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General Procedure for Test of Hypotheses for Population Mean (Population Variance is Unknown): Procedure Steps, Example Solved Problems

**TEST OF HYPOTHESES FOR POPULATION MEAN ( POPULATION VARIANCE IS
UNKNOWN)**

**Step 1 :** Let *µ _{x}* and

(i) H1: µ ≠ µ0 (ii) H1: µ > µ0 (iii) H1: µ < µ0

**Step 2 : **Let (X_{1}, X_{2}, …, X_{n}) be a random
sample of *n* observations drawn from
the population, where *n* is large (*n* ≥ 30).

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

**Step 4 : **Consider the test statistic Z = under H_{0}, where and S are the sample mean and sample standard deviation respectively. It may be noted that the above
test statistic is obtained from *Z* considered in the test described in
Section 1.9 by substituting *S* for *σ*.

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

**Step 5 : **Calculate the value of *Z* for the given sample (*x _{1},
x_{2}, ..., x_{n}*) as . Here, and s are respectively the
values of and S calculated for the given sample.

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

**Step
7:**
Decide on H_{0} choosing the suitable rejection rule from the following
table corresponding to H_{1}.

A motor vehicle manufacturing company desires to introduce a new
model motor vehicle. The company claims that the mean fuel consumption of its
new model vehicle is lower than that of the existing model of the motor
vehicle, which is 27 kms/litre. A sample of 100 vehicles of the new model
vehicle is selected randomly and their fuel consumptions are observed. It is
found that the mean fuel consumption of the 100 new model motor vehicles is 30
kms/litre with a standard deviation of 3 kms/litre. Test the claim of the
company at 5% level of significance.

**Step 1 : **Let the fuel consumption of the new model motor vehicle be assumed
to be distributed** **according to a distribution with mean and standard
deviation respectively *μ* and *σ*. The null and alternative
hypotheses are

**Null hypothesis ***H*_{0}:** ***μ*** **= 27

*i.e*., the average fuel consumption of the company’s new model motor
vehicle is not* *significantly different from that of the existing model.

**Alternative hypothesis ***H*_{1}:** ***μ >*** **27

*i.e*., the average fuel consumption of the company’s new model motor
vehicle is* *significantly lower than that of the existing model. In other
words, the number of kms by the new model motor vehicle is significantly more
than that of the existing model motor vehicle.

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

The given sample information are

Size of the sample (*n*) = 100. Hence, it is a large sample.

Sample mean ( )= 30

Sample standard deviation(s) = 3

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

α= 5%

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

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

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

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

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

**Step 6 : ****Critical Value**

Since *H _{1}* is a one-sided (right) alternative
hypothesis, the critical value at

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

Since *H _{1}* is a one-sided (right) alternative,
elements of the critical region are defined 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 Population Mean (Population Variance is Unknown) | Procedure Steps, Example Solved Problems | Statistics

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