Hypothesis:

It is an assumption of population parameter ( mean, proportion, variance)

There are two types of hypothesis :

1) 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.

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

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 : It is the assumption which we wish to test and whose validity is tested for possible rejection on the basis of sample information.It is denoted by Ho

Alternative Hypothesis : It is the hypothesis which differs from the null hypothesis. It is not tested It is denoted by H1 or Ha

Level of significance : It is the maximum probability of making a type I error when the null hypothesis is true as an equality.

It is usually expressed as % and is denoted by symbol α( called alpha)It is used as a guide in decision making. It is used to indicate the upper limit of the size of the critical region TEST STATISTIC

Test Statistic Used for

Z test For test of hypothesis involving large samples i.e. > 30

t-test For test of hypothesis involving small samples i.e. < 30And if σ is unknown

χ2 For testing the discrepancy between observed frequency and expected frequency without any reference to population parameter

Critical Region or Rejection Region :Critical region is the region which corresponds to a pre-determined level of significance. The set of value of the test statistic which leads to rejection of the null hypothesis is called region of rejection or Critical region of the test. Conversely the set of values of the test statistic which leads to the acceptance of Ho is called region of acceptance.

Critical Value :It is that value of statistics which separate 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.

Type I and type II error:

The decision to accept or reject null hypothesis Ho is made on the basis of the information supplied by the sample data. There is always a chance of committing an error

Type I error:

This is an 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 an error committed by the test in accepting a false null hypothesis. The probability of committing type II error is denoted by ‘β’.

Degrees of freedom:

It means the number of variables for which one has freedom to choose.

Need for degree of freedom:

Influence the value of discrepancy between the observed and expected values

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

Case Formula for d.o.f Example

In case of one sample [ say n=50]

Sample size (n)-1 50-1=49

In case of two sample[say n1 =50,n2=60]

(n1 - 1) + (n2 -1 )Orn1 + n2 – 2

(50-1)+(60-1) = 108Or(50+60-2) = 108

P –value : It is the probability , computed using the test statistic, that measure the support ( or lack of support) provided by the sample for the null hypothesis.

Critical Region

• Set of all values of the test statistic that would cause a rejection of the null hypothesis

CriticalRegion

Critical Region

• Set of all values of the test statistic that would cause a rejection of the

• null hypothesis

CriticalRegion

Critical Region

• Set of all values of the test statistic that would cause a rejection of the

null hypothesis

CriticalRegions

Critical ValueValue (s) that separates the critical region

from the values that would not lead to a rejection of H 0

Critical Value( z score )

Left-tailed TestH0: µ ³ 200

H1: µ < 200

200

Values that differ significantly

from 200

Fail to reject H0Reject H0

Points Left

Right-tailed TestH0: µ £ 200

H1: µ > 200

Values that differ significantly

from 200200

Fail to reject H0 Reject H0

Points Right

Two-tailed TestH0: µ = 200H1: µ ¹ 200

Means less than or greater than

Fail to reject H0Reject H0 Reject H0

200

Values that differ significantly from 200

a is divided equally between the two tails of the critical

region

Lower Tail Test

Upper Tail Test

Two-Tail Test

Hypothesis Ho : μ ≥ μoH1 : μ < μo

Ho : μ ≤ μoH1 : μ > μo

Ho : μ = μoH1 : μ ≠ μo

Test Z Z ZRejection Rule p-value approach

Reject Ho if p-value ≤ α

Reject Ho if p-value ≤ α

Reject Ho if p-value ≤ α

Rejection rule critical value approach

Reject Ho if Z ≤ -Zα

Reject Ho if Z ≥ Z α

Reject Ho if Z ≤ -Zα or Z ≥ Z α

If the distribution of a population is essentially normal, then the distribution of

is a Student t Distribution for all samples of size n. It is often referred to as a t distribution and is used to find critical values denoted byta/2.

t =x - µ

sn

Student t Distribution

t-distribution (DF = 24)Assume the conjecture is true!

t = x – µxS / n

Test Statistic:

Critical value = 1.71 * 8000/5 + 30000 = 32736

30 K( t = 0)

Fail to reject H0 Reject H0

32.7 k( t = 1.71 )

Margin of Error E for Estimate of (With σ Not Known)

Formula 7-6

where ta/2 has n – 1 degrees of freedom.

nsE = ta/2

Table lists values for tα/2

= population mean = sample means = sample standard deviationn = number of sample valuesE = margin of errorta/2 = critical t value separating an area of a/2 in the right

tail of the t distribution

Notation

x

Important Properties of the Student t Distribution

1. The Student t distribution is different for different sample sizes (see the following slide, for the cases n = 3 and n = 12).

2. The Student t distribution has the same general symmetric bell shape as the standard normal distribution but it reflects the greater variability (with wider distributions) that is expected with small samples.

3. The Student t distribution has a mean of t = 0 (just as the standard normal distribution has a mean of z = 0).

4. As the sample size n gets larger, the Student t distribution gets closer to the normal distribution.

Application of t-distribution

1) To test the significance of the mean of random sample

2) To test the significance of the difference between the mean of two independent samples

3) To test the significance of the difference between the mean of two dependent samples of the paired observation

4) To test the significance of an observed correlation coefficient.

Choosing the Appropriate Distribution

Use the normal (z) distribution

known and normally distributed populationor known and n > 30

Use t distribution not known and normally distributed populationor not known and n < 30

Chi Square Method

Use a nonparametric method or bootstrapping

Population is not normally distributed

𝛘2 Defined :It is pronounced as Chi-Square test is one of the simplest and most widely used non-parametric tests in statistical work. It describes the magnitude of the discrepancy between theory and observation. It is defined as

𝛘2 = Where O= observed frequencies ,E= Expected frequencies

Rejection Rule : p- value approach : reject Ho if p-value ≤ αcritical value approach :reject Ho if 𝛘2 ≥ 𝛘α2

where α is the level of significance with n rows and n columns provide (n-1)(m-1) degrees of freedom

Properties of the Distribution of the Chi-Square Statistic

The chi-square distribution is not symmetric, unlike the normal and Student t distributions.

Chi-Square Distribution Chi-Square Distribution for df = 10 and df = 20

As the number of degrees of freedom increases, thedistribution becomes more symmetric.

1. The values of chi-square can be zero or positive, but they cannot be negative.

2. The chi-square distribution is different for each number of degrees of freedom, which is d.f = n – 1. As the number of degrees of freedom increases, the chi-square distribution approaches a normal distribution.

In Table, each critical value of 2 corresponds to an area given in the top row of the table, and that area represents the cumulative area located to the right of the critical value.

Properties of the Distribution of the Chi-Square Statistic

Condition when applying chi-square test

1. In the first place N must be reasonably large to ensure the similarity between theoretically correct distribution and our sampling distribution of 𝛘2 it is difficult to say what constitutes largeness, but as a general rule 𝛘2 test should not be used when N is less than 50, however few the cells

2. No theoretical cell frequency should be small when the expected frequency are too small., the value of 𝛘2

will be overestimated and will result in too many restrictions of the null hypothesis

3. The constraints on the cell frequencies if any should be linear