procedure of testing a Hypothesis.

Q.6 (a) Define the term ‘Hypothesis’. Explain in detail the procedure of testing a Hypothesis. ( Mcom-l 2024 )
Meaning / Definition of Hypothesis
A hypothesis is a tentative statement or assumption about a population parameter which we want to test on the basis of sample information.
It is a logical guess about the value of a population mean, proportion, difference of means, etc., framed in a way that it can be tested statistically. procedure of testing a Hypothesis.
Example: “The average monthly income of teachers is ₹40,000” is a hypothesis about population mean.
Procedure / Steps of Testing a Hypothesis
- Formulation of hypotheses
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First of all two hypotheses are framed:
(i) Null hypothesis (H₀) – It is a statement of no difference / no effect / no change.
Example: H₀ : μ = 40,000 (average income is ₹40,000).
(ii) Alternative hypothesis (H₁ or Hₐ) – It is a statement that contradicts H₀ and represents what we want to prove.
Example: H₁ : μ ≠ 40,000 (average income is not ₹40,000).
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First of all two hypotheses are framed:
- Selection of level of significance (α)
- Decide the maximum probability of rejecting a true H₀ which the researcher is willing to take.
- Common levels: 5% (0.05) or 1% (0.01).
- Smaller α means stricter test and lesser chance of Type I error. procedure of testing a Hypothesis.
- Selection of an appropriate test statistic
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According to the nature of the problem, size of sample and type of data, choose a suitable test:
- Z-test, t-test, chi-square test, F-test, etc.
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Define the test statistic formula, e.g.
for testing a population mean when σ is known. procedure of testing a Hypothesis.
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According to the nature of the problem, size of sample and type of data, choose a suitable test:
- Determination of sampling distribution and critical region
- Identify the sampling distribution of the test statistic under H₀ (normal, t, chi-square, F).
- For the chosen α, obtain the critical value(s) from statistical tables.
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Decide whether the test is:
- Two-tailed (H₁: parameter ≠ hypothesised value), or
- Left-tailed (H₁: parameter < value), or
- Right-tailed (H₁: parameter > value).
- The critical region (rejection region) consists of those values of the test statistic for which H₀ will be rejected. procedure of testing a Hypothesis.
- Collection of sample data and computation of test statistic
- Draw a random sample from the population.
- Using sample observations, calculate the value of the test statistic (Z, t, χ², F etc.) according to the selected formula.
- Decision regarding H₀
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Compare the calculated value of the test statistic with the tabulated (critical) value:
- If the calculated value falls in the critical region, reject H₀.
- If the calculated value falls in the acceptance region, do not reject H₀ (i.e., H₀ is accepted at the chosen level of significance).
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Compare the calculated value of the test statistic with the tabulated (critical) value:
- Conclusion / Interpretation
- Express the statistical decision in simple words relating to the problem. procedure of testing a Hypothesis.
- Example: “At 5% level of significance, the hypothesis that the average monthly income of teachers is ₹40,000 is rejected; therefore, the average income is significantly different from ₹40,000.”
Q.6 (b) Differentiate Null and Alternative Hypothesis giving examples.
1. Meaning
- Null Hypothesis (H₀):
A statement that there is no difference, no effect, or no relationship in the population. It is the hypothesis to be tested and is assumed to be true unless evidence suggests otherwise. - Alternative Hypothesis (H₁ / Hₐ):
A statement that contradicts H₀. It expresses the presence of a difference, effect or relationship and represents what the researcher aims to support.
2. Nature of statement
- H₀ usually includes equality sign (=, ≤, ≥).
- H₁ includes inequality sign (≠, >, <).
3. Role in testing
- H₀ is the basis of the testing procedure; all calculations (sampling distribution, standard error, etc.) are made on the assumption that H₀ is true. procedure of testing a Hypothesis.
- H₁ is accepted only when H₀ is rejected. It is supported by the sample evidence.
4. Attitude of researcher
- H₀: The researcher tries to find whether there is sufficient evidence against it.
- H₁: The researcher actually wishes to prove or support this hypothesis.
5. Symbol
- Null hypothesis is denoted by H₀.
- Alternative hypothesis is denoted by H₁ or Hₐ. procedure of testing a Hypothesis.
6. Example (two-tailed test)
Suppose a company claims that the mean life of its bulbs is 1,000 hours. A researcher wants to test this claim.
- H₀: μ = 1,000 hours (mean life is 1,000 hours).
- H₁: μ ≠ 1,000 hours (mean life is not 1,000 hours).
Example (right-tailed test)
A new teaching method is believed to increase the average marks of students beyond 60.
- H₀: μ ≤ 60
- H₁: μ > 60
Example (left-tailed test)
A machine is supposed to fill bottles with at least 500 ml of liquid. procedure of testing a Hypothesis.
- H₀: μ ≥ 500 ml
- H₁: μ < 500 ml
In every case, we first assume H₀ is true, perform the test, and then decide whether to reject H₀ and accept H₁, or to continue to accept H₀ at the chosen level of significance.
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