Hypothesis Testing

Introduction

A statistical hypothesis is an assumption about an unknown population parameter.It is a well defined procedure which helps us to decide objectively whether to accept or reject the hypothesis based on the information available from the sample.In statistical analysis, we use the concept of probability to specify a probability level at which a researchers concludes that the observed difference between the sample statistics and population parameter is not due to chance.

Hypothesis Testing Procedure

Step 1: - Set Null and Alternative Hypothesis

The null hypothesis is denoted by Ho, is the hypothesis which is tested for the possible rejection under the assumption that it is true.Theoretically, Ho is set as no difference considered true, until and unless it is proved wrong by the collected sample data.The alternative hypothesis is denoted by H1 or Hα, is a logical opposite of the Ho.

Step 2: - Determine the appropriate Statistical Test

After setting he hypothesis, the researches has to

decide on an appropriate statistical test that will be

tested for the statistical analysis.

The statistic used in the study (mean, proportion,

variance etc.) must also be considered when a

researchers decides on appropriate statistical test,

which can be applied for hypothesis testing in order

to obtain the best results.

Step 3: -set the level of significance

The level of significance is denoted by α is the

probability, which is attached to a null hypothesis, which

may be rejected even when it is true.

The level of significance also known as the size of the

rejection region or the size of the critical region.

Level of significance must be determined before we draw

samples, so that the obtained result is free from the bias of

a decision maker.

0.01, 0.05, 0.010

Step 4: - Set the decision Rule

If the computed value of the test statistic falls in the

Next step is to establish a critical region, which is

the area under the normal curve . These regions are

termed as acceptance region (when the Ho is

accepted) and the rejection region or critical region.

acceptance region , the null hypo is accepted .

Otherwise Ho is rejected.

Step 5: - Collect the sample data

In this stage data are collected and appropriate

sample statistics are computed.

The first 4 steps should be completed before

collecting the data for the study.

Step 6: - Analyze the data

In this step the researcher has to compute the test

statistic. This involves selection of appropriate

probability distribution for a particular test.

For Example- When the sample is small, then t-

distribution is used. If sample size is large then use Z-

test.

Some commonly used testing procedures are F, t, Z, chi

square.

Step 7: - Arrive at a statistical conclusion

In this step the researcher draw a conclusion.

A statistical conclusion is a decision to accept

or reject a Ho. This depends whether the

computed statistic falls in the acceptance

region or rejection region.

Critical Region

The critical region (or rejection region ) is the set of all values of the test statistic that cause us to reject the null hypothesis.

Significance Level

The significance level (denoted by ) is the probability that the test statistic will fall in the critical region when the null hypothesis is actually true. Common choices for are 0.05, 0.01, and 0.10.

Critical Value

A critical value is any value separating the critical region (where we reject the H0) from the values of the test statistic that does not lead to rejection of the null hypothesis, the sampling distribution that applies, and the significance level .

F-TEST

The name was coined by George W. Snedecor, in honour of Sir Ronald A.isher.

Any statistical test in which the test statistic has an F-distribution under the null hypothesis.

It is most often used when comparing statistical models that have been fit to a data set, in order to identify the model that best fits the population from which the data were sampled.

The F-test is designed to test if two population variances are equal. It does this by comparing the ratio of two variances. So, if the variances are equal, the ratio of the variances will be 1.

CALCULATION OF F-TEST

The F-tests arise by assuming decomposition of variability while collecting data with sums of squares.

These sums of squares are developed together such that the statistics tends greater in a condition when the null hypothesis is false.

When using the F-test, you again require a hypothesis to compare standard deviations. That is, you will test the null hypothesis Ho:σ12 = σ22 against an appropriate alternate hypothesis.

You calculate the value of the F-test as the ratio of the two variance where s12 ≥ s22, so that F ≥ 1. The degrees of freedom for the numerator and denominator are n1-1 and n2-1, respectively.

The degrees of freedom for the numerator and denominator are n1-1 and n2-1, respectively.

As with the t-test, you compare Fcalc to a tabulated value Ftab, to see if you should accept or reject the null hypothesis.

PROBLEM DEFINITION

XYZ constructions is a leading company in the construction

sector in Delhi . It wants to construct flats in two areas,

Indrapuram & Pritampura . The company wants to estimate

the amount that customers are willing to spend on purchasing

a flat in the two areas . It randomly selected 21 potential

customers from Pritampura and 27 from Indrapuram and

posed the question, “how much are you willing to spend on

a flat?” The data collected from the two areas is shown in

table below.

The company assumes that the intention to purchase of the

customers is normally distributed with equal variance in the

two areas taken for the study. On the basis of the samples

taken for the study, estimate the difference in population

means taking 95% as the confidence level.

Proposed Exp. On flats by customers from Indrapuram (in thousand Rs.)

125 165

130 170

126 130

127 145

150 130

135 140

140 150

160 160

120 140

150 145

155 165

145

140

165

135

130

Proposed Exp. On flats by customers from Pritampura (in thousand Rs.)

185 135

165 185

160 180

170 190

180 145

190 160

170

150

155

160

145

150

155

160

145

140