available from similar tests on the same type of bridge. Which type of test is appropriate? If the maximum load bearing capacity of this bridge must be 10 tons, what are the null & alternative hypotheses?,[object Object]
One Sample & Two Sample Tests,[object Object],One Sample Test,[object Object],	When we want to draw inferences about the population on the basis of given sample.,[object Object],Two Sample Test,[object Object],	When we want to compare and draw inferences about 2 populations on the basis of given samples.,[object Object]
Independent & Paired Samples,[object Object],Independent Samples,[object Object],Drawn randomly from different populations.,[object Object],Paired Samples,[object Object],When the data for the two samples relate to the same group of respondents.,[object Object]
Types of Hypotheses,[object Object],Research hypotheses.,[object Object],Logical hypotheses.,[object Object],Null hypothesis (Ho).,[object Object],Alternative hypothesis (Ha).,[object Object],Statistical hypotheses.,[object Object]
Research Hypotheses.,[object Object],Statement in words as to what the investigator expects to find.,[object Object],Example.,[object Object],	Students who drink caffeine will be able to memorise information faster than students who do not drink caffeine.,[object Object]
Logical Hypotheses,[object Object],Stated in terms of null & alternate hypotheses.,[object Object],Null Hypothesis (Ho).,[object Object],	Students who drink caffeine will be not be able to memorise information faster than students who do not drink caffeine.,[object Object],Alternative Hypothesis (Ha).,[object Object],	Students who drink caffeine will be able to memorise information faster than students who do not drink caffeine.,[object Object]
Statistical Hypotheses,[object Object],Statement in statistical terms as to what would be found if the research hypothesis is true.,[object Object],A sales manager has asked her salespeople to observe a limit on travelling expenses. The manager hope to keep expenses to an average of $ 100 per salesman per day. What will be H0 and Ha?,[object Object]
Steps in Hypotheses Testing,[object Object],Formulation of the null and alternate hypothesis,[object Object],Definition of a test statistic,[object Object],Determination of the distribution of the test statistic,[object Object],Definition of critical region of the test statistic,[object Object],Testing whether the calculated value of the test statistic falls within the acceptance region.,[object Object]
1: Formulation of H0,[object Object],The Null hypothesis assumes a certain specific value for the unknown population parameter.,[object Object],Defined as an inequality – greater than or less than.,[object Object],For example, if the mean of a population is considered, then,[object Object],H0: μ ≤ μ0,[object Object],H0: μ = μ0,[object Object],H0: μ ≥ μ0,[object Object]
2: Formulation of Ha,[object Object],The alternate hypothesis assigns the values to the population parameter that is not contained in the null hypothesis.,[object Object],For example,,[object Object],Ha: μ > μ0,[object Object],Ha: μ ≠ μ0,[object Object],Ha: μ < μ0,[object Object],The null hypothesis is accepted or rejected on the basis of the information provided by the sample.,[object Object]
3: Definition of a Test Statistic,[object Object],A test statistic must be defined to test the validity of the hypothesis.,[object Object],The test statistic is computed from sample information.,[object Object],A number calculated to represent the match between a set of data and the expectation under the null hypothesis,[object Object],A test that uses the z-score as a test statistic is called a z-test.,[object Object]
4: Determination of the distribution of the test statistic,[object Object],The probability distribution of the test statistic depends on the null hypothesis assumed, the parameter to be tested, and the sample size. Commonly used ones are the Normal, Student’s “t”, Chi-square and F-distributions.,[object Object]
5: Definition of the critical region for the test statistic,[object Object],The set of values of the test statistic that leads to the rejection of H0 in favour of Ha is called the rejection region or critical region. ,[object Object],Depends upon whether the testing is one-sided or two-sided. ,[object Object]
6: Decision rule,[object Object],A decision rule is used to accept or reject the null hypothesis.,[object Object],P- value,[object Object],	P < α,[object Object],	Reject the null hypothesis,[object Object],	Statistically significant,[object Object],Test statistic,[object Object],	Test statistic (calculated value) < Table value of α,[object Object],	Accept H0,[object Object],	Statistically insignificant,[object Object]
7: Outcome,[object Object],The acceptance or rejection of the hypothesis will lead to the following possible outcomes:,[object Object],To accept H0 when H0 is true- Correct decision,[object Object],To reject H0 when H0 is false- Correct decision,[object Object],To reject H0 when H0 is true- Type I error,[object Object],To accept H0 when H0 is false- Type II error,[object Object]
8: Error probabilities,[object Object],The inferences made on the basis of the sample information would always have some degree of error. ,[object Object],	α = Type I error = Rejecting H0 when H0 is true,[object Object],	β = Type II error = Accepting H0 when H0 is false,[object Object],Rejecting a null hypothesis (Type I error) is more serious than accepting it when it is false. Therefore, the error probability α is referred to as the significance level. ,[object Object]
Parametric & Non Parametric Tests,[object Object],Parametric Test,[object Object],Statistical procedures that use interval or ratio scaled data and assume populations or sampling distributions with normal distributions.,[object Object],Non Parametric Test,[object Object],Statistical procedures that use nominal or ordinal scaled data and make no assumptions about the distribution of the population.,[object Object]
Parametric Tests,[object Object],z-test,[object Object],t-test,[object Object],F-test,[object Object],Chi square test,[object Object]
Conditions for using Tests,[object Object]
t-Test,[object Object],Used when sample size is ≤ 30.,[object Object],Given by W.S.Gosset (pen name Student),[object Object],Also called Student’s t distribution.,[object Object],Based on t distribution.,[object Object],The relevant test statistic is t.,[object Object]
t-Test,[object Object],Conditions,[object Object],Sample should be small.,[object Object],Population standard deviation must be unknown.,[object Object],Assumption,[object Object],Normal or approximately normal population.,[object Object]
t Distribution,[object Object],Characteristics,[object Object],Flatter than normal distribution.,[object Object],Lower at mean, higher at tails than normal distribution.,[object Object],As degrees of freedom increase, t-distribution approaches the standard normal distribution (df=8 or more),[object Object],Degrees of Freedom: No. of observations minus the no. of constraints or assumptions needed to calculate a statistical term.,[object Object]
t-Test,[object Object],Confidence Interval,[object Object],Degrees of Freedom: n-1,[object Object]
t-Test,[object Object],The specimen of copper wires drawn from a large lot to have the following breaking strength (in Kg weight),[object Object],	578, 572, 570, 568, 572, 578, 570, 572, 596, 544,[object Object],Test whether the mean breaking strength of the lot may be taken to be 578 kg, at 5% significance level.,[object Object]
t-Test,[object Object],Given a sample mean of 83, a sample standard deviation of 12.5, & a sample size of 22, test the hypothesis that the value of population mean is 70 against the alternative that is more than 70. Use 0.025 significance level.,[object Object]
z -Test,[object Object],Based on the normal distribution.,[object Object],Mostly used for judging the significance level of mean.,[object Object],The relevant test statistic is z. ,[object Object],The value of z is calculated & compared with its probable value.,[object Object],If calculated value is less than table value- accept H0,[object Object]
z -Test,[object Object],[object Object],[object Object]
z -Test,[object Object],Hinton press hypothesizes that the average life of its largest web press is 14,500 hours. They know that the standard deviation of press life is 2100 hours. From a sample of 36 presses , the company finds a sample mean of 13,000 hours. At a 0.01 significance level, should the company conclude that the average life of the presses is less than the hypothesized 14,500 hours?,[object Object]
F-Test,[object Object],Based on F-Distribution,[object Object],Used to compare variance of 2 independent samples.,[object Object],Relevant Test statistic is F.,[object Object],Larger the F value, greater the possibility of  having statistically significant results.,[object Object]
F-Test,[object Object],Characteristics,[object Object],Family of distributions .,[object Object],Has 2 degrees of freedom.,[object Object]

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