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General Procedure for Test of Hypotheses For Equality of Means of Two Populations (Population Variances are Unknown): Procedure Steps, Example Solved Problems

**TEST OF HYPOTHESES FOR EQUALITY OF MEANS OF TWO POPULATIONS ( POPULATION
VARIANCES ARE UNKNOWN)**

**Step-1 : **Let** ***µ _{X}*

Frame the null hypothesis as *H _{0}: µ_{X}*

(i) *H _{1}: µ_{X}*

**Step 2 : **Let (*X _{1}, X_{2}, …, X_{m}*) be a
random sample of

**Step 3 : **Specify the level of significance,** ***α*.

**Step 4 : **Consider the test statistic

*i.e*., the
above test statistic is obtained from Z considered in the test described in* *Section 1.11 by substituting *S _{X}*

The approximate sampling distribution of the test
statistic under *H _{0 }*is
the

**Step 5 : **Calculate the value of** ***Z*** **for the given samples
(*x _{1}, x_{2}, ...,x_{m}*) and (

Here and are respectively the values of and for the given samples.

Also,
s_{x}^{2} and s_{y}^{2} are respectively the
values of S_{X}^{2} and S_{Y}^{2} for the given
samples.

**Step 6 : **Find the critical value,** ***z _{e}*,
corresponding to α and

**Step 7 : **Make decision on** ***H _{0}*

A Model Examination was conducted to XII Standard students in the
subject of Statistics. A District Educational Officer wanted to analyze the
Gender-wise performance of the students using the marks secured by randomly
selected boys and girls. Sample measures were calculated and the details are presented
below:

Test, at 5% level of significance, whether performance of the
students differ significantly with respect to their gender.

**Step 1 :** Let *μ _{X}* and

**Null hypothesis: ***H _{0}*:

*i.e*., there is no significant difference in the performance of the
students with respect to* *their gender.

**Alternative hypothesis: ***H*_{1}** **:** ***µ*** **_{X}** **≠** ***µ*_{Y}

*i.e*., performance of the students differ significantly with the
respect to the gender. It is
a two-sided alternative hypothesis.

**Step 2 : ****Data**

The given sample information are

Since *m ≥* 30 and *n ≥* 30, both the samples are large.

**Step 3 :**** Level of significance**

α= 5%

**Step 4 :** **Test statistic**

The test statistic under *H _{0}* is

The sampling distribution of *Z* under *H _{0}*
is the

**Step 5 :**** Calculation of the Test Statistic**

The value of *Z* is calculated for the given sample
informations from

**Step 6 :**** Critical value**

Since *H _{1}* is a two-sided alternative, the critical
value at 5% level of significance is

**Step 7 :**** Decision**

Since *H _{1}* is a two-sided alternative, elements of
the critical region are determined by the rejection rule |

Tags : Procedure Steps, Example Solved Problems | Statistics , 12th Statistics : Chapter 1 : Tests of Significance - Basic Concepts and Large Sample Tests

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12th Statistics : Chapter 1 : Tests of Significance - Basic Concepts and Large Sample Tests : Test of Hypotheses For Equality of Means of Two Populations (Population Variances are Unknown) | Procedure Steps, Example Solved Problems | Statistics

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