23 Hypothesis Testing and Decision Making

Prof. Pankaj Madan

 

  • Introduction
  • Hypothesis Testing
  • Conditions for using the distributions in testing hypothesis
  • Standard error
  • Procedures in Hypothesis Testing
  • Errors in Hypothesis Tests
  • Concept of one sample t-test
  • Summary
  • Self-Check Exercise with solutions

 

Quadrant-I

 

Hypothesis Testing and Decision Making

 

Learning Objectives:

  • After the completion of this module the student will understand:
  • Hypothesis Testing
  • Conditions for using the distributions in testing hypothesis
  • Standard error
  • Procedures in Hypothesis Testing
  • Errors in Hypothesis Tests
  • Concept of one sample t-test

 

 

1.        Introduction

 

Setting up and testing hypothesis is an essential part of statistical inference. In order to formulate such a test, some theory is to be put forward, either because it is believed to be true or because it is to be used as a basis for argument, but has not been proved. For example, claiming that a new drug is better than the current drug for treatment of the same symptoms; claim of chamber of commerce about the mean household income.

 

In each of the example stated, the question of interest is the two competing claims / hypothesis between which we have to make a choice; the null hypothesis, denoted H0, against the alternative hypothesis, denoted H1. These two competing claims (hypothesis) are not however treated on an equal basis: special consideration is given to the null hypothesis. A hypothesis is a claim, a statement, or an assumption regarding a population parameter that may or may not be true, but needs to be verified by a random sample from a single population or two different populations or even multiple populations.

 

We have two common situations:

 

1.      The experiment has been carried out in an attempt to disprove or reject a particular hypothesis, the null hypothesis, thus we give that one priority so it cannot be rejected unless the evidence against it is sufficiently strong. For example,

 

H0: Mileage of Mahindra Scorpio is 12.5 Km/l Against

 

H1: Mileage of Mahindra Scorpio is NOT (or less than) 12.5 Km/l

 

 

2.  If one of the two hypotheses is ‘simpler’ we give it priority so that a more ‘complicated’ theory is not adopted unless there is sufficient evidence against the simpler one. For example, it is ‘simpler’ to claim that mileage of Mahindra Scorpio is 12.5 Km/l but in most of the cases when it is not a claim rather it’s an assumption or statement, Null hypothesis is considered ‘negative’ or null. In this case the population is Mahindra Scorpio Cars and sample taken is from single population (i.e. only Mahindra Scorpio Cars).

 

The hypothesis are often statements about population parameters like expected value and variance; for example H0 might be that the expected value of the height of ten year old boys in the Indian population is not different from that of ten year old girls (this may be comparing two populations i.e. boys and girls). A hypothesis might also be a statement about the distributional form of a characteristic of interest, for example that the height of ten year old boys is normally distributed within the Indian population.

 

A null hypothesis is denoted by H0 (read ‘H-naught) and an alternative hypothesis is denoted by HA (read ‘H – a’). Both H0 and HA are claim and counter-claim regarding a population parameter, and we test H0 against HA

 

The outcome of a hypothesis test is “Reject H0 in favor of H1” or “Do not reject H0“.

 

Hypothesis Testing

 

Whenever, we have a decision to make about a population characteristic, we make a hypothesis. Some examples are:> 50     or  50

 

Suppose that we want to test the hypothesis that 50. Then we can think of our opponent suggesting that = 50. We call the opponent’s hypothesis the null hypothesis and write:

 

H0:  = 50

and our hypothesis- the alternative hypothesis and write

 

H1: 50

 

For the null hypothesis, we always use equality, since we are comparing with a previously determined mean.

 

For the alternative hypothesis, we have the choices: <,>, or . Tagging the two competing claims regarding a parameter as H0 (Null Hypothesis) and H1 (Alternative Hypothesis) is a matter of semantics and it does not affect the final decision on which claim is accepted and which one is rejected.

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2. CONDITIONS FOR USING THE DISTRIBUTIONS IN TESTING HYPOTHESIS

 

1.   For Testing about Means

 

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Why use STANDARD ERROR instead of Standard Deviation

 

The standard deviation of the sampling distribution or (distribution of a sample status) is called the standard error.

Where

 

: Population Standard Deviation

 

x: Sample Size.

 

When x is large, the probability of a sample value of the statistic deviating from its mean by more than 3 times its standard error is very small. We, therefore apply this formula to find out the degree of reliability of the statistic where inference drawn about the population based on data of a sample. Consequently, the determination of standard errors of statistic is a matter of importance in the sampling of large samples.

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UTILITY CONCEPT OF STANDARD ERROR

 

1.   To test a given hypothesis

 

2.   It provides an idea about the unreliability of the sample.

 

3.   To determine limits within which the parameters are expected to lie.

 

e.g.        Standard Error for Binomial Distribution = npq

 

n = Size of the sample

 

p = Number of success trial.

 

q = Number of failure

 

 

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Procedures in Hypothesis Testing

 

When we test a hypothesis we proceed as follows:

 

Formulate the null and alternative hypothesis.

 

  1. Choose a level of significance (if the difference between the hypothesized mean and sample mean is 5% or 0.05 then value 0.05 is called level of significance ( = 0.05). If nothing is mentioned about , then convention is to take = 0.05)
  2. Determine the sample size (Same as confidence intervals)
  3. Collect data.
  4. Calculate z (or t) score.
  5. Utilize the table to determine if the zcal or tcal score falls within the acceptance region (ztab or ttab is the cutoff point against which the test statistic value (zcal or tcal or ∆ is compared and it is also known as the critical value for the test statistic)
  6. Decide to

 

a. Reject the null hypothesis and therefore accept the alternative hypothesis or

 

b. Fail to reject the null hypothesis and therefore state that there is not enough evidence to suggest the truth of the alternative hypothesis.

 

 

∆  is the test statistic value against which all calculated values\

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Errors in Hypothesis Tests

 

Type I Error

 

In a hypothesis test, a type I error occurs when the null hypothesis is rejected when it is in fact true; that is, H0 is wrongly rejected.

 

For example, in a clinical trial of a new drug, the null hypothesis might be that the new drug is no better, on average, than the current drug; i.e.

 

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H0: there is no difference between the two drugs on average.

 

A type I error would occur if we conclude that the two drugs produced different effects when in fact there was no difference between them.

 

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The following table gives a summary of possible results of any hypothesis test:

 

Decision
Reject H0 Don’t reject H0
H0 Type I Error Right decision
Truth
H1 Right decision Type II Error

 

A type I error is often considered to be more serious, and therefore more important to avoid, than a type II error. The hypothesis test procedure is therefore adjusted so that there is a guaranteed ‘low’ probability of rejecting the null hypothesis wrongly; this probability is never 0. This probability of a type I error can be precisely computed as

 

P (type I error) = significance level =α

 

The exact probability of a type II error is generally unknown.

 

If we do not reject the null hypothesis, it may still be false (a type II error) as the sample may not be big enough to identify the falseness of the null hypothesis (especially if the truth is very close to hypothesis).

 

For any given set of data, type I and type II errors are inversely related; the smaller the risk of one, the higher the risk of the other.

 

A type I error can also be referred to as an error of the first kind.

 

 

Type II Error

 

In a hypothesis test, a type II error occurs when the null hypothesis H0, is not rejected when it is in fact false. For example, in a clinical trial of a new drug, the null hypothesis might be that the new drug is no better, on average, than the current drug; i.e.

 

H0: there is no difference between the two drugs on average.

 

A type II error would occur if it was concluded that the two drugs produced the same effect, i.e. there is no difference between the two drugs on average, when in fact they produced different ones.

 

A type II error is frequently due to sample sizes being too small.

 

The probability of a type II error is generally unknown, but is symbolized by β and written

 

P (type II error) = β

 

A type II error can also be referred to as an error of the second kind.

 

Compare type I error.

 

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One Sample t-test

 

A one sample t-test is a hypothesis test for answering questions about the mean where the data are a random sample of independent observations from an underlying normal distribution N(µ,), where variance is unknown.

 

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The null hypothesis for the one sample t-test is:

 

H0: µ = µ0, where µ0 is known.

 

That is, the sample has been drawn from a population of given mean and unknown variance (which therefore has to be estimated from the sample).

 

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This null hypothesis, H0 is tested against one of the following alternative hypotheses, depending on the question posed:

 

H1: µ is not equal to µ0

 

H1: µ > µ0

 

H1: µ < µ0

 

Self-Check Exercise with solutions

 

 

Ques: For a retail shop in a mall, the target customer footfall is 50 but is not known. The sample measurements of customers that visit this shop in a day are 45, 54, 51, 47, 52, 50, 41, 51, 43 and 53. Test H0: µ = 50 against H1: < 50, with = 0.05.

 

 

Ans. Since is not known, we will make use of the sample standard deviation and take the help of t-statistic for constructing the rejection region.

 

The sample standard deviation(s) is given by

 

where

 

 

The t-statistics is given by

 

 

hence   = S

 

From the table of t-distribution, the value of t (0.05) for degree of freedom n-1 = 9 is 1.833, so the rejection region R; ttab -1.833

 

Since the calculated value of tcal = -0.924, falls in the acceptance region, H0 is accepted at 5% level of significance.

 

Ques: Suppose that you are a business analyst that is trying to establish that a country has been unfair with regard to its GDP growth rate figure. Suppose the mean GDP growth rate per year is 7%. What hypotheses would you set?

Ans: You set the null hypothesis to be

 

H0:     = .07

 

H1:    < .07

 

Ques: What is a type I error?

 

Ans: World bank does not sanction loan to the country, when it is making correct GDP projections.

 

Ques: What is a type II error?

 

Ans. World bank give loan to the country in spite of its wrong GDP projections.

 

Note: Larger results in a smaller  , and smaller     results in a larger  .

 

Summary

 

This module provides an opportunity to the students to develop an understanding about the basic concepts related to the hypothesis testing because it is an essential part of statistical inference. It is related to statistical estimation. Hypothesis testing consists of decision rules which are required in drawing probability inferences about the population. Mostly, we use the hypothesis testing to prove the null hypothesis or alternative hypothesis. In order to formulate such a test, usually some theory has been put forward, either because it is believed to be true or because it is to be used as a basis for argument, but has not been proved, i.e. claiming that mileage of a new model of a car is better than the existing model for same traffic conditions.

 

In most of the business probl ems, before framing the two competing claims hypotheses between which we have a choice; the null hypothesis, denoted H0, against the alternative hypothesis, denoted H1, we need to check the past data or facts of the business situation in the past. Althoughthese two competing hypotheses are to be treated on equal basis, yet special consideration is given to the null hypothesis that is framed based on experience or facts