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

**TEST OF HYPOTHESES FOR EQUALITY OF MEANS OF TWO POPULATIONS (Population
variances are known)**

**Step-1 :** Let *µ _{X}* and

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

*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 under *H*_{0}, where and are respectively the means of the two samples described in Step-2.

The approximate sampling
distribution of the test statistic Z = under *H*_{0}
(i.e., µ_{X} = µ_{Y}) is the *N(0,1)* distribution.

It may be noted that the test statistic, when .

**Step-5 : **Calculate the value of Z for the given samples (*x*_{1}, *x*_{2},* x*_{3},….* x*_{m}) and (*y*_{1}, *y*_{ 2},* y*_{ 3},….* y*_{m}) as .

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

**Step-6 :**
Find the critical value, *z*_{e},
corresponding to *α* and *H _{1}* from the following table

**Step-7 : **Make decision on *H _{0}* choosing the suitable
rejection rule from the following table corresponding to

Performance of students of X Standard in a national level talent
search examination was studied. The scores secured by randomly selected
students from two districts, *viz*., *D _{1}* and

**Step 1 : **Let** ***μ _{X}*

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

*i.e*., average scores secured by the students from the study districts
are not significantly* *different.

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

*i.e*., average scores secured by the students from the study districts
are significantly* *different. It is a two-sided alternative.

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

The given sample information are

Size of the Sample-1 (*m*) = 500

Size of the Sample-2 (*n*) = 800. Hence, both the samples are
large.

Mean of Sample-1 ( ) = 58

Mean of Sample-2 ( ) = 57

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

α= 5%

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

The test statistic under the null hypothesis *H _{0}*
is

Since both *m*
and *n* are large, the sampling
distribution of *Z* under *H _{0}* is the

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

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

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

Since *H _{1}* is a two-sided alternative hypothesis,
the critical value at

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

Since *H*_{1} is a two-sided alternative, elements of
the critical region are defined by the rejection rule |*z*_{0} | ≥
*z* * _{e}* =

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 Known) | Procedure Steps, Example Solved Problems | Statistics

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