Critical Region

Critical Region Hypothesis Testing Null Hypothesis Statistical Significance P Value
In statistical hypothesis testing, the critical region refers to the set of values for the test statistic that leads to the rejection of the null hypothesis.
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Critical Region

Definition

In statistical hypothesis testing, the critical region (or rejection region) is the set of values for the test statistic that, if the observed value falls within it, leads to the rejection of the null hypothesis (\(H_0\)). This region is determined prior to the analysis, based on the significance level (\(\alpha\)) chosen for the test. The critical region represents thresholds beyond which observed data are considered sufficiently rare under the null hypothesis to conclude that the observed effect is statistically significant.

Examples

  1. Z-Test for a Population Mean:

    • Null Hypothesis (\(H_0\)): The population mean is equal to a specified value (\(\mu_0\)).
    • Alternative Hypothesis (\(H_a\)): The population mean is not equal to \(\mu_0\).
    • Significance Level (\(\alpha\)): 0.05.
    • Critical Region: For a two-tailed test, the critical regions would be the two tails of the normal distribution, typically represented by the values for \(Z\) below \(-1.96\) or above \(1.96\).

  2. Chi-Square Test for Independence:

    • Null Hypothesis (\(H_0\)): Two categorical variables are independent.
    • Alternative Hypothesis (\(H_a\)): Two categorical variables are not independent.
    • Degrees of Freedom: Calculated from the contingency table.
    • Significance Level (\(\alpha\)): 0.01.
    • Critical Region: Values of the chi-square statistic that are greater than a certain threshold, determined by the chi-square distribution table and degrees of freedom.

Frequently Asked Questions (FAQs)

Q1: What determines the size of the critical region?

  • A: The size of the critical region is determined by the significance level (\(\alpha\)), which reflects the probability of rejecting the null hypothesis when it is actually true (Type I error).

Q2: How are critical values related to the critical region?

  • A: Critical values are the boundaries of the critical region. These values are determined based on the chosen significance level and the sampling distribution of the test statistic.

Q3: Can the critical region be used for both one-tailed and two-tailed tests?

  • A: Yes, critical regions can be defined for both one-tailed and two-tailed tests, but the specific boundaries will differ based on the directionality stated in the alternative hypothesis.

Q4: What happens if the test statistic does not fall within the critical region?

  • A: If the test statistic does not fall within the critical region, the null hypothesis cannot be rejected. This means there is not enough evidence to support the alternative hypothesis.

Q5: Is the critical region always symmetric?

  • A: No, the critical region is not always symmetric. The symmetry depends on the nature of the test and the distribution of the test statistic.

Null Hypothesis (\(H_0\)): A statement asserting that there is no effect or no difference, often the hypothesis that researchers aim to test against.

Significance Level (\(\alpha\)): The threshold used to determine if a test statistic’s value is extreme enough to reject the null hypothesis. Common choices are 0.05, 0.01, and 0.10.

Alternative Hypothesis (\(H_a\)): The hypothesis that contradicts the null hypothesis, suggesting that there is an effect or a difference.

P-Value: The probability of obtaining a test statistic at least as extreme as the one observed during the study, assuming the null hypothesis is true.

Type I Error: The error of rejecting the null hypothesis when it is actually true.

Type II Error: The error of failing to reject the null hypothesis when it is actually false.

Online References

Suggested Books for Further Studies

  1. Statistical Methods for Research Workers by R. A. Fisher
  2. Introduction to the Practice of Statistics by David S. Moore, George P. McCabe, and Bruce A. Craig
  3. The Elements of Statistical Learning by Trevor Hastie, Robert Tibshirani, and Jerome Friedman
  4. Principles of Statistics by M.G. Bulmer

Fundamentals of Critical Region: Statistics Basics Quiz

### What is the critical region in hypothesis testing? - [ ] A region on a map where most test samples are taken. - [ ] The set of values where the null hypothesis is accepted. - [ ] A pre-determined area on a diagram. - [x] The set of values for a test statistic that leads to rejecting the null hypothesis. > **Explanation:** The critical region is the set of values for the test statistic that leads to rejecting the null hypothesis (\\(H_0\\)). ### What does the significance level (\\(\alpha\\)) represent? - [x] The probability of making a Type I error. - [ ] The probability of making a Type II error. - [ ] The power of the test. - [ ] None of the above. > **Explanation:** The significance level (\\(\alpha\\)) represents the probability of rejecting the null hypothesis when it is true, thus making a Type I error. ### In a two-tailed z-test with an \\(\alpha\\) of 0.05, what are the critical values? - [ ] -1.64 and 1.64 - [ ] -1.28 and 1.28 - [ ] -2.33 and 2.33 - [x] -1.96 and 1.96 > **Explanation:** For a two-tailed z-test with \\(\alpha\\) of 0.05, the critical values are -1.96 and 1.96, corresponding to the tails of the standard normal distribution. ### What happens if the test statistic is in the critical region? - [x] The null hypothesis is rejected. - [ ] The null hypothesis is accepted. - [ ] No conclusion can be made. - [ ] The test is deemed invalid. > **Explanation:** If the test statistic falls within the critical region, the null hypothesis is rejected. ### What is the alternative hypothesis (\\(H_a\\))? - [ ] A statement about the population parameter being different from the null hypothesis. - [x] The hypothesis that there is an effect or a difference. - [ ] The hypothesis to be proven true. - [ ] None of the above. > **Explanation:** The alternative hypothesis (\\(H_a\\)) is the hypothesis suggesting there is an effect or a difference, opposing the null hypothesis. ### How is the size of the critical region related to the significance level (\\(\alpha\\))? - [ ] Directly proportional - [x] Inversely proportional - [ ] There is no relation - [ ] Dependent on sample size only > **Explanation:** The size of the critical region is inversely proportional to the significance level (\\(\alpha\\)). A lower \\(\alpha\\) results in a smaller critical region. ### What is a Type I error? - [ ] Failing to reject the null hypothesis when it is false. - [ ] Accepting the null hypothesis when it is true. - [ ] Accepting the alternative hypothesis when it is true. - [x] Rejecting the null hypothesis when it is true. > **Explanation:** A Type I error occurs when the null hypothesis is rejected when it is true. ### Which of the following affects the positions of critical values in hypothesis testing? - [ ] Effect size - [ ] Sample size - [x] Significance level - [ ] Power of the test > **Explanation:** The significance level (\\(\alpha\\)) affects the positions of the critical values. A smaller \\(\alpha\\) makes it harder to reject the null hypothesis. ### In a one-tailed test with \\(\alpha\\) of 0.01, what percentage of the distribution lies in the critical region? - [ ] 0.1% - [ ] 5% - [x] 1% - [ ] 10% > **Explanation:** In a one-tailed test with \\(\alpha\\) of 0.01, 1% of the distribution lies in the critical region. ### What do you conclude if the p-value is less than the significance level (\\(\alpha\\))? - [ ] Fail to reject the null hypothesis - [x] Reject the null hypothesis - [ ] Test is inconclusive - [ ] Accept the null hypothesis > **Explanation:** If the p-value is less than the significance level, you reject the null hypothesis as the results are considered statistically significant.

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Wednesday, August 7, 2024
Critical-Path Method (CPM)
Critical Path Method (CPM)
P-Value January 1, 1
Alternative Hypothesis January 1, 1
Null Hypothesis (H0) January 1, 1
Statistically Significant January 1, 1
T-statistic January 1, 1
Test Statistic January 1, 1
Type 2 Error January 1, 1
Type I Error January 1, 1

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