Hypothesis Testing Theory-munu

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General Description

There are two types of statistical inferences: estimation of population parameters

and hypothesis testing. Hypothesis testing is one of the most important tools of

application of statistics to real life problems. Most often, decisions are required to

be made concerning populations on the basis of sample information. Statisticaltests are used in arriving at these decisions.

There are five ingredients to any statistical test :

(a) Null Hypothesis

(b) Alternate Hypothesis

(c) Test Statistic

(d) Rejection/Critical Region

(e) Conclusion

In attempting to reach a decision, it is useful to make an educated guess or assumption about the population involved, such as the type of distribution.

Statistical Hypotheses : They are defined as assertion or conjecture about the

parameter or parameters of a population, for example the mean or the variance of a

normal population. They may also concern the type, nature or probability

distribution of the population.

Statistical hypotheses are based on the concept of proof by contradiction. For

example, say, we test the mean (Q) of a population to see if an experiment has

caused an increase or decrease in Q. We do this by proof of contradiction byformulating a null hypothesis.

Null Hypothesis : It is a hypothesis which states that there is no difference

between the procedures and is denoted by H 0. For the above example the

corresponding H0 would be that there has been no increase or decrease in the mean.

Always the null hypothesis is tested, i.e., we want to either accept or reject the null

hypothesis because we have information only for the null hypothesis.

Alternative Hypothesis : It is a hypothesis which states that there is a difference

between the procedures and is denoted by H A.

Table 1. Various types of H0 and H A

Case Null Hypothesis H 0 Alternate Hypothesis H A

1 Q = Q Q ¿Q



2 Q Q Q " Q

3 Q " Q Q Q

Test Statistic : It is the random variable X whose value is tested to arrive at adecision. The Central Limit Theorem states that for large sample sizes (n > 30)

drawn randomly from a population, the distribution of the means of those samples

will approximate normality, even when the data in the parent population are not

distributed normally. A z statistic is usually used for large sample sizes (n > 30),

but often large samples are not easy to obtain, in which case the t -distribution can

be used. The population standard deviation W is estimated by the sample standard

deviation, s. The t curves are bell shaped and distributed around t=0. The exact

shape on a given t-curve depends on the degrees of freedom. In case of performing

multiple comparisons by one way Anova, the F-statistic is normally used.It is

defined as the ratio of the mean square due to the variability between groups to themean square due to the variability within groups. The critical value of F is read off

from tables on the F-distribution knowing the Type-I error Eand the degrees of

freedom between & within the groups.

R ejection R egion : It is the part of the sample space (critical region) where the

null hypothesis H0 is rejected. The size of this region, is determined by the

probability (E) of the sample point falling in the critical region when H0 is

true. E is also known as the level of significance, the probability of the value of the

random variable falling in the critical region. Also it should be noted that the term

"Statistical significance" refers only to the rejection of a null hypothesis at some

level E.It implies only that the observed difference between the sample statistic andthe mean of the sampling distribution did not occur by chance alone.

Conclusion : If the test statistic falls in the rejection/critical region, H0 is rejected,

else H0 is accepted.

Go to Table of Contents

T

ypes of T

ests Tests of hypothesis can be carried out on one or two samples. One sample tests are

used to test if the population parameter (Q) is different from a specified value. Two

sample tests are used to detect the difference between the parameters of two

populations (Q1 and Q2).



Two sample tests can further be classified as unpaired or paired two sample tests.

While in unpaired two sample tests the sample data are not related, in paired two

sample tests the sample data are paired according to some identifiable

characteristic. For example, when testing hypothesis about the effect of a treatment

on (say) a landfill, we would like to pair the data taken at different points before

and after implementation of the treatment.

Both one sample and two sample tests can be classified as :

One tailed test : Here the alternate hypothesis HA is one-sided and we test whether

the test statistic falls in the critical region on only one side of the distribution.

1. One sample test: For example, we are measuring the concentration of a lake

and we need to know if the mean concentration of the lake is greater than a

specified value of 10mg/L.

Hence, H0: Q¿ 10 mg/L, vs, HA: Q > 10 mg/L.

2. Two sample test: In Table1, cases 2 and 3 are illustrations of two sample,one tailed tests. In case 2 we want to test whether the population mean of the

first sample is lesser than that of the second sample.

Hence, H0: Q1 ¿ Q2 , vs, HA: Q1 < Q2.

Two tailed test : Here the alternate hypothesis HA is formulated to test for

difference in either direction, i.e., for either an increase or a decrease in the random

variable. Hence the test statistic is tested for occurrence within either of the two

critical regions on the two extremes of the distribution.

1. One sample test: For the lake example we need to know if the mean

concentration of the lake is the same as or different from a specified value of

10 mg/L.

Hence, H0: Q¿ 10 mg/L, vs, HA: Q = 10 mg/L.

2. Two sample test: In Table 1, case 1 is an illustration of a two sample two

tailed test. In case 1 we want to test whether the population mean of the first

sample (Q1) is the same as or different from the mean of the second sample

(Q2).Hence H0: Q!Q2 , vs, HA: Q¿Q2.

Given the same level of significance the two tailed test is more conservative, i.e., it

is more rigorous than the one-tailed test because the rejection point is farther out inthe tail. It is more difficult to reject H 0 with a two-tailed test than with a one-tailed

test.

When using probability to decide whether a statistical test provides evidence for or

against our predictions, there is always a chance of driving the wrong conclusions.

Even when choosing a probability level of 95%, there is always a 5% chance that



one rejects the null hypothesis when it was actually correct. This is called Type I

error, represented by the Greek letter E.

It is possible to err in the opposite way if one fails to reject the null hypothesis

when it is, in fact, incorrect. This is called Type II error, represented by the Greek

letter F. These two errors are represented in the following chart.

Table 2. Types of error

Type of decision H0 true H0 false

Reject H0 Type I error (E) Correct decision (1- F)

Accept H0 Correct decision (1-E) Type II error ( F)

A related concept is power, which is the probability of rejecting the null hypothesis

when it is actually false. Power is simply 1 minus the Type II error rate, and is

usually expressed as 1- F.

When choosing the probability level of a test, it is possible to control the risk of

committing a Type I error by choosing an appropriate E.

This also affects Type II error, since they are are inversely related: as one

increases, the other decreases. To appreciate this in a diagram, follow this link:

Choice of E

There is little control on the risk of committing Type II error, because it also

depends on the actual difference being evaluated, which is usually unknown. The

following link leads to a diagram that illustrates how at a fixed Evalue, the F value

changes according to the actual distribution of the population:

Changes in F

The consequences of these different types of error are very different. For example,

if one tests for the significant presence of a pollutant, incorrectly deciding that a

site is polluted (Type I error) will cause a waste of resources and energy cleaning

up a site that does not need it. On the other hand, failure to determine presence of



pollution (Type II error) can lead to environmental deterioration or health problems

in the nearby community.

Go to Table of Contents

Steps in Hypothesis Testing

1

Identify the null hypothesis H0 and the alternate hypothesis HA.

2

Choose E. The value should be small, usually less than 10%. It is important toconsider the consequences of both types of errors.

3

Select the test statistic and determine its value from the sample data. This value iscalled the observed value of the test statistic. Remember that a t statistic is usually

appropriate for a small number of samples; for larger number of samples, a z

statistic can work well if data are normally distributed.

4

Compare the observed value of the statistic to the critical value obtained for the

chosenE.

5



Make a decision.

If the test statistic falls in the criticalregion:

Reject H0 in favour of HA.

If the test statistic does not fall in thecritical region:

Conclude that there is not enough evidence to reject H0.

Hypothesis Testing Theory-munu

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