Statistical Inference and Hypothesis Testing Notes

Statistical Inference

  • A statistical procedure of drawing conclusions about the characteristics of a population through sample data.
  • Decision-making is crucial in the statistical process.

Steps in Performing Hypothesis Testing

  1. Formulation of Null and Alternative Hypotheses
  2. Select the Level of Significance (α\alpha)
  3. Determine the Test Statistic to Use
  4. Define the Area of Rejection or the Critical Region
  5. Compute the value of the statistical test.
  6. Decision: Reject H<em>0H<em>0 or fail to reject H</em>0H</em>0, citing the level of significance used.
  7. Interpretation or Conclusion

Step 1: Formulation of Null and Alternative Hypotheses

Null Hypothesis (H0H_0)

  • Refers to a claim or assertion about the population parameter.
  • The hypothesis intended to be rejected during hypothesis testing.
  • Known as the "NULL" hypothesis due to insufficient evidence to warrant its truthfulness.
  • States that the independent variable has no effect on the dependent variable.

Alternative Hypothesis (HaH_a)

  • An assertion or claim that contradicts the null hypothesis.
  • States that the independent variable has an effect on the dependent variable.

Hypothesis Symbols/Signs

HypothesisSymbol/SignWord(s)
Null Hypothesis (H0H_0)=equal to, same as, not changed from, is
Alternative Hypothesis (HaH_a)≠not equal, different from, changed from, is not, not the same as
>above, greater than, bigger than, higher than, longer than, more than, increased, at least
<below, less than, smaller than, shorter than, reduced from, decreased, at most

Step 2: Select the Level of Significance (α\alpha)

  • Level of Confidence: The degree of assurance that a statistical statement is correct under specified conditions. Represented as (1−α)(1 - \alpha).
  • Level of Significance: The degree of uncertainty about the statistical statement under the same conditions used to determine the confidence level. Common values for α\alpha are 0.01, 0.05, and 0.10.

Type I and Type II Errors

  • Type I Error: Rejecting the null hypothesis when it is TRUE.
  • Type II Error: Failing to reject the null hypothesis when it is FALSE.

Critical Values of Z

Level of SignificanceOne-tail TestTwo-tail Test
0.10±1.28±1.645
0.05±1.645±1.96
0.01±2.33±2.58

Step 3: Determine the Test Statistic to Use

Z-test

  • A parametric test concerning the mean (one or two population means).
  • Used when:
    • The probability distribution of the random variable is normal, and the standard deviation is known or assumed.
    • The population standard deviation is estimated from the sample standard deviation.
    • The sample size is large (n≥30n \geq 30).
  • Types:
    • Z-test for One-sample Mean
    • Z-test for Two Independent Means

T-test

  • Similar to the z-test, but used under different assumptions.
  • Used when:
    • The probability distribution of the random variable is approximately normal.
    • The sample size is small (n < 30).
  • Types:
    • T-test for One-sample Mean
    • T-test between Two Independent Means
  • Degrees of freedom (df):
    • For One-sample Mean: df=n−1df = n - 1
    • For Two Independent Means: df=n<em>1+n</em>2−2df = n<em>1 + n</em>2 - 2
    • When sample sizes are different use the smaller of the two: df=smaller<br/>between(n<em>1−1)or(n</em>2−1)df = smaller <br />\newline between (n<em>1 - 1) or (n</em>2 - 1)

When to Use Z-test or T-test

ConditionDecision
Is σ\sigma known?Yes: Use z-distribution, regardless of sample size.
No, Is n≥30n \geq 30?Yes: Use z-distribution and ss in place of σ\sigma in the formula.
No, Is σ\sigma known?No: Use t-distribution and ss in the formula.

Step 4: Define the Area of Rejection or the Critical Region

  • Area of Rejection: Also known as the critical region. The area under the normal curve where the null hypothesis is rejected based on the decision rule.
  • Critical Value (CV): The value that separates the area of rejection from the area where the null hypothesis is not rejected under the normal curve.

Types of Tests

  • Two-Tailed Test (Non-Directional)
    • The critical regions are the left and right tails of the normal curve, with both negative and positive critical values.
    • Decision Rule: Reject H<em>0H<em>0 if the computed value is > +Tabular Value or < -Tabular Value. Otherwise, do not reject H</em>0H</em>0.
  • One-Tailed Test (Directional)
    • The critical value is either negative or positive.
    • Left Tail Test (H<em>aH<em>a is
    • Right Tail Test (H<em>aH<em>a is >): Reject H</em>0H</em>0 if the computed value is > +CV. Otherwise, do not reject.

Step 5: Compute the Value of the Statistical Test

  • This step involves calculating the test statistic using the appropriate test formula (z-test or t-test).

Step 6: Decision

  • Reject H<em>0H<em>0 or fail to reject H</em>0H</em>0, citing the level of significance used in the study.
  • Sample Format: Since the computed -value (___) is __ the tabular value (___). Therefore, ______ the null hypothesis (H0H_0) at ____ level of significance.

Step 7: Interpretation

  • (Rejection or Non-rejection) of the null hypothesis (H<em>0H<em>0) means that (State the paragraph form of the symbolic form in step 1; If “Rejection of the H</em>0H</em>0” state the H<em>aH<em>a and if “Non-rejection of the H</em>0H</em>0” state the H0H_0) based on the sample of (nn) using (0.01, 0.05 or 0.10) level of significance.
  • The rejection of the null hypothesis can lead to a conclusion that the alternative or the research hypothesis is true. In contrary, non-rejection of H0H_0 will direct to the conclusion that the claim is true, or it can be concluded that there is no sufficient evidence to support the alternative hypothesis.
  • The conclusion should be affirmed in the context of the problem, and the level of significance and sample size(s) used must be stated.