Project Report

On

TESTS OF HYPOTHESIS

Submitted To: Submitted By: Simi Mam Sumeet Singh MBA (Gen)

Roll. No.= 89 Sec = B

Contents

Sr.No. Subject Covered Page No.

1 Introduction 3

2 Decision Errors: 5

3 Decision Rules: 5

4 One-Tailed and Two-Tailed

Tests:

7

5 A General Procedure for

Conducting Hypothesis Tests:

8-10

2

6Common test statistics

11-13

7 Importance: 14

8 Criticism: 14

Introduction:

A statistical hypothesis test is a method of making statistical

decisions using experimental data. In statistics, a result is called

statistically significant if it is unlikely to have occurred by

chance. The phrase "test of significance" was coined by Ronald

Fisher: "Critical tests of this kind may be called tests of

significance, and when such tests are available we may

discover whether a second sample is or is not significantly

different from the first.

Hypothesis testing is sometimes called confirmatory data

analysis, in contrast to exploratory data analysis. In frequency

probability, these decisions are almost always made using null-

hypothesis tests; that is, ones that answer the question

Assuming that the null hypothesis is true, what is the

probability of observing a value for the test statistic that is at

least as extreme as the value that was actually observed? One

use of hypothesis testing is deciding whether experimental

results contain enough information to cast doubt on

conventional wisdom.

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Statistical hypothesis testing is a key technique of frequentist

statistical inference, and is widely used, but also much

criticized. The main alternative to statistical hypothesis testing

is Bayesian inference.

The critical region of a hypothesis test is the set of all outcomes

which, if they occur, will lead us to decide that there is a

difference. That is, cause the null hypothesis to be rejected in

favour of the alternative hypothesis. The critical region is

usually denoted by C.

A statistical hypothesis is an assumption about a population

parameter. This assumption may or may not be true. The best

way to determine whether a statistical hypothesis is true would

be to examine the entire population. Since that is often

impractical, researchers typically examine a random sample

from the population. If sample data are consistent with the

statistical hypothesis, the hypothesis is accepted; if not, it is

rejected.

There are two types of statistical hypotheses.

1. Null hypothesis . The null hypothesis, denoted by H0, is

usually the hypothesis that sample observations result

purely from chance.

2. Alternative hypothesis . The alternative hypothesis,

denoted by H1 or Ha, is the hypothesis that sample

observations are influenced by some non-random cause.

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For example, suppose we wanted to determine whether a coin

was fair and balanced. A null hypothesis might be that half the

flips would result in Heads and half, in Tails. The alternative

hypothesis might be that the number of Heads and Tails would

be very different. Symbolically, these hypotheses would be

expressed as

H0: P = 0.5

Ha: P ≠ 0.5

Suppose we flipped the coin 50 times, resulting in 40 Heads and

10 Tails. Given this result, we would be inclined to reject the

null hypothesis and accept the alternative hypothesis.

Decision Errors:

Two types of errors can result from a hypothesis test.

1. Type I error . A Type I error occurs when the researcher

rejects a null hypothesis when it is true. The probability of

committing a Type I error is called the significance level.

This probability is also called alpha, and is often denoted

by α.

2. Type II error . A Type II error occurs when the researcher

accepts a null hypothesis that is false. The probability of

committing a Type II error is called Beta, and is often

denoted by β. The probability of not committing a Type II

error is called the Power of the test.

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Decision Rules:

The analysis plan includes decision rules for accepting or

rejecting the null hypothesis. In practice, statisticians describe

these decision rules in two ways - with reference to a P-value or

with reference to a region of acceptance.

1. P-value. The strength of evidence in support of a null

hypothesis is measured by the P-value. Suppose the test

statistic is equal to S. The P-value is the probability of

observing a test statistic as extreme as S, assuming the

null hypothesis is true. If the P-value is less than the

significance level, we reject the null hypothesis.

2. Region of acceptance. The region of acceptance is a

range of values. If the test statistic falls within the region

of acceptance, the null hypothesis is accepted. The region

of acceptance is defined so that the chance of making a

Type I error is equal to the significance level.

The set of values outside the region of acceptance is called the

region of rejection. If the test statistic falls within the region

of rejection, the null hypothesis is rejected. In such cases, we

say that the hypothesis has been rejected at the α level of

significance.

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These approaches are equivalent. Some statistics texts use the

P-value approach; others use the region of acceptance

approach. In subsequent lessons .

One-Tailed and Two-Tailed Tests:

1. One-Tailed Test : A test of a statistical hypothesis, where

the region of rejection is on only one side of the sampling

distribution, is called a one-tailed test.

For example, suppose the null hypothesis states that the

mean is less than or equal to 10. The alternative hypothesis

would be that the mean is greater than 10. The region of

rejection would consist of a range of numbers located located

on the right side of sampling distribution; that is, a set of

numbers greater than 10.

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2. Two-Tailed Test: A test of a statistical hypothesis, where

the region of rejection is on both sides of the sampling

distribution, is called a two-tailed test.

For example, suppose the null hypothesis states that the

mean is equal to 10. The alternative hypothesis would be

that the mean is less than 10 or greater than 10. The region

of rejection would consist of a range of numbers located

located on both sides of sampling distribution; that is, the

region of rejection would consist partly of numbers that were

less than 10 and partly of numbers that were greater than 10

A General Procedure for Conducting

Hypothesis Tests:

All hypothesis tests are conducted the same way. The

researcher states a hypothesis to be tested, formulates an

analysis plan, analyzes sample data according to the plan, and

accepts or rejects the null hypothesis, based on results of the

analysis.

1. State the hypotheses. Every hypothesis test requires

the analyst to state a null hypothesis and an alternative

hypothesis. The hypotheses are stated in such a way that

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they are mutually exclusive. That is, if one is true, the

other must be false; and vice versa.

2. Formulate an analysis plan. The analysis plan describes

how to use sample data to accept or reject the null

hypothesis. It should specify the following elements.

a. Significance level . Often, researchers choose

significance levels equal to 0.01, 0.05, or 0.10; but

any value between 0 and 1 can be used.

b. Test method. Typically, the test method involves a

test statistic and a sampling distribution. Computed

from sample data, the test statistic might be a mean

score, proportion, difference between means,

difference between proportions, z-score, t-score, chi-

square, etc. Given a test statistic and its sampling

distribution, a researcher can assess probabilities

associated with the test statistic. If the test statistic

probability is less than the significance level, the null

hypothesis is rejected.

3. Analyze sample data. Using sample data, perform

computations called for in the analysis plan.

a. Test statistic. When the null hypothesis involves a

mean or proportion, use either of the following

equations to compute the test statistic.

Test statistic = (Statistic - Parameter) / (Standard deviation of statistic)

Test statistic = (Statistic - Parameter) / (Standard error of statistic)

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where Parameter is the value appearing in the null

hypothesis, and Statistic is the point estimate of Parameter.

As part of the analysis, you may need to compute the

standard deviation or standard error of the statistic.

Previously, we presented common formulas for the standard

deviation and standard error.

When the parameter in the null hypothesis involves

categorical data, you may use a chi-square statistic as the

test statistic. Instructions for computing a chi-square test

statistic are presented in the lesson on the chi-square

goodness of fit test.

b. P-value . The P-value is the probability of observing a

sample statistic as extreme as the test statistic,

assuming the null hypothesis is true.

4. Interpret the results. If the sample findings are unlikely,

given the null hypothesis, the researcher rejects the null

hypothesis. Typically, this involves comparing the P-value

to the significance level, and rejecting the null hypothesis

when the P-value is less than the significance level.

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Common test statistics

In the table below, the symbols used are defined at the bottom of the table. Many other tests can be found in other articles.

Name Formula Assumptions or notes

One-sample z-test

(Normal population or n > 30) and σ known.

(z is the distance from the mean in relation to the standard deviation of the mean). For non-normal distributions it is possible to calculate a minimum proportion of a population that falls within k standard deviations for any k (see: Chebyshev's inequality).

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Two-sample z-test

Normal population and independent observations and σ1 and σ2 are known

One-sample t-test

(Normal population or n > 30) and σ unknown

Paired t-test(Normal population of differences or n > 30) and σ unknown

Two-sample pooled t-test, equal variances*

[5]

(Normal populations or n1 + n2 > 40) and independent observations and σ1 = σ2 and σ1 and σ2 unknown

Two-sample unpooled t-test, unequal variances*

[6]

(Normal populations or n1 + n2 > 40) and independent observations and σ1 ≠ σ2 and σ1 and σ2 unknown

One-proportion z-test

n .p0 > 10 and n (1 − p0) > 10 and it is a SRS (Simple Random Sample).

Two-proportion z-test, pooled

n1 p1 > 5 and n1(1 − p1) > 5 and n2 p2 > 5 and n2(1 − p2) > 5 and independent observations

Two-proportion z-test, unpooled

n1 p1 > 5 and n1(1 − p1) > 5 and n2 p2 > 5 and n2(1 − p2) > 5 and independent observations

One-sample chi-square test

One of the following

• All expected counts are at least 5

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• All expected counts are > 1 and no more that 20% of expected counts are less than 5

*Two-sample F test for equality of variances

Arrange so > and reject H0 for F > F(α / 2,n1 − 1,n2 − 1)[7]

α, the probability of Type I error (rejecting a null hypothesis when it is in fact true)

n = sample size

n1 = sample 1 size

n2 = sample 2 size = sample mean μ0 = hypothesized

population mean μ1 = population 1

mean μ2 = population 2

mean σ = population

standard deviation

σ2 = population variance

s = sample standard deviation

s2 = sample variance

s1 = sample 1 standard deviation

s2 = sample 2 standard deviation

t = t statistic

df = degrees of freedom

= sample mean of differences

d0 = hypothesized population mean difference

sd = standard deviation of differences

= x/n = sample proportion, unless specified otherwise

p0 = hypothesized population proportion

p1 = proportion 1

p2 = proportion 2

min{n1,n2} = minimum of n1 and n2

x1 = n1p1

x2 = n2p2

χ2 = Chi-squared statistic

F = F statistic

In general, the subscript 0 indicates a value taken from the null hypothesis, H0, which should be used as much as possible in constructing its test statistic.

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Importance:

Statistical hypothesis testing plays an important role in the

whole of statistics and in statistical inference. For example,

Lehmann (1992) in a review of the fundamental paper by

Neyman and Pearson (1933) says: "Nevertheless, despite

their shortcomings, the new paradigm formulated in the 1933

paper, and the many developments carried out within its

framework continue to play a central role in both the theory

and practice of statistics and can be expected to do so in the

foreseeable future".

Criticism:

Some statisticians have commented that pure "significance

testing" has what is actually a rather strange goal of

detecting the existence of a "real" difference between two

populations. In practice a difference can almost always be

found given a large enough sample, the typically more

relevant goal of science is a determination of causal effect

size. The amount and nature of the difference, in other

words, is what should be studied. Many researchers also feel

that hypothesis testing is something of a misnomer. In

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practice a single statistical test in a single study never

"proves" anything.

Rejection of the null hypothesis at some effect size has no

bearing on the practical significance at the observed effect

size. A statistically significantly not be relevant in practice

due to other, larger effects of more concern, whilst a true

effect of practical significance may not appear statistically

significant if the test lacks the power to detect it. Appropriate

specification of both the hypothesis and the test of said

hypothesis is therefore important to provide inference of

practical utility.

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